Appendix
Proof of Lemma 1
First we put \(n=0\) and show that there exists positive and unique \((F_{1},K_{1},S_{1},H_{1},Y_{1},Z_{1})\). From Eq. (11) we have
$$ F_{1}-F_{0}-\phi ( h ) \varTheta ( F_{1} ) + \phi ( h ) \varLambda (F_{1},H_{0})=0. $$
We define a function \(\chi _{1}(F)\) as follows:
$$ \chi _{1}(F)=F-F_{0}-\phi ( h ) \varTheta ( F ) + \phi ( h ) \varLambda (F,H_{0})=0. $$
According to Conditions C1–C2 we have \(\chi _{1}\) is a strictly increasing function in F and
$$\begin{aligned} &\chi _{1}(0) =-F_{0}-\phi ( h ) \varTheta ( 0 ) < 0, \\ &\lim_{F\rightarrow \infty }\chi _{1}(F) =\infty . \end{aligned}$$
Hence, there exists unique \(F_{1}>0\) such that \(\chi _{1}(F_{1})=0\).
From Eq. (12) we have
$$ K_{1}-K_{0}-\phi ( h ) ( 1-\varepsilon ) e ^{-\mu _{1}\tau _{1}}\varLambda (F_{-m_{1}+1},H_{-m_{1}})+\phi ( h ) ( \alpha +m ) \digamma _{1} ( K_{1} ) =0. $$
Define a function \(\chi _{2}(K)\) as follows:
$$ \chi _{2}(K)=K-K_{0}-\phi ( h ) ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}}\varLambda (F_{-m_{1}+1},H _{-m_{1}})+\phi ( h ) ( \alpha +m ) \digamma _{1} ( K ) =0. $$
Conditions C2–C3 imply that \(\chi _{2}\) is a strictly increasing function in K. In addition,
$$\begin{aligned} &\chi _{2}(0) =-K_{0}-\phi ( h ) ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}}\varLambda (F_{-m_{1}+1},H _{-m_{1}})< 0, \\ &\lim_{K\rightarrow \infty }\chi _{2}(K) =\infty . \end{aligned}$$
It follows that there exists unique \(K_{1}\in (0,\infty )\) such that \(\chi _{2}(K_{1})=0\).
Since \(\digamma _{2} ( S_{1} ) =S_{1}\), \(\digamma _{3} ( H_{1} ) =H_{1}\), \(\digamma _{4} ( Y_{1} ) =Y_{1}\), and \(\digamma _{5} ( Z_{1} ) =Z_{1}\), then from Eqs. (13) and (16) we get
$$\begin{aligned} Z_{1} =&Z_{0}+\frac{\phi ( h ) g [ S_{0}+\phi ( h ) \varepsilon e^{-\mu _{2}\tau _{2}}\varLambda ( F_{-m_{2}+1},H _{-m_{2}} ) +\phi ( h ) m\digamma _{1} ( K_{1} ) ] }{1+\phi ( h ) ( a+\lambda Z _{1} ) }Z_{1} \\ &{}-\phi ( h ) \xi Z_{1}. \end{aligned}$$
Then
$$ A_{1}Z_{1}^{2}+B_{1}Z_{1}+C_{1}=0, $$
(27)
where
$$\begin{aligned} &A_{1} = \bigl( 1+\phi ( h ) \xi \bigr) \phi ( h ) \lambda , \\ &B_{1} = \bigl( 1+\phi ( h ) \xi \bigr) \bigl( 1+ \phi ( h ) a \bigr) -\phi ( h ) \lambda Z _{0} \\ &\hphantom{B_{1} =}{} -\phi ( h ) g \bigl[ S_{0}+\phi ( h ) e ^{-\mu _{2}\tau _{2}}\varLambda ( F_{-m_{2}+1},H_{-m_{2}} ) + \phi ( h ) m\digamma _{1} ( K_{1} ) \bigr] , \\ &C_{1} =- \bigl( 1+\phi ( h ) a \bigr) Z_{0}. \end{aligned}$$
Since \(A_{1}>0\), \(C_{1}<0\), then \(B_{1}^{2}-4A_{1}C_{1}>0\), and hence there exists a unique positive root of Eq. (27) \(Z_{1}>0\). It follows from Eq. (13)
$$ S_{1}=\frac{S_{0}+\phi ( h ) \varepsilon e^{-\mu _{2}\tau _{2}}\varLambda ( F_{-m_{2}+1},H_{-m_{2}} ) +\phi ( h ) m\digamma _{1} ( K_{1} ) }{1+\phi ( h ) ( a+\lambda Z_{1} ) }>0. $$
Then we have \(S_{1}>0\).
Now we show that \(Y_{1}>0\). From Eqs. (14)–(15) we get
$$ Y_{1}=Y_{0}+\frac{\phi ( h ) q ( H_{0}+\phi ( h ) \theta e^{-\mu _{3}\tau _{3}}S_{-m_{3}+1} ) }{1+\phi ( h ) ( c+dY_{1} ) }Y_{1}- \phi ( h ) \eta Y_{1}. $$
Then we get
$$ A_{2}Y_{1}^{2}+B_{2}Y_{1}+C_{2}=0, $$
(28)
where
$$\begin{aligned} &A_{2} = \bigl( 1+\phi ( h ) \eta \bigr) \phi ( h ) d, \\ &B_{2} = \bigl( 1+\phi ( h ) \eta \bigr) \bigl( 1+ \phi ( h ) c \bigr) -\phi ( h )\,dY_{0}- \phi ( h ) q \bigl[ H_{0}+\phi ( h ) \theta e ^{-\mu _{3}\tau _{3}}\digamma _{2} ( S_{-m_{3}+1} ) \bigr] , \\ &C_{2} =- \bigl( 1+\phi ( h ) c \bigr) Y_{0}. \end{aligned}$$
Since \(A_{2}>0\), \(C_{2}<0\), then \(B_{2}^{2}-4A_{2}C_{2}>0\), and hence there exists a unique positive root of Eq. (28) \(Y_{1}>0\).
From Eq. (14) we get
$$ H_{1}=\frac{H_{0}+\phi ( h ) \theta e^{-\mu _{3}\tau _{3}}S _{-m_{3}+1}}{1+\phi ( h ) ( c+dY_{1} ) }>0. $$
Therefore, the solution \((F_{1},K_{1},S_{1},H_{1},Y_{1},Z_{1})\) exists uniquely and is positive.
Repeating the above process for \(n=1\), we can prove that \((F_{2},K _{2},S_{2},H_{2},Y_{2},Z_{2})\) exists uniquely and is positive. Therefore, mathematical induction yields that, for all \(n\in \mathbb{N}\), \((F_{n},K_{n},S_{n},H_{n},Y_{n},Z_{n})\) exists uniquely and is positive.
By induction, we obtain \(F_{n}>0\), \(K_{n}>0\), \(S_{n}>0\), \(H_{n}>0\), \(Y_{n}>0\), and \(Z_{n}>0\)
\(\forall n\geq 0\). To investigate the boundedness of solution, from Eq. (11) we have
$$ \frac{F_{n+1}-F_{n}}{\phi ( h ) }\leq \varTheta ( F _{n+1} ) \leq b- \overline{b}F_{n+1}. $$
Hence
$$ F_{n+1}\leq \frac{F_{n}}{1+\phi ( h ) \overline{b}}+\frac{ \phi ( h ) b}{1+\phi ( h ) \overline{b}}. $$
By Lemma 2.2 in [67] we have
$$ F_{n}\leq \biggl( \frac{1}{1+\phi ( h ) \overline{b}} \biggr) ^{n}F_{0}+\frac{b}{\overline{b}} \biggl[ 1- \biggl( \frac{1}{1+\phi ( h ) \overline{b}} \biggr) ^{n} \biggr] , $$
which implies that \(\lim_{n\rightarrow \infty }\sup F_{n} \leq b/\overline{b}\leq \vartheta _{1}\). Define
$$ \varOmega _{n}= ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}}F_{n-m _{1}}+ \varepsilon e^{-\mu _{2}\tau _{2}}F_{n-m_{2}}+K_{n}+S_{n}+ \frac{ \lambda }{g}Z_{n}. $$
Then
$$\begin{aligned} \varOmega _{n+1}-\varOmega _{n} ={}& ( 1-\varepsilon ) e^{-\mu _{1} \tau _{1}} ( F_{n-m_{1}+1}-F_{n-m_{1}} ) +\varepsilon e^{- \mu _{2}\tau _{2}} ( F_{n-m_{2}+1}-F_{n-m_{2}} ) \\ &{} +K_{n+1}-K_{n}+S_{n+1}-S_{n}+ \frac{\lambda }{g} ( Z_{n+1}-Z _{n} ) \\ ={}& ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}}\phi ( h ) \bigl[ \varTheta ( F_{n-m_{1}+1} ) -\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \bigr] \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}}\phi ( h ) \bigl[ \varTheta ( F_{n-m_{2}+1} ) -\varLambda ( F_{n-m_{2}+1},H _{n-m_{2}} ) \bigr] \\ &{} +\phi ( h ) \bigl[ (1-\varepsilon )e^{-\mu _{1}\tau _{1}}\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) -(\alpha +m) \digamma _{1} ( K_{n+1} ) \bigr] \\ &{} +\phi ( h ) \bigl[ \varepsilon e^{-\mu _{2}\tau _{2}} \varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) +m\digamma _{1} ( K_{n+1} ) -a\digamma _{2} ( S_{n+1} ) \\ &{} -\lambda \digamma _{2} ( S_{n+1} ) \digamma _{5} ( Z_{n+1} ) \bigr] \\ & {}+\phi ( h ) \frac{\lambda }{g} \bigl[ g\digamma _{2} ( S_{n+1} ) \digamma _{5} ( Z_{n+1} ) -\xi \digamma _{5} ( Z_{n+1} ) \bigr] \\ ={}&\phi ( h ) \biggl[ ( 1-\varepsilon ) e ^{-\mu _{1}\tau _{1}}\varTheta ( F_{n-m_{1}+1} ) +\varepsilon e^{-\mu _{2}\tau _{2}}\varTheta ( F_{n-m_{2}+1} ) -\alpha \digamma _{1} ( K_{n+1} ) \\ & {}-aS_{n+1}-\frac{\lambda \xi }{g}Z_{n+1} \biggr] . \end{aligned}$$
According to Conditions C1 and C3, we have
$$\begin{aligned} \varOmega _{n+1}-\varOmega _{n} \leq{}& \phi ( h ) \biggl[ (1- \varepsilon )e^{-\mu _{1}\tau _{1}} ( b-\overline{b}F_{n-m_{1}+1} ) + \varepsilon e^{-\mu _{2}\tau _{2}} ( b-\overline{b}F_{n-m_{2}+1} ) -\alpha \upsilon _{1}K_{n+1} \\ & {} -aS_{n+1}-\frac{\lambda \xi }{g}Z_{n+1} \biggr] . \end{aligned}$$
We have
$$ ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}}b+\varepsilon e ^{-\mu _{2}\tau _{2}}b\leq b ( 1- \varepsilon ) +b\varepsilon =b. $$
Then
$$\begin{aligned} &\varOmega _{n+1}-\varOmega _{n} \\ &\quad \leq \phi ( h ) b-\phi ( h ) \sigma _{1} \biggl[ ( 1- \varepsilon ) e^{-\mu _{1}\tau _{1}}F_{n-m_{1}+1}+\varepsilon e^{-\mu _{2}\tau _{2}}F_{n-m_{2}+1}+K _{n+1}+S_{n+1}+\frac{\lambda }{g}Z_{n+1} \biggr] \\ &\quad =\phi ( h ) b-\phi ( h ) \sigma _{1}\varOmega _{n+1}. \end{aligned}$$
Hence
$$ \varOmega _{n+1}\leq \frac{\varOmega _{n}}{1+\phi ( h ) \sigma _{1}}+\frac{\phi ( h ) b}{1+\phi ( h ) \sigma _{1}}. $$
Consequently, we get \(\lim_{n\rightarrow \infty }\sup \varOmega _{n}\leq \vartheta _{1}\), \(\lim_{n\rightarrow \infty }\sup K_{n}\leq \vartheta _{1}\), \(\lim_{n\rightarrow \infty }\sup S_{n}\leq \vartheta _{1}\), and \(\lim_{n\rightarrow \infty }\sup Z_{n}\leq \vartheta _{2}\). Thus, for any \(\varpi >0\), there exists an integer \(\varrho _{\varpi }>0\) such that \(S_{n}\leq \vartheta _{1}+\varpi \) for \(n\geq \varrho _{\varpi }\). We define
$$ \varPsi _{n}=H_{n}+\frac{d}{q}Y_{n}. $$
Then
$$\begin{aligned} \varPsi _{n+1}-\varPsi _{n} ={}&H_{n+1}-H_{n}+ \frac{d}{q} ( Y_{n+1}-Y_{n} ) \\ ={}&\phi ( h ) \biggl( \theta e^{-\mu _{3}\tau _{3}}S_{n-m _{3}+1}-cH_{n+1}- \frac{d\eta }{q}Y_{n+1} \biggr) \\ \leq{}& \phi ( h ) \biggl( \theta e^{-\mu _{3}\tau _{3}} ( \vartheta _{1}+\varpi ) -cH_{n+1}-\frac{d\eta }{q}Y_{n+1} \biggr) \\ \leq{}& \phi ( h ) \bigl( \theta e^{-\mu _{3}\tau _{3}} ( \vartheta _{1}+\varpi ) -\sigma _{2}\varPsi _{n+1} \bigr) \quad \text{for }n\geq \varrho _{\varpi }+m_{3}. \end{aligned}$$
Then \(\lim_{n\rightarrow \infty }\sup \varPsi _{n}\leq \frac{ \theta e^{-\mu _{3}\tau _{3}} ( \vartheta _{1}+\varpi ) }{ \sigma _{2}}\leq \frac{\theta ( \vartheta _{1}+\varpi ) }{ \sigma _{2}}\). The arbitrariness of ϖ yields that \(\lim_{n \rightarrow \infty }\sup \varPsi _{n}\leq \frac{\theta \vartheta _{1}}{\sigma _{2}}=\vartheta _{3}\). Hence, \(\lim_{n\rightarrow \infty }\sup H_{n}\leq \vartheta _{3}\) and \(\lim_{n\rightarrow \infty }\sup Y_{n}\leq \vartheta _{4}\). Therefore, the solution \((F_{n},K_{n},S_{n},H_{n},Y_{n},Z_{n})\) converges to Γ as \(n\rightarrow \infty\). □
Proof of Lemma 2
The equilibria of system (11)–(16) satisfy:
$$\begin{aligned} &\varTheta ( F ) -\varLambda ( F,H ) =0, \end{aligned}$$
(29)
$$\begin{aligned} & ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}}\varLambda ( F,H ) - ( \alpha +m ) \digamma _{1} ( K ) =0, \end{aligned}$$
(30)
$$\begin{aligned} &\varepsilon e^{-\mu _{2}\tau _{2}}\varLambda ( F,H ) +m \digamma _{1} ( K ) -a\digamma _{2} ( S ) -\lambda \digamma _{2} ( S ) \digamma _{5} ( Z ) =0, \end{aligned}$$
(31)
$$\begin{aligned} &\theta e^{-\mu _{3}\tau _{3}}\digamma _{2} ( S ) -c\digamma _{3} ( H ) -d\digamma _{3} ( H ) \digamma _{4} ( Y ) =0, \end{aligned}$$
(32)
$$\begin{aligned} &q\digamma _{3} ( H ) \digamma _{4} ( Y ) -\eta \digamma _{4} ( Y ) =0, \end{aligned}$$
(33)
$$\begin{aligned} &g\lambda \digamma _{2} ( S ) \digamma _{5} ( Z ) -\xi \digamma _{5} ( Z ) =0. \end{aligned}$$
(34)
From Eq. (33), either \(\digamma _{4} ( Y ) =0\) or \(\digamma _{4} ( Y ) \neq 0\)
\(( \digamma _{3} ( H ) =\frac{\eta }{q} ) \). By solving Eq. (34), we get \(\digamma _{5} ( Z ) =0\) or \(\digamma _{5} ( Z ) \neq 0\)
\(( \digamma _{2} ( S ) =\frac{\xi }{g} ) \). If Condition C3 \(\digamma _{4} ( Y ) =0\) and \(\digamma _{5} ( Z ) =0\) imply that \(Y=0\) and \(Z=0\), thus we have the following possibilities:
1.
\(Y=Z=0\), then Condition C3 implies that \(\digamma _{i}^{-1}\), \(i=1,\ldots,5\), exist and they are strictly increasing functions. From Eqs. (29)–(32) we get
$$\begin{aligned} &K =\digamma _{1}^{-1} \biggl( \frac{ ( 1-\varepsilon ) e ^{-\mu _{1}\tau _{1}}\varTheta ( F ) }{\alpha +m} \biggr) =\pi _{1} ( F ) , \end{aligned}$$
(35)
$$\begin{aligned} &S =\digamma _{2}^{-1} \biggl( \frac{\gamma e^{\mu _{3}\tau _{3}}\varTheta ( F ) }{a} \biggr) =\pi _{2} ( F ) , \end{aligned}$$
(36)
$$\begin{aligned} &H =\digamma _{3}^{-1} \biggl( \frac{\theta \gamma \varTheta ( F ) }{ac} \biggr) =\pi _{3} ( F ) , \end{aligned}$$
(37)
where
$$ \gamma =\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{\alpha +m}+\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}. $$
(38)
Obviously, \(\pi _{i} ( 0 ) >0\) and \(\pi _{i} ( F^{0} ) =0\), \(i=1,2,3\). Substituting Eq. (37) into Eq. (32), we obtain
$$ \frac{\theta \gamma }{a}\varLambda \bigl( F,\pi _{3} ( F ) \bigr) -c \digamma _{3} \bigl( \pi _{3} ( F ) \bigr) =0. $$
(39)
Equation (39) admits a solution \(F=F^{0}\), which gives \(K=S=H=0\) and leads to the pathogen-free equilibrium \(Q^{0}=(F^{0},0,0,0,0,0)\). Let
$$ \psi _{1} ( F ) =\frac{\theta \gamma }{a}\varLambda \bigl( F, \pi _{3} ( F ) \bigr) -c\digamma _{3} \bigl( \pi _{3} ( F ) \bigr) =0. $$
Then from Conditions C1–C3 we have
$$\begin{aligned} &\psi _{1} ( 0 ) =-c\digamma _{3} \bigl( \pi _{3} ( 0 ) \bigr) < 0, \\ &\psi _{1} \bigl( F^{0} \bigr) =0. \end{aligned}$$
Moreover,
$$\begin{aligned} &\psi _{1}^{\prime } ( F ) =\frac{\theta \gamma }{a} \biggl[ \frac{\partial \varLambda }{\partial F}+\pi _{3}^{\prime } ( F ) \frac{\partial \varLambda }{\partial H} \biggr] -c\pi _{3}^{\prime } ( F ) \digamma _{3}^{\prime } \bigl( \pi _{3} ( F ) \bigr) , \\ &\psi _{1}^{\prime } \bigl( F^{0} \bigr) = \frac{\theta \gamma }{a} \biggl[ \frac{\partial \varLambda ( F^{0},0 ) }{\partial F}+ \pi _{3}^{\prime } \bigl( F^{0} \bigr) \frac{\partial \varLambda ( F^{0},0 ) }{\partial H} \biggr] -c\pi _{3}^{\prime } \bigl( F ^{0} \bigr) \digamma _{3}^{\prime } ( 0 ) . \end{aligned}$$
Condition C2 implies that \(\frac{\partial \varLambda ( F ^{0},0 ) }{\partial F}=0\). Also, from Condition C3, we have \(\digamma _{3}^{\prime } ( 0 ) >0\), then
$$\begin{aligned} \psi _{1}^{\prime } \bigl( F^{0} \bigr) & =c\pi _{3}^{\prime } \bigl( F ^{0} \bigr) \digamma _{3}^{\prime } ( 0 ) \biggl( \frac{ \theta \gamma }{ac\digamma _{3}^{\prime } ( 0 ) } \frac{ \partial \varLambda ( F^{0},0 ) }{\partial H}-1 \biggr) \\ & =\frac{\theta \gamma \varTheta ^{\prime } ( F^{0} ) }{a} \biggl( \frac{\theta \gamma }{ac\digamma _{3}^{\prime } ( 0 ) }\frac{\partial \varLambda ( F^{0},0 ) }{\partial H}-1 \biggr) . \end{aligned}$$
From Condition C1, we have \(\varTheta ^{\prime } ( F^{0} ) <0\). Therefore, if
$$ \frac{\theta \gamma }{ac\digamma _{3}^{\prime } ( 0 ) }\frac{ \partial \varLambda ( F^{0},0 ) }{\partial H}>1, $$
hence \(\psi _{1}^{\prime } ( F^{0} ) <0\) and there exists \(F^{\ast }\in ( 0,F^{0} ) \) such that \(\psi _{1} ( F ^{\ast } ) =0\). From Eqs. (35)–(37) we obtain \(K^{\ast }=\pi _{1} ( F^{\ast } ) >0\), \(S^{\ast }=\pi _{2} ( F^{\ast } ) >0\), and \(H^{\ast }=\pi _{3} ( F^{ \ast } ) >0\). Therefore, a persistent infection equilibrium without immune response \(Q^{\ast } ( F^{\ast },K^{\ast },H^{ \ast },S^{\ast },0,0 ) \) exists when \(\frac{\theta \gamma }{ac \digamma _{3}^{\prime } ( 0 ) }\frac{\partial \varLambda ( F^{0},0 ) }{\partial H}>1\). Let us define
$$ \mathcal{R}_{0}=\frac{\theta \gamma }{ac\digamma _{3}^{\prime } ( 0 ) }\frac{\partial \varLambda ( F^{0},0 ) }{\partial H}. $$
2.
\(Y\neq 0\) and \(Z=0\), we have \(\overline{H}=\digamma _{3} ^{-1} ( \frac{\eta }{q} ) >0\). Let \(H=\overline{H}\) in Eq. (29) and define \(\psi _{2}\) as follows:
$$ \psi _{2} ( F ) =\varTheta ( F ) -\varLambda ( F, \overline{H} ) =0. $$
According to Conditions C1 and C2, we have
$$ \psi _{2} ( 0 ) =\varTheta ( 0 ) >0\quad \text{and}\quad \psi _{2} \bigl( F^{0} \bigr) =-\varLambda \bigl( F^{0},\overline{H} \bigr) < 0. $$
Since \(\psi _{2} ( F ) \) is a strictly decreasing function of F, then there exists unique \(\overline{F}\in ( 0,F^{0} ) \) such that \(\psi _{2} ( \overline{F} ) =0\). Now from Eqs. (32), (35), and (36) we obtain
$$ \overline{K}=\pi _{1} ( \overline{F} ) ,\qquad \overline{S}=\pi _{2} ( \overline{F} ) ,\qquad \overline{Y}=\digamma _{4}^{-1} \biggl( \frac{c}{d} \biggl( \frac{\theta \gamma \varLambda ( \overline{F},\overline{H} ) }{ac\digamma _{3} ( \overline{H} ) }-1 \biggr) \biggr) . $$
Clearly, \(\overline{K}>0\) and \(\overline{S}>0\); moreover, \(\overline{Y}>0\) when \(\frac{\theta \gamma \varLambda ( \overline{F},\overline{H} ) }{ac\digamma _{3} ( \overline{H} ) }>1\). Now we define
$$ \mathcal{R}_{1}^{Y}=\frac{\theta \gamma \varLambda ( \overline{F}, \overline{H} ) }{ac\digamma _{3} ( \overline{H} ) }. $$
Hence, Y̅ can be rewritten as \(\overline{Y}=\digamma _{4} ^{-1} ( \frac{c}{d} ( \mathcal{R}_{1}^{Y}-1 ) ) \). It follows that there exists a persistent infection equilibrium with only humoral immune response \(\overline{Q} ( \overline{F}, \overline{K},\overline{S},\overline{H},\overline{Y},0 ) \) if \(\mathcal{R}_{1}^{Y}>1\).
3.
\(Z\neq 0\) and \(Y=0\), we have \(\widehat{S}=\digamma _{2} ^{-1} ( \frac{\xi }{g} ) >0\). Let \(S=\widehat{S}\) in Eq. (32), then we have
$$ \widehat{H}=\digamma _{3}^{-1} \biggl( \frac{\theta e^{-\mu _{3}\tau _{3}} \xi }{cg} \biggr) >0. $$
Let \(H=\widehat{H}\) in Eq. (29) and define \(\psi _{3}\) as follows:
$$ \psi _{3} ( F ) =\varTheta ( F ) -\varLambda ( F, \widehat{H} ) =0. $$
Clearly,
$$ \psi _{3} ( 0 ) =\varTheta ( 0 ) >0\quad \text{and}\quad \psi _{3} \bigl( F^{0} \bigr) =-\varLambda \bigl( F^{0},\widehat{H} \bigr) < 0. $$
According to Conditions C1 and C2, there exists unique \(\widehat{F}\in ( 0,F^{0} ) \) such that \(\psi _{3} ( \widehat{F} ) =0\). From Eq. (35) we conclude that \(\widehat{K}=\pi _{1} ( \widehat{F} ) >0\). Now from Eqs. (30)–(32) we have \(\widehat{Z}=\digamma _{5}^{-1} ( \frac{a}{ \lambda } ( \mathcal{R}_{1}^{Z}-1 ) ) \), where
$$ \mathcal{R}_{1}^{Z}=\frac{\theta \gamma \varLambda ( \widehat{F}, \widehat{H} ) }{ac\digamma _{3} ( \widehat{H} ) }. $$
Consequently, there exists a persistent infection equilibrium with only CTL immune response \(\widehat{Q} ( \widehat{F},\widehat{K}, \widehat{S},\widehat{H},0,\widehat{Z} ) \) if \(\mathcal{R}_{1} ^{Z}>1\).
4.
\(Z\neq 0\) and \(Y\neq 0\), we have \(\widetilde{H}= \overline{H}=\digamma _{3}^{-1} ( \frac{\eta }{q} ) >0\) and \(\widetilde{S}=\widehat{S}=\digamma _{2}^{-1} ( \frac{\xi }{g} ) >0\). Let \(H=\widetilde{H}\) in Eq. (29) and define \(\psi _{4}\) as follows:
$$ \psi _{4} ( F ) =\varTheta ( F ) -\varLambda ( F, \widetilde{H} ) =0. $$
Clearly,
$$ \psi _{4} ( 0 ) =\varTheta ( 0 ) >0\quad \text{and}\quad \psi _{4} \bigl( F^{0} \bigr) =-\varLambda \bigl( F^{0},\widetilde{H} \bigr) < 0. $$
According to C1 and C2, there exists unique \(\widetilde{F}\in ( 0,F^{0} ) \) such that \(\psi _{4} ( \widetilde{F} ) =0\). Thus, we conclude from Eq. (35) that \(\widetilde{K}=\pi _{1} ( \widetilde{F} ) >0\). Now from Eqs. (30)–(32) we have
$$ \widetilde{Z}=\digamma _{5}^{-1} \biggl( \frac{a}{\lambda } \bigl( \mathcal{R}_{2}^{Z}-1 \bigr) \biggr)\quad \text{and}\quad \widetilde{Y}=\digamma _{4}^{-1} \biggl( \frac{c}{d} \bigl( \bigl( \mathcal{R}_{1}^{Y}/\mathcal{R}_{2}^{Z} \bigr) -1 \bigr) \biggr) >0, $$
where
$$ \mathcal{R}_{2}^{Z}=\frac{e^{\mu _{3}\tau _{3}}\gamma \varLambda ( \widetilde{F},\widetilde{H} ) }{a\digamma _{2} ( \widetilde{S} ) }\quad \text{and}\quad \mathcal{R}_{1}^{Y}/ \mathcal{R}_{2}^{Z}=\frac{\theta \digamma _{2} ( \widetilde{S} ) \varLambda ( \overline{F}, \overline{H} ) }{c\digamma _{3} ( \widetilde{H} ) \varLambda ( \widetilde{F},\widetilde{H} ) }= \frac{\theta e ^{-\mu _{3}\tau _{3}}\xi q}{cg\eta }. $$
It follows that there exists a persistent infection equilibrium with both humoral and CTL immune responses \(\widetilde{Q} ( \widetilde{F},\widetilde{K},\widetilde{S},\widetilde{H},\widetilde{Y}, \widetilde{Z} ) \) if \(\mathcal{R}_{1}^{Y}>\mathcal{R}_{2}^{Z}>1\). □
Proof of Theorem 1
Define
$$\begin{aligned} \mathcal{L}_{n} ={}&\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n}-F^{0}- \int _{F^{0}}^{F_{n}}\lim_{H\rightarrow 0^{+}} \frac{ \varLambda ( F^{0},H ) }{\varLambda ( \varsigma ,H ) }\,d\varsigma \biggr) +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m}K_{n}+e ^{-\mu _{3}\tau _{3}}S_{n}+\frac{a}{\theta }H_{n} \\ & {} +\frac{ad}{\theta q}Y_{n}+\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g}Z_{n} \biggr] +\frac{ac}{\theta }\digamma _{3} ( H_{n} ) + \frac{ad\eta }{\theta q}\digamma _{4} ( Y_{n} ) + \frac{\lambda \xi e^{-\mu _{3}\tau _{3}}}{g}\digamma _{5} ( Z_{n} ) \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\sum_{j=n-m_{1}}^{n-1} \varLambda ( F_{j+1},H _{j} ) +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}} \sum_{j=n-m_{2}}^{n-1} \varLambda ( F_{j+1},H_{j} ) \\ &{} +ae^{-\mu _{3}\tau _{3}}\sum_{j=n-m_{3}}^{n-1} \digamma _{2} ( S _{j+1} ) , \end{aligned}$$
where γ is defined by Eq. (38). Hence, \(\mathcal{L} _{n}>0\) for all \(F_{n},K_{n},S_{n},H_{n},Y_{n},Z_{n}>0\), and \(\mathcal{L}_{n}=0\) if and only if \(F_{n}=F^{0}\), \(K_{n}=0\), \(S_{n}=0\), \(H_{n}=0\), \(Y_{n}=0\), and \(Z_{n}=0\). We compute the difference \(\Delta \mathcal{L}_{n}=\mathcal{L}_{n+1}-\mathcal{L}_{n}\) as follows:
$$\begin{aligned} &\Delta \mathcal{L}_{n} =\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n+1}-F^{0}- \int _{F^{0}}^{F_{n+1}} \lim_{H\rightarrow 0^{+}} \frac{\varLambda ( F^{0},H ) }{ \varLambda ( \varsigma ,H ) }\,d\varsigma \biggr) \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} +\frac{me ^{-\mu _{3}\tau _{3}}}{\alpha +m}K_{n+1}+e^{-\mu _{3}\tau _{3}}S_{n+1}+ \frac{a}{ \theta }H_{n+1} \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} +\frac{ad}{\theta q}Y_{n+1}+ \frac{\lambda e^{-\mu _{3}\tau _{3}}}{g}Z_{n+1} \biggr] +\frac{ac}{\theta }\digamma _{3} ( H_{n+1} ) +\frac{ad\eta }{\theta q}\digamma _{4} ( Y_{n+1} ) \\ &\hphantom{\Delta \mathcal{L}_{n} =}{}+\frac{\lambda \xi e^{-\mu _{3}\tau _{3}}}{g}\digamma _{5} ( Z_{n+1} ) \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\sum_{j=n-m_{1}+1}^{n} \varLambda ( F_{j+1},H _{j} ) \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\sum _{j=n-m_{2}+1} ^{n}\varLambda ( F_{j+1},H_{j} ) +ae^{-\mu _{3}\tau _{3}} \sum_{j=n-m_{3}+1}^{n} \digamma _{2} ( S_{j+1} ) \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} -\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n}-F ^{0}- \int _{F^{0}}^{F_{n}}\lim_{H\rightarrow 0^{+}} \frac{\varLambda ( F^{0},H ) }{\varLambda ( \varsigma ,H ) }\,d\varsigma \biggr) +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m}K_{n} \\ &\hphantom{\Delta \mathcal{L}_{n} =}{}+e ^{-\mu _{3}\tau _{3}}S_{n}+\frac{a}{\theta }H_{n}+ \frac{ad}{\theta q}Y _{n} \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g}Z_{n} \biggr] - \frac{ac}{ \theta }\digamma _{3} ( H_{n} ) - \frac{ad\eta }{\theta q} \digamma _{4} ( Y_{n} ) - \frac{\lambda \xi e^{-\mu _{3}\tau _{3}}}{g}\digamma _{5} ( Z_{n} ) \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} -\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\sum_{j=n-m_{1}}^{n-1} \varLambda ( F_{j+1},H _{j} ) \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} -\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\sum _{j=n-m_{2}} ^{n-1}\varLambda ( F_{j+1},H_{j} ) -ae^{-\mu _{3}\tau _{3}} \sum_{j=n-m_{3}}^{n-1} \digamma _{2} ( S_{j+1} ), \\ &\Delta \mathcal{L}_{n} =\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n+1}-F_{n}- \int _{F_{n}}^{F_{n+1}} \lim_{H\rightarrow 0^{+}} \frac{\varLambda ( F^{0},H ) }{ \varLambda ( \varsigma ,H ) }\,d\varsigma \biggr) +\frac{me ^{-\mu _{3}\tau _{3}}}{\alpha +m} ( K_{n+1}-K_{n} ) \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} +e^{-\mu _{3}\tau _{3}} ( S_{n+1}-S_{n} ) +\frac{a}{ \theta } ( H_{n+1}-H_{n} ) + \frac{ad}{\theta q} ( Y _{n+1}-Y_{n} ) \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} + \frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} ( Z _{n+1}-Z_{n} ) \biggr] \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} +\frac{ac}{\theta } \bigl[ \digamma _{3} ( H_{n+1} ) - \digamma _{3} ( H_{n} ) \bigr] + \frac{ad\eta }{\theta q} \bigl[ \digamma _{4} ( Y_{n+1} ) - \digamma _{4} ( Y _{n} ) \bigr] \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} +\frac{\lambda \xi e^{-\mu _{3}\tau _{3}}}{g} \bigl[ \digamma _{5} ( Z_{n+1} ) -\digamma _{5} ( Z_{n} ) \bigr] \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m} \Biggl( \sum _{j=n-m_{1}+1}^{n}\varLambda ( F _{j+1},H_{j} ) -\sum_{j=n-m_{1}}^{n-1}\varLambda ( F_{j+1},H _{j} ) \Biggr) \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}} \Biggl( \sum _{j=n-m_{2}+1}^{n}\varLambda ( F_{j+1},H_{j} ) - \sum_{j=n-m_{2}}^{n-1}\varLambda ( F_{j+1},H_{j} ) \Biggr) \\ &\hphantom{\Delta \mathcal{L}_{n} =}{} +ae^{-\mu _{3}\tau _{3}} \Biggl( \sum _{j=n-m_{3}+1}^{n}\digamma _{2} ( S_{j+1} ) -\sum_{j=n-m_{3}}^{n-1} \digamma _{2} ( S _{j+1} ) \Biggr) . \end{aligned}$$
Using Lemma 3.1 [68], we get
$$\begin{aligned} \lim_{H\rightarrow 0^{+}} \frac{\varLambda (F^{0},H)}{\varLambda (F_{n+1},H)} ( F_{n+1}-F_{n} ) \leq& \int _{F_{n}}^{F_{n+1}}\lim_{H\rightarrow 0^{+}} \frac{ \varLambda (F^{0},H)}{\varLambda (\varsigma ,H)}\,d\varsigma \\ \leq& \lim_{H\rightarrow 0^{+}}\frac{\varLambda (F^{0},H)}{\varLambda (F_{n},H)} ( F_{n+1}-F_{n} ) . \end{aligned}$$
Hence
$$\begin{aligned} \Delta \mathcal{L}_{n} \leq{}& \frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( 1-\lim_{H\rightarrow 0^{+}}\frac{\varLambda (F^{0},H)}{ \varLambda (F_{n+1},H)} \biggr) ( F_{n+1}-F_{n} ) \\ &{}+\frac{me ^{-\mu _{3}\tau _{3}}}{\alpha +m} ( K_{n+1}-K_{n} ) +e^{-\mu _{3}\tau _{3}} ( S_{n+1}-S_{n} ) \\ & {}+\frac{a}{\theta } ( H_{n+1}-H_{n} ) + \frac{ad}{ \theta q} ( Y_{n+1}-Y_{n} ) + \frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} ( Z_{n+1}-Z_{n} ) \biggr] \\ &{}+ \frac{ac}{\theta } \bigl[ \digamma _{3} ( H_{n+1} ) - \digamma _{3} ( H _{n} ) \bigr] \\ & {}+\frac{ad\eta }{\theta q} \bigl[ \digamma _{4} ( Y_{n+1} ) -\digamma _{4} ( Y_{n} ) \bigr] + \frac{\lambda \xi e^{-\mu _{3}\tau _{3}}}{g} \bigl[ \digamma _{5} ( Z_{n+1} ) - \digamma _{5} ( Z_{n} ) \bigr] \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m} \bigl[ \varLambda ( F_{n+1},H_{n} ) - \varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \bigr] \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}} \bigl[ \varLambda ( F_{n+1},H_{n} ) -\varLambda ( F_{n-m_{2}+1},H_{n-m _{2}} ) \bigr] \\ &{} +ae^{-\mu _{3}\tau _{3}} \bigl[ \digamma _{2} ( S_{n+1} ) - \digamma _{2} ( S_{n-m_{3}+1} ) \bigr] . \end{aligned}$$
From Eqs. (11)–(16) we have
$$\begin{aligned} \Delta \mathcal{L}_{n} \leq{}& \gamma \biggl( 1-\lim _{H\rightarrow 0^{+}}\frac{ \varLambda (F^{0},H)}{\varLambda (F_{n+1},H)} \biggr) \bigl(\varTheta ( F_{n+1} ) -\varLambda ( F_{n+1},H_{n} ) \bigr) \\ &{} +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \bigl[ ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}}\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) - ( \alpha +m ) \digamma _{1} ( K_{n+1} ) \bigr] \\ &{} +e^{-\mu _{3}\tau _{3}} \bigl[ \varepsilon e^{-\mu _{2}\tau _{2}} \varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) +m\digamma _{1} ( K_{n+1} ) -a\digamma _{2} ( S_{n+1} ) \\ &{} -\lambda \digamma _{2} ( S_{n+1} ) \digamma _{5} ( Z_{n+1} ) \bigr] \\ &{} +\frac{a}{\theta } \bigl[ \theta e^{-\mu _{3}\tau _{3}}\digamma _{2} ( S_{n-m_{3}+1} ) -c\digamma _{3} ( H_{n+1} ) -d \digamma _{3} ( H_{n+1} ) \digamma _{4} ( Y_{n+1} ) \bigr] \\ &{} +\frac{ad}{\theta q} \bigl[ q\digamma _{3} ( H_{n+1} ) \digamma _{4} ( Y_{n+1} ) -\eta \digamma _{4} ( Y_{n+1} ) \bigr] \\ &{} +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} \bigl[ g \digamma _{2} ( S_{n+1} ) \digamma _{5} ( Z_{n+1} ) -\xi \digamma _{5} ( Z_{n+1} ) \bigr] \\ &{} +\frac{ac}{\theta } \bigl[ \digamma _{3} ( H_{n+1} ) - \digamma _{3} ( H_{n} ) \bigr] + \frac{ad\eta }{\theta q} \bigl[ \digamma _{4} ( Y_{n+1} ) - \digamma _{4} ( Y _{n} ) \bigr] \\ &{} +\frac{\lambda \xi e^{-\mu _{3}\tau _{3}}}{g} \bigl[ \digamma _{5} ( Z_{n+1} ) -\digamma _{5} ( Z _{n} ) \bigr] \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m} \bigl[ \varLambda ( F_{n+1},H_{n} ) - \varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \bigr] \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}} \bigl[ \varLambda ( F_{n+1},H_{n} ) -\varLambda ( F_{n-m_{2}+1},H_{n-m _{2}} ) \bigr] \\ & {}+ae^{-\mu _{3}\tau _{3}} \bigl[ \digamma _{2} ( S_{n+1} ) - \digamma _{2} ( S_{n-m_{3}+1} ) \bigr] \\ ={}&\gamma \biggl( 1-\lim_{H\rightarrow 0^{+}}\frac{\varLambda (F^{0},H)}{ \varLambda (F_{n+1},H)} \biggr) \varTheta ( F_{n+1} ) +\gamma \lim_{H\rightarrow 0^{+}} \frac{\varLambda (F^{0},H)}{\varLambda (F_{n+1},H)}\varLambda ( F_{n+1},H _{n} ) \\ &{} -\frac{ac}{\theta }\digamma _{3} ( H_{n} ) - \frac{ad \eta }{\theta q}\digamma _{4} ( Y_{n} ) - \frac{\lambda \xi e^{-\mu _{3}\tau _{3}}}{g}\digamma _{5} ( Z_{n} ) . \end{aligned}$$
Using \(\varTheta ( F^{0} ) =0\), we obtain
$$\begin{aligned} \Delta \mathcal{L}_{n} \leq{}& \gamma \biggl( 1- \frac{\partial \varLambda (F^{0},0)/\partial H}{\partial \varLambda (F_{n+1},0)/\partial H} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta \bigl( F^{0} \bigr) \bigr) +\gamma \frac{\partial \varLambda (F^{0},0)/\partial H}{\partial \varLambda (F_{n+1},0)/\partial H}\varLambda ( F_{n+1},H_{n} ) \\ &{} -\frac{ac}{\theta }\digamma _{3} ( H_{n} ) - \frac{ad \eta }{\theta q}\digamma _{4} ( Y_{n} ) - \frac{\lambda \xi e^{-\mu _{3}\tau _{3}}}{g}\digamma _{5} ( Z_{n} ) \\ ={}&\gamma \biggl( 1-\frac{\partial \varLambda (F^{0},0)/\partial H}{ \partial \varLambda (F_{n+1},0)/\partial H} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta \bigl( F^{0} \bigr) \bigr) \\ & {}+\frac{ac}{\theta } \biggl( \frac{\gamma \theta }{ac}\frac{\partial \varLambda (F^{0},0)/\partial H}{\partial \varLambda (F_{n+1},0)/\partial H} \frac{ \varLambda ( F_{n+1},H_{n} ) }{\digamma _{3} ( H_{n} ) }-1 \biggr) \digamma _{3} ( H_{n} ) \\ &{} -\frac{ad\eta }{\theta q}\digamma _{4} ( Y_{n} ) - \frac{ \lambda \xi e^{-\mu _{3}\tau _{3}}}{g}\digamma _{5} ( Z_{n} ) . \end{aligned}$$
From Condition C4 we have
$$ \frac{\varLambda (F_{n+1},H_{n})}{\digamma _{3} ( H_{n} ) } \leq \lim_{H\rightarrow 0^{+}}\frac{\varLambda (F_{n+1},H)}{\digamma _{3} ( H ) }= \frac{\partial \varLambda (F_{n+1},0)/\partial H}{ \digamma _{3}^{\prime } ( 0 ) }. $$
Then we get
$$\begin{aligned} \Delta \mathcal{L}_{n} \leq {}&\gamma \biggl( 1- \frac{\partial \varLambda (F^{0},0)/\partial H}{\partial \varLambda (F_{n+1},0)/\partial H} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta \bigl( F^{0} \bigr) \bigr) \\ &{} +\frac{ac}{\theta } \biggl( \frac{\gamma \theta }{ac}\frac{ \partial \varLambda (F^{0},0)/\partial H}{\digamma _{3}^{\prime } ( 0 ) }-1 \biggr) \digamma _{3} ( H_{n} ) \\ & {}-\frac{ad\eta }{\theta q}\digamma _{4} ( Y_{n} ) - \frac{ \lambda \xi e^{-\mu _{3}\tau _{3}}}{g}\digamma _{5} ( Z_{n} ) \\ ={}&\gamma \biggl( 1-\frac{\partial \varLambda (F^{0},0)/\partial H}{ \partial \varLambda (F_{n+1},0)/\partial H} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta \bigl( F^{0} \bigr) \bigr) + \frac{ac}{ \theta } ( \mathcal{R}_{0}-1 ) \digamma _{3} ( H_{n} ) \\ &{}-\frac{ad\eta }{\theta q}\digamma _{4} ( Y_{n} ) -\frac{\lambda \xi e^{-\mu _{3}\tau _{3}}}{g}\digamma _{5} ( Z _{n} ) . \end{aligned}$$
Conditions C1 and C2 imply that
$$ \biggl( 1-\frac{\partial \varLambda (F^{0},0)/\partial H}{\partial \varLambda (F_{n+1},0)/\partial H} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta \bigl( F^{0} \bigr) \bigr) \leq 0. $$
Hence, if \(\mathcal{R}_{0}\leq 1\), then we have \(\Delta \mathcal{L} _{n}\leq 0\) for all \(n\geq 0\). Obviously, \(\lim_{n\rightarrow \infty }\Delta \mathcal{L}_{n}=0\) if \(\lim_{n\rightarrow \infty }F_{n}=F^{0}\), \(\lim_{n\rightarrow \infty }( \mathcal{R}_{0}-1)\digamma _{3} ( H_{n} ) =0\), \(\lim_{n \rightarrow \infty }\digamma _{4} ( Y_{n} ) =0\), and \(\lim_{n\rightarrow \infty }\digamma _{5} ( Z_{n} ) =0\). We have two cases:
If \(\mathcal{R}_{0}<1\), then \(\lim_{n\rightarrow \infty }\digamma _{3} ( H_{n} ) =0\), \(\lim_{n \rightarrow \infty }\digamma _{4} ( Y_{n} ) =0\), \(\lim_{n\rightarrow \infty }\digamma _{5} ( Z_{n} ) =0\), and from Condition C3 we get \(\lim_{n\rightarrow \infty }H_{n}=0\), \(\lim_{n\rightarrow \infty }Y_{n}=0\), and \(\lim_{n \rightarrow \infty }Z_{n}=0\), then we get from Eqs. (13)–(14) \(\lim_{n\rightarrow \infty }K_{n}=0\) and \(\lim_{n\rightarrow \infty }S_{n}=0\).
If \(\mathcal{R}_{0}=1\), then \(\lim_{n\rightarrow \infty }\Delta K_{n}=0\) when \(\lim_{n\rightarrow \infty }Y_{n}=0\), \({\lim_{n\rightarrow \infty }Z_{n}}=0\), and \(\lim_{n\rightarrow \infty } F_{n}=F^{0}\), and from Eq. (11) we obtain \(\lim_{n \rightarrow \infty }\varLambda (F^{0},H_{n})=0\), then \(\lim_{n \rightarrow \infty }H_{n}=0\). Moreover, from Eqs. (13)–(14) we get \(\lim_{n\rightarrow \infty }K_{n}=0\), \(\lim_{n\rightarrow \infty }S_{n}=0\). Hence \(Q^{0}\) is G.A.S.
□
Proof of Lemma 3
From Conditions C1 and C2, for \(F^{\ast },\overline{F},\widehat{F},S^{\ast }, \widehat{S},H^{\ast },\overline{H},\widehat{H}>0\), we have
$$\begin{aligned} &\bigl( F^{\ast }-\overline{F} \bigr) \bigl( \varTheta ( \overline{F} ) -\varTheta \bigl( F^{\ast } \bigr) \bigr) >0, \end{aligned}$$
(40)
$$\begin{aligned} &\bigl( \overline{F}-F^{\ast } \bigr) \bigl( \varLambda ( \overline{F}, \overline{H} ) -\varLambda \bigl( F^{\ast }, \overline{H} \bigr) \bigr) >0, \end{aligned}$$
(41)
$$\begin{aligned} &\bigl( \overline{H}-H^{\ast } \bigr) \bigl( \varLambda \bigl( F^{ \ast },\overline{H} \bigr) -\varLambda \bigl( F^{\ast },H^{\ast } \bigr) \bigr) >0, \end{aligned}$$
(42)
$$\begin{aligned} &\bigl( \overline{H}-H^{\ast } \bigr) \bigl( \varLambda ( \overline{F}, \overline{H} ) -\varLambda \bigl( \overline{F},H^{ \ast } \bigr) \bigr) >0. \end{aligned}$$
(43)
Using Condition C4, we get
$$ \bigl( H^{\ast }-\overline{H} \bigr) \biggl( \frac{\varLambda ( F ^{\ast },\overline{H} ) }{\digamma _{3} ( \overline{H} ) }- \frac{\varLambda ( F^{\ast },H^{\ast } ) }{\digamma _{3} ( H^{\ast } ) } \biggr) >0. $$
(44)
Suppose that \(\operatorname{sgn} ( \overline{F}-F^{\ast } ) =\operatorname{sgn} ( \overline{H}-H^{\ast } ) \). For the equilibria \(Q^{\ast }\) and Q̅, we have
$$\begin{aligned} \varTheta ( \overline{F} ) -\varTheta \bigl( F^{\ast } \bigr) & = \varLambda ( \overline{F},\overline{H} ) -\varLambda \bigl( F^{\ast },H^{\ast } \bigr) \\ & = \bigl( \varLambda ( \overline{F},\overline{H} ) -\varLambda \bigl( F^{\ast },\overline{H} \bigr) \bigr) + \bigl( \varLambda \bigl( F^{\ast },\overline{H} \bigr) -\varLambda \bigl( F^{\ast },H ^{\ast } \bigr) \bigr) . \end{aligned}$$
Therefore, from inequalities (40)–(43) we get
$$ \operatorname{sgn} \bigl( F^{\ast }-\overline{F} \bigr) = \operatorname{sgn} \bigl( \overline{F}-F^{\ast } \bigr) , $$
which leads to a contradiction. Thus, \(\operatorname{sgn} ( \overline{F}-F^{\ast } ) =\operatorname{sgn} ( H^{\ast }- \overline{H} ) \). Using the equilibrium conditions for \(Q^{\ast }\), we have \(\frac{\theta \gamma \varLambda ( F^{\ast },H ^{\ast } ) }{ac\digamma _{3} ( H^{\ast } ) }=1\), then
$$\begin{aligned} \mathcal{R}_{1}^{Y}-1 & =\frac{\theta \gamma \varLambda ( \overline{F},\overline{H} ) }{ac\digamma _{3} ( \overline{H} ) }- \frac{\theta \gamma \varLambda ( F^{\ast },H ^{\ast } ) }{ac\digamma _{3} ( H^{\ast } ) } \\ & =\frac{\theta \gamma }{ac} \biggl[ \frac{\varLambda ( \overline{F},\overline{H} ) }{\digamma _{3} ( \overline{H} ) }-\frac{\varLambda ( F^{\ast },H^{\ast } ) }{\digamma _{3} ( H^{\ast } ) } \biggr] \\ & =\frac{\theta \gamma }{ac} \biggl[ \frac{1}{\digamma _{3} ( \overline{H} ) } \bigl( \varLambda ( \overline{F}, \overline{H} ) -\varLambda \bigl( F^{\ast },\overline{H} \bigr) \bigr) +\frac{\varLambda ( F^{\ast },\overline{H} ) }{ \digamma _{3} ( \overline{H} ) }-\frac{\varLambda ( F ^{\ast },H^{\ast } ) }{\digamma _{3} ( H^{\ast } ) } \biggr] . \end{aligned}$$
Thus, from inequalities (41)–(44) we get \(\operatorname{sgn} ( \mathcal{R}_{1}^{Y}-1 ) = \operatorname{sgn} ( H^{\ast }-\overline{H} ) \). Similarly, one can show that \(\operatorname{sgn} ( \widehat{F}-F^{\ast } ) =\operatorname{sgn} ( H^{\ast }- \widehat{H} ) =\operatorname{sgn} ( \mathcal{R}_{1}^{Z}-1 ) \). Moreover, we have
$$ \digamma _{2} \bigl( S^{\ast } \bigr) -\digamma _{2} ( \widehat{S} ) =\frac{ce^{\mu _{3}\tau _{3}}}{\theta } \bigl( \digamma _{3} \bigl( H^{\ast } \bigr) -\digamma _{3} ( \widehat{H} ) \bigr) , $$
which gives us \(\operatorname{sgn} ( H^{\ast }-\widehat{H} ) =\operatorname{sgn} ( S^{\ast }-\widehat{S} ) \). □
Proof of Theorem 2
Consider a function \(\mathcal{U}_{n}(F _{n},K_{n},S_{n},H_{n},Y_{n},Z_{n})\) as follows:
$$\begin{aligned} \mathcal{U}_{n} ={}&\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n}-F^{\ast }- \int _{F^{\ast }}^{F_{n}}\frac{\varLambda (F^{ \ast },H^{\ast })}{\varLambda (\varsigma ,H^{\ast })}\,d\varsigma \biggr) +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K_{n}-K^{\ast }- \int _{K^{\ast }}^{K_{n}}\frac{\digamma _{1}(K^{\ast })}{\digamma _{1}( \varsigma )}\,d\varsigma \biggr) \\ &{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n}-S^{\ast }- \int _{S^{ \ast }}^{S_{n}}\frac{\digamma _{2}(S^{\ast })}{\digamma _{2}(\varsigma )}\,d\varsigma \biggr) \\ &{}+\frac{a}{\theta } \biggl( H_{n}-H^{\ast }- \int _{H^{\ast }}^{H_{n}}\frac{\digamma _{3}(H^{\ast })}{\digamma _{3}( \varsigma )}\,d\varsigma \biggr) +\frac{ad}{\theta q}Y_{n}+\frac{ \lambda e^{-\mu _{3}\tau _{3}}}{g}Z_{n} \biggr] \\ &{} +\frac{ac}{\theta }\digamma _{3} \bigl( H^{\ast } \bigr) G \biggl( \frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3}(H^{\ast })} \biggr) \\ &{}+\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{ \alpha +m}\varLambda \bigl(F^{\ast },H^{\ast }\bigr)\sum _{j=n-m_{1}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (F^{\ast },H ^{\ast })} \biggr) \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{ \ast },H^{\ast } \bigr)\sum_{j=n-m_{2}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (F^{\ast },H^{\ast })} \biggr) \\ &{} +ae ^{-\mu _{3}\tau _{3}}\digamma _{2} \bigl( S^{\ast } \bigr) \sum_{j=n-m _{3}}^{n-1}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{ \digamma _{2} ( S^{\ast } ) } \biggr) . \end{aligned}$$
We have \(\mathcal{U}_{n}(F_{n},K_{n},S_{n},H_{n},Y_{n},Z_{n})>0\) for all \(F_{n},K_{n},S_{n},H_{n},Y_{n},Z_{n}>0\); moreover, \(\mathcal{U}_{n}(F ^{\ast },K^{\ast }, S^{\ast },H^{\ast },0,0)=0\). We compute \(\Delta \mathcal{U}_{n}=\mathcal{U}_{n+1}-\mathcal{U}_{n}\) as follows:
$$\begin{aligned} &\triangle \mathcal{U}_{n} =\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n+1}-F^{\ast }- \int _{F^{\ast }}^{F_{n+1}}\frac{ \varLambda (F^{\ast },H^{\ast })}{\varLambda (\varsigma ,H^{\ast })}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K_{n+1}-K ^{\ast }- \int _{K^{\ast }}^{K_{n+1}}\frac{\digamma _{1}(K^{\ast })}{ \digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n+1}-S^{\ast }- \int _{S^{\ast }} ^{S_{n+1}}\frac{\digamma _{2}(S^{\ast })}{\digamma _{2}(\varsigma )}\,d \varsigma \biggr) +\frac{a}{\theta } \biggl( H_{n+1}-H^{\ast }- \int _{H^{\ast }}^{H_{n+1}}\frac{\digamma _{3}(H^{\ast })}{\digamma _{3}( \varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +\frac{ad}{\theta q}Y_{n+1}+\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g}Z_{n+1} \biggr] +\frac{ac}{\theta }\digamma _{3} \bigl( H^{ \ast } \bigr) G \biggl( \frac{\digamma _{3} ( H_{n+1} ) }{ \digamma _{3}(H^{\ast })} \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda \bigl(F^{\ast },H^{\ast } \bigr)\sum_{j=n-m_{1}+1} ^{n}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (F ^{\ast },H^{\ast })} \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{ \ast },H^{\ast } \bigr)\sum_{j=n-m_{2}+1}^{n}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (F^{\ast },H^{\ast })} \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{}+ae ^{-\mu _{3}\tau _{3}}\digamma _{2} \bigl( S^{\ast } \bigr) \sum_{j=n-m_{3}+1}^{n}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( S^{\ast } ) } \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} -\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n}-F ^{\ast }- \int _{F^{\ast }}^{F_{n}}\frac{\varLambda (F^{\ast },H^{\ast })}{ \varLambda (\varsigma ,H^{\ast })}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K_{n}-K^{\ast }- \int _{K^{\ast }}^{K _{n}}\frac{\digamma _{1}(K^{\ast })}{\digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n}-S^{\ast }- \int _{S^{ \ast }}^{S_{n}}\frac{\digamma _{2}(S^{\ast })}{\digamma _{2}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{}+\frac{a}{\theta } \biggl( H_{n}-H^{\ast }- \int _{H^{\ast }}^{H_{n}}\frac{\digamma _{3}(H^{\ast })}{\digamma _{3}( \varsigma )}\,d\varsigma \biggr) +\frac{ad}{\theta q}Y_{n}+\frac{ \lambda e^{-\mu _{3}\tau _{3}}}{g}Z_{n} \biggr] \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} -\frac{ac}{\theta }\digamma _{3} \bigl( H^{\ast } \bigr) G \biggl( \frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3}(H^{\ast })} \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} -\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{ \alpha +m}\varLambda \bigl(F^{\ast },H^{\ast }\bigr)\sum _{j=n-m_{1}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (F^{\ast },H ^{\ast })} \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} -\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{ \ast },H^{\ast } \bigr)\sum_{j=n-m_{2}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (F^{\ast },H^{\ast })} \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} -ae ^{-\mu _{3}\tau _{3}}\digamma _{2} \bigl( S^{\ast } \bigr) \sum_{j=n-m _{3}}^{n-1}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{ \digamma _{2} ( S^{\ast } ) } \biggr), \\ &\triangle \mathcal{U}_{n} =\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n+1}-F_{n}- \int _{F_{n}}^{F_{n+1}}\frac{\varLambda (F ^{\ast },H^{\ast })}{\varLambda (\varsigma ,H^{\ast })}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K_{n+1}-K_{n}- \int _{K_{n}}^{K_{n+1}}\frac{\digamma _{1}(K^{\ast })}{\digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n+1}-S_{n}- \int _{S_{n}}^{S_{n+1}}\frac{ \digamma _{2}(S^{\ast })}{\digamma _{2}(\varsigma )}\,d\varsigma \biggr) +\frac{a}{\theta } \biggl( H_{n+1}-H_{n}- \int _{H_{n}}^{H_{n+1}}\frac{ \digamma _{3}(H^{\ast })}{\digamma _{3}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +\frac{ad}{\theta q} ( Y_{n+1}-Y_{n} ) + \frac{ \lambda e^{-\mu _{3}\tau _{3}}}{g} ( Z_{n+1}-Z_{n} ) \biggr] \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} + \frac{ac}{\theta }\digamma _{3} \bigl( H^{\ast } \bigr) \biggl[ G \biggl( \frac{\digamma _{3} ( H_{n+1} ) }{\digamma _{3}(H ^{\ast })} \biggr) -G \biggl( \frac{\digamma _{3} ( H_{n} ) }{ \digamma _{3}(H^{\ast })} \biggr) \biggr] \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda \bigl(F^{\ast },H^{\ast } \bigr) \biggl[ G \biggl( \frac{ \varLambda ( F_{n+1},H_{n} ) }{\varLambda (F^{\ast },H^{\ast })} \biggr) \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} -G \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda (F^{\ast },H^{ \ast })} \biggr) \biggr] \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{ \ast },H^{\ast } \bigr) \biggl[ G \biggl( \frac{\varLambda ( F_{n+1},H_{n} ) }{\varLambda (F^{\ast },H^{\ast })} \biggr) -G \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) }{ \varLambda (F^{\ast },H^{\ast })} \biggr) \biggr] \\ &\hphantom{\triangle \mathcal{U}_{n} =}{} +ae^{-\mu _{3}\tau _{3}}\digamma _{2} \bigl( S^{\ast } \bigr) \biggl[ G \biggl( \frac{\digamma _{2} ( S_{n+1} ) }{\digamma _{2} ( S^{\ast } ) } \biggr) -G \biggl( \frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S^{\ast } ) } \biggr) \biggr] . \end{aligned}$$
We have
$$\begin{aligned} &\biggl( 1-\frac{\varLambda (F^{\ast },H^{\ast })}{\varLambda (F_{n},H^{ \ast })} \biggr) ( F_{n+1}-F_{n} ) \leq F_{n+1}-F_{n}- \int _{F_{n}}^{F_{n+1}}\frac{\varLambda (F^{\ast },H^{\ast })}{\varLambda ( \varsigma ,H^{\ast })}\,d\varsigma \\ &\hphantom{\biggl( 1-\frac{\varLambda (F^{\ast },H^{\ast })}{\varLambda (F_{n},H^{ \ast })} \biggr) ( F_{n+1}-F_{n} )} \leq \biggl( 1-\frac{\varLambda (F^{\ast },H^{\ast })}{\varLambda (F_{n+1},H ^{\ast })} \biggr) ( F_{n+1}-F_{n} ) , \end{aligned}$$
(45)
$$\begin{aligned} &\biggl( 1-\frac{\digamma _{i}(\rho ^{\ast })}{\digamma _{i}(\rho _{n})} \biggr) ( \rho _{n+1}-\rho _{n} ) \leq \rho _{n+1}-\rho _{n}- \int _{\rho _{n}}^{\rho _{n+1}}\frac{\digamma _{i}(\rho ^{\ast })}{\digamma _{i}(\varsigma )}\,d\varsigma \\ &\hphantom{\biggl( 1-\frac{\digamma _{i}(\rho ^{\ast })}{\digamma _{i}(\rho _{n})} \biggr) ( \rho _{n+1}-\rho _{n} )} \leq \biggl( 1-\frac{\digamma _{i}(\rho ^{\ast })}{\digamma _{i}( \rho _{n+1})} \biggr) ( \rho _{n+1}-\rho _{n} ) , \end{aligned}$$
(46)
\(i=1,\ldots,5\), \(\rho ^{\ast }\in \{ K^{\ast },S^{\ast },H^{\ast } \} \).
Then
$$\begin{aligned} \Delta \mathcal{U}_{n} \leq {}&\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( 1-\frac{\varLambda (F^{\ast },H^{\ast })}{\varLambda (F_{n+1},H ^{\ast })} \biggr) ( F_{n+1}-F_{n} ) +\frac{me^{-\mu _{3} \tau _{3}}}{\alpha +m} \biggl( 1-\frac{\digamma _{1}(K^{\ast })}{\digamma _{1}(K_{n+1})} \biggr) ( K_{n+1}-K_{n} ) \\ &{} +e^{-\mu _{3}\tau _{3}} \biggl( 1-\frac{\digamma _{2}(S^{\ast })}{ \digamma _{2}(S_{n+1})} \biggr) ( S_{n+1}-S_{n} ) +\frac{a}{ \theta } \biggl( 1- \frac{\digamma _{3}(H^{\ast })}{\digamma _{3}(H_{n+1})} \biggr) ( H_{n+1}-H_{n} ) \\ &{} + \frac{ad}{\theta q} ( Y_{n+1}-Y_{n} ) \\ &{} +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} ( Z_{n+1}-Z _{n} ) \biggr] + \frac{ac}{\theta }\digamma _{3} \bigl( H^{ \ast } \bigr) \biggl[ \frac{\digamma _{3} ( H_{n+1} ) }{ \digamma _{3}(H^{\ast })}-\frac{\digamma _{3} ( H_{n} ) }{ \digamma _{3}(H^{\ast })}+\ln \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda \bigl(F^{\ast },H^{\ast } \bigr) \biggl[ \frac{ \varLambda ( F_{n+1},H_{n} ) }{\varLambda (F^{\ast },H^{\ast })}-\frac{ \varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda (F^{ \ast },H^{\ast })} \\ &{} +\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m _{1}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr] \\ & {}+\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{ \ast },H^{\ast } \bigr) \biggl[ \frac{\varLambda ( F_{n+1},H_{n} ) }{ \varLambda (F^{\ast },H^{\ast })}-\frac{\varLambda ( F_{n-m_{2}+1},H _{n-m_{2}} ) }{\varLambda (F^{\ast },H^{\ast })} \\ &{}+\ln \biggl( \frac{ \varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) }{\varLambda ( F _{n+1},H_{n} ) } \biggr) \biggr] \\ &{} +ae^{-\mu _{3}\tau _{3}}\digamma _{2} \bigl( S^{\ast } \bigr) \biggl[ \frac{\digamma _{2} ( S_{n+1} ) }{\digamma _{2} ( S^{ \ast } ) }-\frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{ \digamma _{2} ( S^{\ast } ) }+\ln \biggl( \frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) \biggr] . \end{aligned}$$
From Eqs. (11)–(16) we have
$$\begin{aligned} \Delta \mathcal{U}_{n} \leq{}& \gamma \biggl( 1- \frac{\varLambda (F^{ \ast },H^{\ast })}{\varLambda (F_{n+1},H^{\ast })} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varLambda ( F_{n+1},H_{n} ) \bigr) \\ &{} +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( 1-\frac{\digamma _{1}(K^{\ast })}{\digamma _{1}(K_{n+1})} \biggr) \bigl[ ( 1- \varepsilon ) e^{-\mu _{1}\tau _{1}}\varLambda ( F_{n-m_{1}+1},H _{n-m_{1}} ) \\ &{}- ( \alpha +m ) \digamma _{1} ( K _{n+1} ) \bigr] \\ &{} +e^{-\mu _{3}\tau _{3}} \biggl( 1-\frac{\digamma _{2}(S^{\ast })}{ \digamma _{2}(S_{n+1})} \biggr) \bigl[ \varepsilon e^{-\mu _{2}\tau _{2}}\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) +m\digamma _{1} ( K_{n+1} ) -a\digamma _{2} ( S_{n+1} ) \\ & {}-\lambda \digamma _{2} ( S_{n+1} ) \digamma _{5} ( Z_{n+1} ) \bigr] \\ & {}+\frac{a}{\theta } \biggl( 1-\frac{\digamma _{3}(H^{\ast })}{\digamma _{3}(H _{n+1})} \biggr) \bigl[ \theta e^{-\mu _{3}\tau _{3}}\digamma _{2} ( S_{n-m_{3}+1} ) -c\digamma _{3} ( H_{n+1} ) -d\digamma _{3} ( H_{n+1} ) \digamma _{4} ( Y_{n+1} ) \bigr] \\ &{} +\frac{ad}{\theta q} \bigl[ q\digamma _{3} ( H_{n+1} ) \digamma _{4} ( Y_{n+1} ) -\eta \digamma _{4} ( Y_{n+1} ) \bigr] \\ &{}+\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} \bigl[ g \digamma _{2} ( S_{n+1} ) \digamma _{5} ( Z_{n+1} ) -\xi \digamma _{5} ( Z_{n+1} ) \bigr] \\ &{} +\frac{ac}{\theta }\digamma _{3} \bigl( H^{\ast } \bigr) \biggl[ \frac{ \digamma _{3} ( H_{n+1} ) }{\digamma _{3}(H^{\ast })}-\frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3}(H^{\ast })}+\ln \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda \bigl(F^{\ast },H^{\ast } \bigr) \biggl[ \frac{ \varLambda ( F_{n+1},H_{n} ) }{\varLambda (F^{\ast },H^{\ast })}-\frac{ \varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda (F^{ \ast },H^{\ast })} \\ & {}+\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m _{1}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr] \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{ \ast },H^{\ast } \bigr) \biggl[ \frac{\varLambda ( F_{n+1},H_{n} ) }{ \varLambda (F^{\ast },H^{\ast })}-\frac{\varLambda ( F_{n-m_{2}+1},H _{n-m_{2}} ) }{\varLambda (F^{\ast },H^{\ast })} \\ &{}+\ln \biggl( \frac{ \varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) }{\varLambda ( F _{n+1},H_{n} ) } \biggr) \biggr] \\ & {}+ae^{-\mu _{3}\tau _{3}}\digamma _{2} \bigl( S^{\ast } \bigr) \biggl[ \frac{\digamma _{2} ( S_{n+1} ) }{\digamma _{2} ( S^{ \ast } ) }-\frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{ \digamma _{2} ( S^{\ast } ) }+\ln \biggl( \frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) \biggr] . \end{aligned}$$
(47)
Collecting terms of Eq. (47), we get
$$\begin{aligned} \Delta \mathcal{U}_{n} \leq{}& \gamma \biggl( 1- \frac{\varLambda (F^{ \ast },H^{\ast })}{\varLambda (F_{n+1},H^{\ast })} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta \bigl( F^{\ast } \bigr) \bigr) +\gamma \varTheta \bigl( F^{\ast } \bigr) \biggl( 1-\frac{ \varLambda (F^{\ast },H^{\ast })}{\varLambda (F_{n+1},H^{\ast })} \biggr) \\ &{} +\gamma \frac{\varLambda (F^{\ast },H^{\ast })}{\varLambda (F_{n+1},H ^{\ast })}\varLambda ( F_{n+1},H_{n} ) -\frac{m ( 1- \varepsilon ) }{\alpha +m}e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}} \varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \frac{\digamma _{1}(K ^{\ast })}{\digamma _{1}(K_{n+1})} \\ &{} +me^{-\mu _{3}\tau _{3}}\digamma _{1}\bigl(K^{\ast }\bigr)- \varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) \frac{\digamma _{2}(S^{\ast })}{ \digamma _{2}(S_{n+1})} \\ &{}-me^{-\mu _{3}\tau _{3}}\digamma _{1} ( K_{n+1} ) \frac{\digamma _{2}(S^{\ast })}{\digamma _{2}(S_{n+1})} \\ &{} +ae^{-\mu _{3}\tau _{3}}\digamma _{2}\bigl(S^{\ast }\bigr)+ \lambda e^{-\mu _{3} \tau _{3}}\digamma _{2}\bigl(S^{\ast }\bigr) \digamma _{5} ( Z_{n+1} ) -ae ^{-\mu _{3}\tau _{3}}\digamma _{2} ( S_{n-m_{3}+1} ) \frac{ \digamma _{3}(H^{\ast })}{\digamma _{3}(H_{n+1})} \\ &{}+ \frac{ac}{\theta } \digamma _{3} \bigl( H^{\ast } \bigr) \\ &{} +\frac{ad}{\theta }\digamma _{3} \bigl( H^{\ast } \bigr) \digamma _{4} ( Y_{n+1} ) -\frac{ad\eta }{\theta q} \digamma _{4} ( Y_{n+1} )-\frac{\lambda \xi e^{-\mu _{3}\tau _{3}}}{g} \digamma _{5} ( Z_{n+1} ) -\frac{ac}{\theta }\digamma _{3} ( H_{n} )\\ &{} +\frac{ac}{\theta }\digamma _{3} \bigl( H^{\ast } \bigr) \ln \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \\ &{} +\frac{m ( 1-\varepsilon ) e ^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{\alpha +m}\varLambda \bigl(F^{\ast },H ^{\ast }\bigr)\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{ \ast },H^{\ast } \bigr)\ln \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H _{n-m_{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \\ &{} +ae ^{-\mu _{3}\tau _{3}}\digamma _{2} \bigl( S^{\ast } \bigr) \ln \biggl( \frac{ \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) . \end{aligned}$$
Using the conditions of \(Q^{\ast }\)
$$ \textstyle\begin{cases} \varTheta ( F^{\ast } ) =\varLambda ( F^{\ast },H^{\ast } ) , \\ ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}}\varLambda ( F ^{\ast },H^{\ast } ) = ( \alpha +m ) \digamma _{1} ( K^{\ast } ) , \\ \varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( F^{ \ast },H^{\ast } ) +me^{-\mu _{3}\tau _{3}}\digamma _{1} ( K ^{\ast } ) =\gamma \varLambda ( F^{\ast },H^{\ast } ) =ae ^{-\mu _{3}\tau _{3}}\digamma _{2} ( S^{\ast } ) , \\ \theta e^{-\mu _{3}\tau _{3}}\digamma _{2} ( S^{\ast } ) =c \digamma _{3} ( H^{\ast } ) , \end{cases} $$
(48)
we get
$$\begin{aligned} \Delta \mathcal{U}_{n} \leq{}& \gamma \biggl( 1- \frac{\varLambda (F^{ \ast },H^{\ast })}{\varLambda (F_{n+1},H^{\ast })} \biggr) \bigl(\varTheta ( F_{n+1} ) -\varTheta \bigl( F^{\ast } \bigr) \bigr) \\ & {}+\gamma \varLambda \bigl(F ^{\ast },H^{\ast }\bigr) \biggl( 1-\frac{\varLambda (F^{\ast },H^{\ast })}{ \varLambda (F_{n+1},H^{\ast })} \biggr) \\ & {}+\gamma \varLambda \bigl(F^{\ast },H^{\ast }\bigr) \frac{\varLambda (F_{n+1},H_{n})}{ \varLambda (F_{n+1},H^{\ast })} \\ &{}-\frac{m ( 1-\varepsilon ) }{ \alpha +m}e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{\ast },H ^{\ast }\bigr)\frac{\digamma _{1}(K^{\ast })\varLambda ( F_{n-m_{1}+1},H _{n-m_{1}} ) }{\digamma _{1}(K_{n+1})\varLambda (F^{\ast },H^{ \ast })} \\ &{} +\frac{m ( 1-\varepsilon ) }{\alpha +m}e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{\ast },H^{\ast } \bigr) \\ &{}-\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{\ast },H^{\ast } \bigr)\frac{ \digamma _{2}(S^{\ast })\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) }{\digamma _{2}(S_{n+1})\varLambda (F^{\ast },H^{\ast })} \\ &{} -\frac{m ( 1-\varepsilon ) }{\alpha +m}e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{\ast },H^{\ast } \bigr)\frac{\digamma _{2}(S ^{\ast })\digamma _{1} ( K_{n+1} ) }{\digamma _{2}(S_{n+1}) \digamma _{1}(K^{\ast })}+\gamma \varLambda \bigl(F^{\ast },H^{\ast } \bigr) \\ &{} -\gamma \varLambda \bigl(F^{\ast },H^{\ast }\bigr) \frac{\digamma _{3}(H^{\ast }) \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H_{n+1}) \digamma _{2}(S^{\ast })}+\gamma \varLambda \bigl(F^{\ast },H^{\ast } \bigr) \\ &{}+\frac{ad}{ \theta } \biggl( \digamma _{3} \bigl(H^{\ast }\bigr)-\frac{\eta }{q} \biggr) \digamma _{4} ( Y_{n+1} ) \\ &{} +\lambda e^{-\mu _{3}\tau _{3}} \biggl( \digamma _{2} \bigl(S^{\ast }\bigr)-\frac{ \xi }{g} \biggr) \digamma _{5} ( Z_{n+1} ) -\gamma \varLambda \bigl(F^{\ast },H^{\ast } \bigr)\frac{\digamma _{3}(H_{n})}{\digamma _{3}(H^{ \ast })} \\ &{}+\gamma \varLambda \bigl(F^{\ast },H^{\ast } \bigr)\ln \biggl( \frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda \bigl(F^{\ast },H^{\ast } \bigr)\ln \biggl( \frac{ \varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda ( F _{n+1},H_{n} ) } \biggr) \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{ \ast },H^{\ast } \bigr)\ln \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H _{n-m_{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \\ &{} + \gamma \varLambda \bigl(F^{\ast },H^{\ast }\bigr)\ln \biggl( \frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) . \end{aligned}$$
It follows that
$$\begin{aligned} \Delta \mathcal{U}_{n} \leq{}& \gamma \biggl( 1- \frac{\varLambda (F^{ \ast },H^{\ast })}{\varLambda (F_{n+1},H^{\ast })} \biggr) \bigl(\varTheta ( F_{n+1} ) -\varTheta \bigl( F^{\ast } \bigr) \bigr) \\ & {}+\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda \bigl(F^{\ast },H^{\ast } \bigr) \\ &{}\times \biggl[ 5-\frac{ \varLambda (F^{\ast },H^{\ast })}{\varLambda (F_{n+1},H^{\ast })}-\frac{ \digamma _{1}(K^{\ast })\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\digamma _{1}(K_{n+1})\varLambda (F^{\ast },H^{\ast })} \\ &{} -\frac{\digamma _{2}(S^{\ast })\digamma _{1} ( K_{n+1} ) }{ \digamma _{2}(S_{n+1})\digamma _{1}(K^{\ast })}-\frac{\digamma _{3}(H ^{\ast })\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H _{n+1})\digamma _{2}(S^{\ast })}-\frac{\varLambda ( F_{n+1},H^{ \ast } ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}(H^{\ast })} \\ &{} +\ln \biggl( \frac{\digamma _{3} ( H_{n} ) \varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3} ( H_{n+1} ) \varLambda ( F_{n+1},H_{n} ) \digamma _{2} ( S_{n+1} ) } \biggr) \biggr] \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{ \ast },H^{\ast } \bigr) \biggl[ 4-\frac{\varLambda (F^{\ast },H^{\ast })}{ \varLambda (F_{n+1},H^{\ast })}-\frac{\digamma _{2}(S^{\ast })\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) }{\digamma _{2}(S_{n+1}) \varLambda (F^{\ast },H^{\ast })} \\ &{} -\frac{\digamma _{3}(H^{\ast })\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H_{n+1})\digamma _{2}(S^{\ast })}-\frac{ \varLambda ( F_{n+1},H^{\ast } ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}(H^{\ast })} \\ &{} +\ln \biggl( \frac{\digamma _{3} ( H_{n} ) \varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3} ( H_{n+1} ) \varLambda ( F_{n+1},H_{n} ) \digamma _{2} ( S_{n+1} ) } \biggr) \biggr] \\ & {}+\frac{ad}{\theta } \bigl( \digamma _{3}\bigl(H^{\ast } \bigr)-\digamma _{3}( \overline{H}) \bigr) \digamma _{4} ( Y_{n+1} ) +\lambda e ^{-\mu _{3}\tau _{3}} \bigl( \digamma _{2}\bigl(S^{\ast }\bigr)-\digamma _{2}( \widehat{S}) \bigr) \digamma _{5} ( S_{n+1} ) \\ &{} +\gamma \varLambda \bigl(F^{\ast },H^{\ast }\bigr) \biggl[ -1+\frac{\varLambda ( F_{n+1},H^{\ast } ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}(H^{\ast })}+\frac{\varLambda (F_{n+1},H _{n})}{\varLambda (F_{n+1},H^{\ast })}-\frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(H^{\ast })} \biggr] \\ ={}&\gamma \biggl( 1-\frac{\varLambda (F^{\ast },H^{\ast })}{\varLambda (F _{n+1},H^{\ast })} \biggr) \bigl(\varTheta ( F_{n+1} ) -\varTheta \bigl( F^{\ast } \bigr) \bigr) \\ &{} -\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda \bigl(F^{\ast },H^{\ast } \bigr) \biggl[ G \biggl( \frac{ \varLambda (F^{\ast },H^{\ast })}{\varLambda (F_{n+1},H^{\ast })} \biggr) \\ &{} +G \biggl( \frac{\digamma _{1}(K^{\ast })\varLambda ( F_{n-m_{1}+1},H _{n-m_{1}} ) }{\digamma _{1}(K_{n+1})\varLambda (F^{\ast },H^{ \ast })} \biggr) \\ &{} +G \biggl( \frac{\digamma _{2}(S^{\ast })\digamma _{1} ( K_{n+1} ) }{\digamma _{2}(S_{n+1})\digamma _{1}(K^{\ast })} \biggr) +G \biggl( \frac{\digamma _{3}(H^{\ast })\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H_{n+1})\digamma _{2}(S^{\ast })} \biggr) \\ &{} +G \biggl( \frac{ \varLambda ( F_{n+1},H^{\ast } ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}(H^{\ast })} \biggr) \biggr] \\ &{} -\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda \bigl(F^{ \ast },H^{\ast } \bigr) \biggl[ G \biggl( \frac{\varLambda (F^{\ast },H^{\ast })}{ \varLambda (F_{n+1},H^{\ast })} \biggr) \\ &{} +G \biggl( \frac{\digamma _{2}(S ^{\ast })\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) }{\digamma _{2}(S_{n+1})\varLambda (F^{\ast },H^{\ast })} \biggr) \\ &{} +G \biggl( \frac{\digamma _{3}(H^{\ast })\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H_{n+1})\digamma _{2}(S^{\ast })} \biggr) +G \biggl( \frac{ \varLambda ( F_{n+1},H^{\ast } ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}(H^{\ast })} \biggr) \biggr] \\ &{} +\frac{ad}{\theta } \bigl( \digamma _{3}\bigl(H^{\ast } \bigr)-\digamma _{3}( \overline{H}) \bigr) \digamma _{4} ( Y_{n+1} ) +\lambda e ^{-\mu _{3}\tau _{3}} \bigl( \digamma _{2}\bigl(S^{\ast }\bigr)-\digamma _{2}( \widehat{S}) \bigr) \digamma _{5} ( Z_{n+1} ) \\ &{} +\gamma \varLambda \bigl(F^{\ast },H^{\ast }\bigr) \biggl( 1-\frac{\varLambda (F _{n+1},H^{\ast })}{\varLambda (F_{n+1},H_{n})} \biggr) \biggl( \frac{ \varLambda (F_{n+1},H_{n})}{\varLambda (F_{n+1},H^{\ast })}- \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3} ( H^{\ast } ) } \biggr) . \end{aligned}$$
(49)
Conditions C1–C4 imply that the first and last terms of Eq. (49) are less than or equal to zero. If \(\mathcal{R}_{1} ^{Y}\leq 1\), then from Lemma 3 we have \(H^{\ast }\leq \overline{H}\) and from Condition C3 we get \(\digamma _{3} ( H^{\ast } ) \leq \digamma _{3} ( \overline{H} ) \). Moreover, if \(\mathcal{R}_{1}^{Z}\leq 1\), then \(\digamma _{2}(S^{\ast })\leq \digamma _{2}(\widehat{S})\). Therefore, \(\Delta \mathcal{U}_{n}\leq 0\), and thus \(\mathcal{U}_{n}\) is a monotone decreasing sequence. Since \(\mathcal{U}_{n}\geq 0\), then there is a limit \(\lim_{n\rightarrow \infty }\mathcal{U}_{n}\geq 0\). Therefore, \(\lim_{n\rightarrow \infty }\Delta \mathcal{U}_{n}=0\), which implies that \(\lim_{n\rightarrow \infty }F_{n}=F^{\ast }\), \(\lim_{n \rightarrow \infty }K_{n}=K^{\ast }\), \(\lim_{n\rightarrow \infty }S_{n}=S^{\ast }\), \(\lim_{n\rightarrow \infty }H_{n}=H^{\ast }\), \(\lim_{n\rightarrow \infty }Y_{n}=0\), and \(\lim_{n\rightarrow \infty }Z_{n}=0\). We have four cases as follows:
\(\mathcal{R}_{1}^{Y}=1\), \(\mathcal{R}_{1}^{Z}=1\), then from Eq. (13)
$$\begin{aligned} 0 =&\varepsilon e^{-\mu _{2}\tau _{2}}\varLambda \bigl( F^{\ast },H^{\ast } \bigr) +m\digamma _{1} \bigl( K^{\ast } \bigr) -a \digamma _{2} \bigl( S^{\ast } \bigr) \\ &{} -\lambda \digamma _{2} \bigl( S ^{\ast } \bigr) \lim_{n\rightarrow \infty } \digamma _{5} ( Z_{n+1} ) . \end{aligned}$$
(50)
Using equilibrium condition (48), we get \(\lim_{n\rightarrow \infty }Z_{n}=0\). Moreover, from Eq. (14) we have
$$ 0=\theta e^{-\mu _{3}\tau _{3}}\digamma _{2} \bigl( S^{\ast } \bigr) -c \digamma _{3} \bigl( H^{\ast } \bigr) -d\digamma _{3} \bigl( H^{\ast } \bigr) \lim_{n\rightarrow \infty } \digamma _{4} ( Y_{n+1} ) . $$
(51)
From Eq. (48) we get \(\lim_{n\rightarrow \infty }Y _{n}=0\).
\(\mathcal{R}_{1}^{Y}=1\), \(\mathcal{R}_{1}^{Z}<1\), and \(\lim_{n\rightarrow \infty }Z_{n}=0\). From Eq. (51) we get \(\lim_{n\rightarrow \infty }Y_{n}=0\).
\(\mathcal{R}_{1}^{Y}<1\), \(\mathcal{R}_{1}^{Z}=1\), \(\lim_{n\rightarrow \infty }Y_{n}=0\). From Eq. (50) we get \(\lim_{n\rightarrow \infty }Z_{n}=0\).
\(\mathcal{R}_{1}^{Y}<1\), \(\mathcal{R}_{1}^{Z}<1\), \(\lim_{n\rightarrow \infty }Y_{n}=0\), and \(\lim_{n \rightarrow \infty }Z_{n}=0\). It follows that if \(\mathcal{R} _{1}^{Z}\leq 1\) and \(\mathcal{R}_{1}^{Y}\leq 1\), then \(\lim_{n \rightarrow \infty }F_{n}=F^{\ast }\), \(\lim_{n\rightarrow \infty }K_{n}=K^{\ast }\), \(\lim_{n\rightarrow \infty }S_{n}=S^{\ast }\), \(\lim_{n\rightarrow \infty }H_{n}=H ^{\ast }\), \(\lim_{n\rightarrow \infty }Y_{n}=0\), and \(\lim_{n\rightarrow \infty }Z_{n}=0\). Then \(Q^{\ast }\) is G.A.S.
□
Proof of Theorem 3
Define \(\mathcal{W}_{n}(F_{n},K_{n},S _{n},H_{n},Y_{n},Z_{n})\)
$$\begin{aligned} \mathcal{W}_{n} ={}&\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n}-\overline{F}- \int _{\overline{F}}^{F_{n}}\frac{\varLambda ( \overline{F},\overline{H})}{\varLambda (\varsigma ,\overline{H})}\,d\varsigma \biggr) +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K_{n}- \overline{K}- \int _{\overline{K}}^{K_{n}}\frac{\digamma _{1}( \overline{K})}{\digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n}-\overline{S}- \int _{\overline{S}} ^{S_{n}}\frac{\digamma _{2}(\overline{S})}{\digamma _{2}(\varsigma )}\,d \varsigma \biggr) +\frac{a}{\theta } \biggl( H_{n}-\overline{H}- \int _{\overline{H}}^{H_{n}}\frac{\digamma _{3}(\overline{H})}{\digamma _{3}(\varsigma )}\,d\varsigma \biggr) \\ &{} +\frac{ad}{\theta q} \biggl( Y_{n}-\overline{Y}- \int _{\overline{Y}}^{Y_{n}}\frac{\digamma _{4}(\overline{Y})}{\digamma _{4}(\varsigma )}\,d\varsigma \biggr) +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g}Z_{n} \biggr] \\ &{} +\frac{a}{\theta } \bigl( c+d\digamma _{4}( \overline{Y}) \bigr) \digamma _{3} ( \overline{H} ) G \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(\overline{H})} \biggr) \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\overline{F},\overline{H})\sum _{j=n-m _{1}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{ \varLambda (\overline{F},\overline{H})} \biggr) \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \overline{F},\overline{H})\sum _{j=n-m_{2}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\overline{F},\overline{H})} \biggr) \\ &{} +ae^{-\mu _{3}\tau _{3}} \digamma _{2} ( \overline{S} ) \sum_{j=n-m_{3}}^{n-1}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \overline{S} ) } \biggr) . \end{aligned}$$
Clearly, \(\mathcal{W}_{n}(F_{n},K_{n},S_{n},H_{n},Y_{n},Z_{n})>0\) for all \(F_{n},K_{n},S_{n},H_{n},Y_{n},Z_{n}>0\) and \(\mathcal{W}_{n}( \overline{F},\overline{K},\overline{S}, \overline{H},\overline{Y},0)=0\). We compute \(\Delta \mathcal{W}_{n}=\mathcal{W}_{n+1}-\mathcal{W}_{n}\) as follows:
$$\begin{aligned} &\triangle \mathcal{W}_{n} =\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n+1}-\overline{F}- \int _{\overline{F}}^{F_{n+1}}\frac{ \varLambda (\overline{F},\overline{H})}{\varLambda (\varsigma ,\overline{H})}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K _{n+1}-\overline{K}- \int _{\overline{K}}^{K_{n+1}}\frac{\digamma _{1}( \overline{K})}{\digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n+1}-\overline{S}- \int _{ \overline{S}}^{S_{n+1}}\frac{\digamma _{2}(\overline{S})}{\digamma _{2}( \varsigma )}\,d\varsigma \biggr) +\frac{a}{\theta } \biggl( H_{n+1}- \overline{H}- \int _{\overline{H}}^{H_{n+1}}\frac{\digamma _{3}( \overline{H})}{\digamma _{3}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +\frac{ad}{\theta q} \biggl( Y_{n+1}-\overline{Y}- \int _{\overline{Y}}^{Y_{n+1}}\frac{\digamma _{4}(\overline{Y})}{ \digamma _{4}(\varsigma )}\,d\varsigma \biggr) +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g}Z_{n+1} \biggr] \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +\frac{a}{\theta } \bigl( c+d\digamma _{4}(\overline{Y}) \bigr) \digamma _{3} ( \overline{H} ) G \biggl( \frac{\digamma _{3} ( H_{n+1} ) }{\digamma _{3}( \overline{H})} \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\overline{F},\overline{H})\sum _{j=n-m _{1}+1}^{n}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{ \varLambda (\overline{F},\overline{H})} \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \overline{F},\overline{H})\sum _{j=n-m_{2}+1}^{n}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\overline{F},\overline{H})} \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +ae^{-\mu _{3}\tau _{3}} \digamma _{2} ( \overline{S} ) \sum_{j=n-m_{3}+1}^{n}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \overline{S} ) } \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} -\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n}- \overline{F}- \int _{\overline{F}}^{F_{n}}\frac{\varLambda (\overline{F}, \overline{H})}{\varLambda (\varsigma ,\overline{H})}\,d\varsigma \biggr) +\frac{me ^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K_{n}-\overline{K}- \int _{\overline{K}}^{K_{n}}\frac{\digamma _{1}(\overline{K})}{\digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n}-\overline{S}- \int _{\overline{S}} ^{S_{n}}\frac{\digamma _{2}(\overline{S})}{\digamma _{2}(\varsigma )}\,d \varsigma \biggr) +\frac{a}{\theta } \biggl( H_{n}-\overline{H}- \int _{\overline{H}}^{H_{n}}\frac{\digamma _{3}(\overline{H})}{\digamma _{3}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +\frac{ad}{\theta q} \biggl( Y_{n}-\overline{Y}- \int _{\overline{Y}}^{Y_{n}}\frac{\digamma _{4}(\overline{Y})}{\digamma _{4}(\varsigma )}\,d\varsigma \biggr) +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g}Z_{n} \biggr] \\ &\hphantom{\triangle \mathcal{W}_{n} =}{}-\frac{a}{\theta } \bigl( c+d\digamma _{4}( \overline{Y}) \bigr) \digamma _{3} ( \overline{H} ) G \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(\overline{H})} \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} -\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\overline{F},\overline{H})\sum _{j=n-m _{1}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{ \varLambda (\overline{F},\overline{H})} \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} -\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \overline{F},\overline{H})\sum _{j=n-m_{2}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\overline{F},\overline{H})} \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} -ae^{-\mu _{3}\tau _{3}} \digamma _{2} ( \overline{S} ) \sum_{j=n-m_{3}}^{n-1}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \overline{S} ) } \biggr) , \\ &\triangle \mathcal{W}_{n} =\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n+1}-F_{n}- \int _{F_{n}}^{F_{n+1}}\frac{\varLambda ( \overline{F},\overline{H})}{\varLambda (\varsigma ,\overline{H})}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K_{n+1}-K _{n}- \int _{K_{n}}^{K_{n+1}}\frac{\digamma _{1}(\overline{K})}{\digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n+1}-S_{n}- \int _{S_{n}}^{S_{n+1}}\frac{ \digamma _{2}(\overline{S})}{\digamma _{2}(\varsigma )}\,d\varsigma \biggr) +\frac{a}{\theta } \biggl( H_{n+1}-H_{n}- \int _{H_{n}}^{H_{n}+1}\frac{ \digamma _{3}(\overline{H})}{\digamma _{3}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +\frac{ad}{\theta q} \biggl( Y_{n+1}-Y_{n}- \int _{Y_{n}}^{Y _{n+1}}\frac{\digamma _{4}(\overline{Y})}{\digamma _{4}(\varsigma )}\,d\varsigma \biggr) +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} ( Z _{n+1}-Z_{n} ) \biggr] \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +\frac{a}{\theta } \bigl( c+d\digamma _{4}(\overline{Y}) \bigr) \digamma _{3} ( \overline{H} ) \biggl[ G \biggl( \frac{ \digamma _{3} ( H_{n+1} ) }{\digamma _{3}(\overline{H})} \biggr) -G \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(\overline{H})} \biggr) \biggr] \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\overline{F},\overline{H}) \Biggl( \sum _{j=n-m_{1}+1}^{n}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\overline{F},\overline{H})} \biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} - \sum_{j=n-m_{1}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\overline{F},\overline{H})} \biggr) \Biggr) \\ & \hphantom{\triangle \mathcal{W}_{n} =}{}+\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \overline{F},\overline{H}) \Biggl( \sum_{j=n-m_{2}+1}^{n}G \biggl( \frac{ \varLambda ( F_{j+1},H_{j} ) }{\varLambda (\overline{F}, \overline{H})} \biggr) -\sum_{j=n-m_{2}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\overline{F},\overline{H})} \biggr) \Biggr) \\ &\hphantom{\triangle \mathcal{W}_{n} =}{} +ae^{-\mu _{3}\tau _{3}}\digamma _{2} ( \overline{S} ) \Biggl( \sum _{j=n-m_{3}+1}^{n}G \biggl( \frac{\digamma _{2} ( S _{j+1} ) }{\digamma _{2} ( \overline{S} ) } \biggr) - \sum_{j=n-m_{3}}^{n-1}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \overline{S} ) } \biggr) \Biggr) . \end{aligned}$$
Using inequalities (45) and (46) by replacing \(F^{\ast }\), \(H^{\ast }\), \(\rho ^{\ast }\) with F̅, H̅, ρ̅, we obtain
$$\begin{aligned} \Delta \mathcal{W}_{n} \leq{}& \frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( 1-\frac{\varLambda (\overline{F},\overline{H})}{\varLambda (F _{n+1},\overline{H})} \biggr) ( F_{n+1}-F_{n} ) +\frac{me ^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( 1-\frac{\digamma _{1}( \overline{K})}{\digamma _{1}(K_{n+1})} \biggr) ( K_{n+1}-K_{n} ) \\ &{} +e^{-\mu _{3}\tau _{3}} \biggl( 1-\frac{\digamma _{2}(\overline{S})}{ \digamma _{2}(S_{n+1})} \biggr) ( S_{n+1}-S_{n} ) +\frac{a}{ \theta } \biggl( 1- \frac{\digamma _{3}(\overline{H})}{\digamma _{3}(H _{n+1})} \biggr) ( H_{n+1}-H_{n} ) \\ &{} +\frac{ad}{\theta q} \biggl( 1-\frac{\digamma _{4}( \overline{Y})}{\digamma _{4}(Y_{n+1})} \biggr) ( Y_{n+1}-Y_{n} ) +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} ( Z _{n+1}-Z_{n} ) \biggr] \\ & {}+\frac{a}{\theta } \bigl( c+d\digamma _{4}(\overline{Y}) \bigr) \digamma _{3} ( \overline{H} ) \biggl[ \frac{\digamma _{3} ( H_{n+1} ) }{\digamma _{3}(\overline{H})}- \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(\overline{H})}+\ln \biggl( \frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\overline{F},\overline{H}) \biggl( \frac{ \varLambda ( F_{n+1},H_{n} ) }{\varLambda (\overline{F}, \overline{H})}-\frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda (\overline{F},\overline{H})} \\ & {}+\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m _{1}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr) \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \overline{F},\overline{H}) \biggl( \frac{\varLambda ( F_{n+1},H_{n} ) }{\varLambda (\overline{F},\overline{H})}-\frac{ \varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) }{\varLambda ( \overline{F},\overline{H})} \\ &{}+\ln \biggl( \frac{\varLambda ( F_{n-m _{2}+1},H_{n-m_{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr) \\ &{} +ae^{-\mu _{3}\tau _{3}}\digamma _{2} ( \overline{S} ) \biggl( \frac{\digamma _{2} ( S_{n+1} ) }{\digamma _{2} ( \overline{S} ) }-\frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( \overline{S} ) }+\ln \biggl( \frac{ \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) \biggr) . \end{aligned}$$
From Eqs. (11)–(16) we have
$$\begin{aligned} \Delta \mathcal{W}_{n} \leq {}&\gamma \biggl( 1- \frac{\varLambda ( \overline{F},\overline{H})}{\varLambda (F_{n+1},\overline{H})} \biggr) \bigl[ \varTheta ( F_{n+1} ) -\varLambda ( F_{n+1},H_{n} ) \bigr] \\ & {}+\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( 1-\frac{\digamma _{1}( \overline{K})}{\digamma _{1}(K_{n+1})} \biggr) \bigl[ ( 1- \varepsilon ) e^{-\mu _{1}\tau _{1}}\varLambda ( F_{n-m_{1}+1},H _{n-m_{1}} ) \\ & {} - ( \alpha +m ) \digamma _{1} ( K _{n+1} ) \bigr] \\ & {}+e^{-\mu _{3}\tau _{3}} \biggl( 1-\frac{\digamma _{2}(\overline{S})}{ \digamma _{2}(S_{n+1})} \biggr) \bigl[ \varepsilon e^{-\mu _{2}\tau _{2}}\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) +m\digamma _{1} ( K_{n+1} ) -a\digamma _{2} ( S_{n+1} ) \\ &{} -\lambda \digamma _{2} ( S_{n+1} ) \digamma _{5} ( Z_{n+1} ) \bigr] \\ &{} +\frac{a}{\theta } \biggl( 1-\frac{\digamma _{3}(\overline{H})}{ \digamma _{3}(H_{n+1})} \biggr) \bigl[ \theta e^{-\mu _{3}\tau _{3}} \digamma _{2} ( S_{n-m_{3}+1} ) -c\digamma _{3} ( H_{n+1} ) -d\digamma _{3} ( H_{n+1} ) \digamma _{4} ( Y_{n+1} ) \bigr] \\ &{} +\frac{ad}{\theta q} \biggl( 1-\frac{\digamma _{4}(\overline{Y})}{ \digamma _{4}(Y_{n+1})} \biggr) \bigl[ q \digamma _{3} ( H_{n+1} ) \digamma _{4} ( Y_{n+1} ) -\eta \digamma _{4} ( Y_{n+1} ) \bigr] \\ &{} +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} \bigl[ g\digamma _{2} ( S_{n+1} ) \digamma _{5} ( Z_{n+1} ) -\xi \digamma _{5} ( Z_{n+1} ) \bigr] \\ & {}+\frac{a}{\theta } \bigl( c+d\digamma _{4}(\overline{Y}) \bigr) \digamma _{3} ( \overline{H} ) \biggl[ \frac{\digamma _{3} ( H_{n+1} ) }{\digamma _{3}(\overline{H})}- \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(\overline{H})}+\ln \biggl( \frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\overline{F},\overline{H}) \biggl( \frac{ \varLambda ( F_{n+1},H_{n} ) }{\varLambda (\overline{F}, \overline{H})}-\frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda (\overline{F},\overline{H})} \\ &{} +\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m _{1}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr) \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \overline{F},\overline{H}) \biggl( \frac{\varLambda ( F_{n+1},H_{n} ) }{\varLambda (\overline{F},\overline{H})}-\frac{ \varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) }{\varLambda ( \overline{F},\overline{H})} \\ & {}+\ln \biggl( \frac{\varLambda ( F_{n-m _{2}+1},H_{n-m_{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr) \\ &{} +ae^{-\mu _{3}\tau _{3}}\digamma _{2} ( \overline{S} ) \biggl( \frac{\digamma _{2} ( S_{n+1} ) }{\digamma _{2} ( \overline{S} ) }-\frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( \overline{S} ) }+\ln \biggl( \frac{ \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) \biggr) \\ ={}&\gamma \biggl( 1-\frac{\varLambda (\overline{F},\overline{H})}{ \varLambda (F_{n+1},\overline{H})} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta ( \overline{F} ) \bigr) +\gamma \varTheta ( \overline{F} ) \biggl( 1-\frac{\varLambda ( \overline{F},\overline{H})}{\varLambda (F_{n+1},\overline{H})} \biggr) \\ &{} +\gamma \frac{\varLambda (\overline{F},\overline{H})}{\varLambda (F_{n+1}, \overline{H})}\varLambda ( F_{n+1},H_{n} ) -\frac{m ( 1- \varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{\alpha +m} \varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \frac{\digamma _{1}( \overline{K})}{\digamma _{1}(K_{n+1})} \\ &{} +me^{-\mu _{3}\tau _{3}}\digamma _{1}(\overline{K})-\varepsilon e^{- \mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( F_{n-m_{2}+1},H_{n-m _{2}} ) \frac{\digamma _{2}(\overline{S})}{\digamma _{2}(S_{n+1})} \\ & {}-me^{-\mu _{3} \tau _{3}}\digamma _{1} ( K_{n+1} ) \frac{\digamma _{2}( \overline{S})}{\digamma _{2}(S_{n+1})} \\ &{} +ae^{-\mu _{3}\tau _{3}}\digamma _{2}(\overline{S})+\lambda e^{-\mu _{3}\tau _{3}}\digamma _{2} ( \overline{S} ) \digamma _{5} ( Z_{n+1} ) -ae^{-\mu _{3}\tau _{3}}\digamma _{2}(S_{n-m_{3}+1})\frac{ \digamma _{3}(\overline{H})}{\digamma _{3}(H_{n+1})} \\ & {}+\frac{ac}{\theta } \digamma _{3} ( \overline{H} ) \\ &{} +\frac{ad}{\theta }\digamma _{3} ( \overline{H} ) \digamma _{4} ( Y_{n+1} ) -\frac{ad\eta }{\theta q}\digamma _{4} ( Y_{n+1} ) +\frac{ad\eta }{\theta q}\digamma _{4}( \overline{Y})-\frac{\lambda e^{-\mu _{3}\tau _{3}}\xi }{g}\digamma _{5} ( Z_{n+1} ) \\ & {}-\frac{a}{\theta } \bigl( c+d\digamma _{4}(\overline{Y}) \bigr) \digamma _{3} ( H_{n} ) +\frac{a}{\theta } \bigl( c+d \digamma _{4}(\overline{Y}) \bigr) \digamma _{3} ( \overline{H} ) \ln \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\overline{F},\overline{H})\ln \biggl( \frac{ \varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda ( F _{n+1},H_{n} ) } \biggr) \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \overline{F},\overline{H})\ln \biggl( \frac{\varLambda ( F_{n-m _{2}+1},H_{n-m_{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \\ & {}+ae^{-\mu _{3}\tau _{3}}\digamma _{2} ( \overline{S} ) \ln \biggl( \frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) . \end{aligned}$$
Using the conditions of Q̅
$$\begin{aligned} &\varTheta ( \overline{F} ) =\varLambda ( \overline{F}, \overline{H} ) , \\ &( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}}\varLambda ( \overline{F},\overline{H} ) = ( \alpha +m ) \digamma _{1} ( \overline{K} ) , \\ &\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \overline{F},\overline{H} ) +me^{-\mu _{3}\tau _{3}}\digamma _{1} ( \overline{K} ) =\gamma \varLambda ( \overline{F}, \overline{H} ) =ae^{-\mu _{3}\tau _{3}}\digamma _{2} ( \overline{S} ) , \\ &\theta e^{-\mu _{3}\tau _{3}}\digamma _{2} ( \overline{S} ) = \bigl( c+d\digamma _{4}(\overline{Y}) \bigr) \digamma _{3} ( \overline{H} ) , \\ &\eta =q\digamma _{3} ( \overline{H} ) , \end{aligned}$$
we get
$$\begin{aligned} &\Delta \mathcal{W}_{n} \leq \gamma \biggl( 1- \frac{\varLambda ( \overline{F},\overline{H})}{\varLambda (F_{n+1},\overline{H})} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta ( \overline{F} ) \bigr)+ \gamma \varLambda (\overline{F},\overline{H}) \biggl( 1- \frac{\varLambda ( \overline{F},\overline{H})}{\varLambda (F_{n+1},\overline{H})} \biggr) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +\gamma \varLambda (\overline{F},\overline{H})\frac{\varLambda (F_{n+1},H _{n})}{\varLambda (F_{n+1},\overline{H})} \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{}- \frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{\alpha +m} \varLambda (\overline{F},\overline{H})\frac{\varLambda ( F_{n-m_{1}+1},H _{n-m_{1}} ) \digamma _{1}(\overline{K})}{\varLambda (\overline{F}, \overline{H})\digamma _{1}(K_{n+1})} \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\overline{F},\overline{H}) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{}-\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda (\overline{F}, \overline{H})\frac{\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) \digamma _{2}(\overline{S})}{\varLambda (\overline{F},\overline{H}) \digamma _{2}(S_{n+1})} \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} -\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\overline{F},\overline{H}) \frac{\digamma _{1}(K_{n+1})\digamma _{2}(\overline{S})}{\digamma _{1}(\overline{K}) \digamma _{2}(S_{n+1})}+\gamma \varLambda (\overline{F},\overline{H}) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{}- \gamma \varLambda (\overline{F},\overline{H})\frac{\digamma _{3}( \overline{H})\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H _{n+1})\digamma _{2}(\overline{S})} \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +\gamma \varLambda (\overline{F},\overline{H})-\gamma \varLambda ( \overline{F},\overline{H})\frac{\digamma _{3}(H_{n})}{\digamma _{3}( \overline{H})}+\lambda e^{-\mu _{3}\tau _{3}} \biggl( \digamma _{2} ( \overline{S} ) -\frac{\xi }{g} \biggr) \digamma _{5} ( Z _{n+1} ) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +\gamma \varLambda (\overline{F},\overline{H})\ln \biggl( \frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{}+\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}- \mu _{3}\tau _{3}}}{\alpha +m}\varLambda (\overline{F}, \overline{H})\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{ \varLambda ( F_{n+1},H_{n} ) } \biggr) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \overline{F},\overline{H})\ln \biggl( \frac{\varLambda ( F_{n-m _{2}+1},H_{n-m_{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +\gamma \varLambda (\overline{F}, \overline{H})\ln \biggl( \frac{ \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) \\ &\hphantom{\Delta \mathcal{W}_{n} } =\gamma \biggl( 1-\frac{\varLambda (\overline{F},\overline{H})}{ \varLambda (F_{n+1},\overline{H})} \biggr) \bigl(\varTheta ( F_{n+1} ) -\varTheta ( \overline{F} ) \bigr) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{}+\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{\alpha +m} \varLambda (\overline{F},\overline{H}) \biggl[ 5-\frac{\varLambda ( \overline{F},\overline{H})}{\varLambda (F_{n+1},\overline{H})} \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} -\frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \digamma _{1}( \overline{K})}{\varLambda (\overline{F},\overline{H})\digamma _{1}(K_{n+1})} \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{}-\frac{ \digamma _{1}(K_{n+1})\digamma _{2}(\overline{S})}{\digamma _{1}( \overline{K})\digamma _{2}(S_{n+1})}-\frac{\digamma _{3}(\overline{H}) \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H_{n+1}) \digamma _{2}(\overline{S})}- \frac{\varLambda ( F_{n+1}, \overline{H} ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H _{n} ) \digamma _{3}(\overline{H})} \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m _{1}} ) \digamma _{2} ( S_{n-m_{3}+1} ) \digamma _{3} ( H_{n} ) }{\varLambda ( F_{n+1},H_{n} ) \digamma _{2}(S_{n+1})\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \overline{F},\overline{H}) \biggl[ 4-\frac{\varLambda (\overline{F}, \overline{H})}{\varLambda (F_{n+1},\overline{H})} \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{}-\frac{\varLambda ( F _{n-m_{2}+1},H_{n-m_{2}} ) \digamma _{2}(\overline{S})}{\varLambda (\overline{F},\overline{H})\digamma _{2}(S_{n+1})}-\frac{\digamma _{3}( \overline{H})\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H _{n+1})\digamma _{2}(\overline{S})} \\ & \hphantom{\Delta \mathcal{W}_{n} \leq}{} -\frac{\varLambda ( F_{n+1},\overline{H} ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}(\overline{H})}+\ln \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H _{n-m_{2}} ) \digamma _{2} ( S_{n-m_{3}+1} ) \digamma _{3} ( H_{n} ) }{\varLambda ( F_{n+1},H_{n} ) \digamma _{2}(S_{n+1})\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +\lambda e^{-\mu _{3}\tau _{3}} \bigl( \digamma _{2} ( \overline{S} ) -\digamma _{2} ( \widetilde{S} ) \bigr) \digamma _{5} ( Z_{n+1} ) \\ & \hphantom{\Delta \mathcal{W}_{n} \leq}{} +\gamma \varLambda (\overline{F},\overline{H}) \biggl[ -1+ \frac{\varLambda ( F_{n+1},\overline{H} ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}(\overline{H})}+\frac{ \varLambda (F_{n+1},H_{n})}{\varLambda (F_{n+1},\overline{H})}-\frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3}(\overline{H})} \biggr], \\ &\Delta \mathcal{W}_{n} \leq \gamma \biggl( 1- \frac{\varLambda ( \overline{F},\overline{H})}{\varLambda (F_{n+1},\overline{H})} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta ( \overline{F} ) \bigr) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{}-\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{ \alpha +m}\varLambda (\overline{F},\overline{H}) \biggl[ G \biggl( \frac{ \varLambda (\overline{F},\overline{H})}{\varLambda (F_{n+1},\overline{H})} \biggr) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +G \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \digamma _{1}(\overline{K})}{\varLambda (\overline{F},\overline{H}) \digamma _{1}(K_{n+1})} \biggr) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{}+G \biggl( \frac{\digamma _{1}(K_{n+1}) \digamma _{2}(\overline{S})}{\digamma _{1}(\overline{K})\digamma _{2}(S _{n+1})} \biggr) +G \biggl( \frac{\digamma _{3}(\overline{H})\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H_{n+1})\digamma _{2}( \overline{S})} \biggr) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +G \biggl( \frac{\varLambda ( F_{n+1},\overline{H} ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}( \overline{H})} \biggr) \biggr] -\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3} \tau _{3}}\varLambda (\overline{F},\overline{H}) \biggl[ G \biggl( \frac{ \varLambda (\overline{F},\overline{H})}{\varLambda (F_{n+1},\overline{H})} \biggr) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +G \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) \digamma _{2}(\overline{S})}{\varLambda (\overline{F},\overline{H})\digamma _{2}(S_{n+1})} \biggr) +G \biggl( \frac{ \digamma _{3}(\overline{H})\digamma _{2} ( S_{n-m_{3}+1} ) }{ \digamma _{3}(H_{n+1})\digamma _{2}(\overline{S})} \biggr) \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +G \biggl( \frac{ \varLambda ( F_{n+1},\overline{H} ) \digamma _{3}(H_{n})}{ \varLambda ( F_{n+1},H_{n} ) \digamma _{3}(\overline{H})} \biggr) \biggr] \\ &\hphantom{\Delta \mathcal{W}_{n} \leq}{} +\lambda e^{-\mu _{3}\tau _{3}}\digamma _{2} ( \widetilde{S} ) \bigl( \mathcal{R}_{2}^{Z}-1 \bigr) \digamma _{5} ( Z_{n+1} ) \\ & \hphantom{\Delta \mathcal{W}_{n} \leq}{} +\gamma \varLambda (\overline{F},\overline{H}) \biggl( 1-\frac{\varLambda (F_{n+1},\overline{H})}{\varLambda (F_{n+1},H_{n})} \biggr) \biggl( \frac{ \varLambda (F_{n+1},H_{n})}{\varLambda (F_{n+1},\overline{H})}-\frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3} ( \overline{H} ) } \biggr) . \end{aligned}$$
(52)
Using Conditions C1–C4, we get that the first and last terms of Eq. (52) are less than or equal to zero. Moreover, if \(\mathcal{R}_{2}^{Z}\leq 1\), we get \(\Delta \mathcal{W}_{n}\leq 0\), and thus \(\mathcal{W}_{n}\) is a monotone decreasing sequence. Since \(\mathcal{W}_{n}\geq 0\), then there is a limit \(\lim_{n\rightarrow \infty }\mathcal{W}_{n}\geq 0\). Therefore, \(\lim_{n\rightarrow \infty }\Delta \mathcal{W}_{n}=0\), which implies that \(\lim_{n\rightarrow \infty }F_{n}=\overline{F}\), \(\lim_{n\rightarrow \infty }K_{n}=\overline{K}\), \(\lim_{n\rightarrow \infty }S_{n}=\overline{S}\), \(\lim_{n\rightarrow \infty }H_{n}=\overline{H}\), and \(\lim_{n\rightarrow \infty } ( \mathcal{R}_{2}^{Z}-1 ) \digamma _{5} ( Z_{n+1} ) =0\) We have two cases:
\(\mathcal{R}_{2}^{Z}=1\), then from Eq. (13)
$$ 0=\varepsilon e^{-\mu _{2}\tau _{2}}\varLambda ( \overline{F,} \overline{H} ) +m \digamma _{1} ( \overline{K} ) -a \digamma _{2} ( \overline{S} ) -\lambda \digamma _{2} ( \overline{S} ) \lim _{n\rightarrow \infty }\digamma _{5} ( Z_{n+1} ) , $$
(53)
we get \(\lim_{n\rightarrow \infty }Z_{n}=0\). From Eq. (14) we get
$$ 0=\theta e^{-\mu _{3}\tau _{3}}\digamma _{2} ( \overline{S} ) -c\digamma _{3} ( \overline{H} ) -d\digamma _{3} ( \overline{H} ) \lim_{n\rightarrow \infty }\digamma _{4} ( Y_{n+1} ) . $$
(54)
This gives \(\lim_{n\rightarrow \infty }Y_{n}=\overline{Y}\).
\(\mathcal{R}_{2}^{Z}<1\), \(\lim_{n\rightarrow \infty }\digamma _{5} ( Z_{n} ) =0\). From Eq. (54) we get \(\lim_{n\rightarrow \infty }Y_{n}=\overline{Y}\). Hence, Q̅ is G.A.S.
□
Proof of Theorem 4
Define \(\mathcal{M}_{n}(F_{n},K_{n},S _{n},H_{n},Y_{n},Z_{n})\):
$$\begin{aligned} \mathcal{M}_{n} = {}&\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n}-\widehat{F}- \int _{\widehat{F}}^{F_{n}}\frac{\varLambda ( \widehat{F},\widehat{H})}{\varLambda (\varsigma ,\widehat{H})}\,d\varsigma \biggr) +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K_{n}- \widehat{K}- \int _{\widehat{K}}^{K_{n}}\frac{\digamma _{1}(\widehat{K})}{ \digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ & {}+e^{-\mu _{3}\tau _{3}} \biggl( S_{n}-\widehat{S}- \int _{\widehat{S}} ^{S_{n}}\frac{\digamma _{2}(\widehat{S})}{\digamma _{2}(\varsigma )}\,d \varsigma \biggr) +\frac{ ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{\theta } \biggl( H_{n}- \widehat{H}- \int _{\widehat{H}}^{H_{n}}\frac{\digamma _{3}(\widehat{H})}{ \digamma _{3}(\varsigma )}\,d\varsigma \biggr) \\ &{} +\frac{d ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{q\theta }Y_{n}+\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} \biggl( Z_{n}- \widehat{Z}- \int _{\widehat{Z}}^{Z_{n}}\frac{\digamma _{5}(\widehat{Z})}{ \digamma _{5}(\varsigma )}\,d\varsigma \biggr) \biggr] \\ &{} +\frac{c}{\theta } \bigl( a+\lambda \digamma _{5}( \widehat{Z}) \bigr) \digamma _{3} ( \widehat{H} ) G \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(\widehat{H})} \biggr) \\ &{}+\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{ \alpha +m}\varLambda (\widehat{F}, \widehat{H})\sum_{j=n-m_{1}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda ( \widehat{F},\widehat{H})} \biggr) \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widehat{F},\widehat{H})\sum _{j=n-m_{2}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widehat{F},\widehat{H})} \biggr) +\gamma \varLambda ( \widehat{F},\widehat{H} ) \sum _{j=n-m_{3}}^{n-1}G \biggl( \frac{ \digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \widehat{S} ) } \biggr) . \end{aligned}$$
Clearly, \(\mathcal{M}_{n}(F_{n},K_{n},S_{n},H_{n},Y_{n},Z_{n})>0\) for all \(F_{n},K_{n},S_{n},H_{n},Y_{n},Z_{n}>0\) and \(\mathcal{M}_{n}( \widehat{F},\widehat{K},\widehat{S},\widehat{H}, 0,\widehat{Z})=0\). We compute \(\Delta \mathcal{M}_{n}=\mathcal{M}_{n}-\mathcal{M}_{n}\) as follows:
$$\begin{aligned} &\triangle \mathcal{M}_{n} =\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n+1}-\widehat{F}- \int _{\widehat{F}}^{F_{n+1}}\frac{ \varLambda (\widehat{F},\widehat{H})}{\varLambda (\varsigma ,\widehat{H})}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K _{n+1}-\widehat{K}- \int _{\widehat{K}}^{K_{n+1}}\frac{\digamma _{1}( \widehat{K})}{\digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n+1}-\widehat{S}- \int _{\widehat{S}} ^{S_{n+1}}\frac{\digamma _{2}(\widehat{S})}{\digamma _{2}(\varsigma )}\,d \varsigma \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\frac{ ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{\theta } \biggl( H_{n+1}- \widehat{H}- \int _{\widehat{H}}^{H_{n+1}}\frac{\digamma _{3}( \widehat{H})}{\digamma _{3}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\frac{d ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{q\theta }Y_{n+1}+\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} \biggl( Z _{n+1}-\widehat{Z}- \int _{\widehat{Z}}^{Z_{n+1}}\frac{\digamma _{5}( \widehat{Z})}{\digamma _{5}(\varsigma )}\,d\varsigma \biggr) \biggr] \\ & \hphantom{\triangle \mathcal{M}_{n} =}{}+\frac{c}{\theta } \bigl( a+\lambda \digamma _{5}( \widehat{Z}) \bigr) \digamma _{3} ( \widehat{H} ) G \biggl( \frac{\digamma _{3} ( H_{n+1} ) }{\digamma _{3}(\widehat{H})} \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widehat{F},\widehat{H})\sum _{j=n-m_{1}+1} ^{n}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda ( \widehat{F},\widehat{H})} \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widehat{F},\widehat{H})\sum _{j=n-m_{2}+1}^{n}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widehat{F},\widehat{H})} \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\gamma \varLambda ( \widehat{F},\widehat{H} ) \sum _{j=n-m_{3}+1}^{n}G \biggl( \frac{ \digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \widehat{S} ) } \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} -\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n}- \widehat{F}- \int _{\widehat{F}}^{F_{n}}\frac{\varLambda (\widehat{F}, \widehat{H})}{\varLambda (\varsigma ,\widehat{H})}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\frac{me ^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K_{n}-\widehat{K}- \int _{\widehat{K}}^{K_{n}}\frac{\digamma _{1}(\widehat{K})}{\digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n}-\widehat{S}- \int _{\widehat{S}} ^{S_{n}}\frac{\digamma _{2}(\widehat{S})}{\digamma _{2}(\varsigma )}\,d \varsigma \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\frac{ ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{\theta } \biggl( H_{n}- \widehat{H}- \int _{\widehat{H}}^{H_{n}}\frac{\digamma _{3}(\widehat{H})}{ \digamma _{3}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\frac{d ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{q\theta }Y_{n}+\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} \biggl( Z_{n}- \widehat{Z}- \int _{\widehat{Z}}^{Z_{n}}\frac{\digamma _{5}(\widehat{Z})}{ \digamma _{5}(\varsigma )}\,d\varsigma \biggr) \biggr] \\ & \hphantom{\triangle \mathcal{M}_{n} =}{}-\frac{c}{\theta } \bigl( a+\lambda \digamma _{5}( \widehat{Z}) \bigr) \digamma _{3} ( \widehat{H} ) G \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(\widehat{H})} \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} -\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widehat{F},\widehat{H})\sum _{j=n-m_{1}} ^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widehat{F},\widehat{H})} \biggr) \\ & \hphantom{\triangle \mathcal{M}_{n} =}{}-\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widehat{F},\widehat{H})\sum _{j=n-m_{2}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widehat{F},\widehat{H})} \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} -\gamma \varLambda ( \widehat{F},\widehat{H} ) \sum _{j=n-m_{3}}^{n-1}G \biggl( \frac{ \digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \widehat{S} ) } \biggr) , \\ &\triangle \mathcal{M}_{n} =\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n+1}-F_{n}- \int _{F_{n}}^{F_{n+1}}\frac{\varLambda ( \widehat{F},\widehat{H})}{\varLambda (\varsigma ,\widehat{H})}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K_{n+1}-K _{n}- \int _{K_{n}}^{K_{n+1}}\frac{\digamma _{1}(\widehat{K})}{\digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n+1}-S_{n}- \int _{S_{n}}^{S_{n+1}}\frac{ \digamma _{2}(\widehat{S})}{\digamma _{2}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\frac{ ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{\theta } \biggl( H_{n+1}-H_{n}- \int _{H_{n}}^{H_{n}+1}\frac{\digamma _{3}( \widehat{H})}{\digamma _{3}(\varsigma )}\,d\varsigma \biggr) \\ & \hphantom{\triangle \mathcal{M}_{n} =}{}+\frac{d ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{q\theta } ( Y_{n+1}-Y_{n} ) + \frac{\lambda e^{-\mu _{3} \tau _{3}}}{g} \biggl( Z_{n+1}-Z_{n}- \int _{Z_{n}}^{Z_{n+1}}\frac{ \digamma _{5}(\widehat{Z})}{\digamma _{5}(\varsigma )}\,d\varsigma \biggr) \biggr] \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\frac{c}{\theta } \bigl( a+\lambda \digamma _{5}( \widehat{Z}) \bigr) \digamma _{3} ( \widehat{H} ) \biggl[ G \biggl( \frac{ \digamma _{3} ( H_{n+1} ) }{\digamma _{3}(\widehat{H})} \biggr) -G \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}( \widehat{H})} \biggr) \biggr] \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widehat{F},\widehat{H}) \Biggl( \sum _{j=n-m_{1}+1}^{n}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widehat{F},\widehat{H})} \biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} - \sum_{j=n-m_{1}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widehat{F},\widehat{H})} \biggr) \Biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widehat{F},\widehat{H}) \Biggl( \sum_{j=n-m_{2}+1}^{n}G \biggl( \frac{ \varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widehat{F}, \widehat{H})} \biggr) -\sum_{j=n-m_{2}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widehat{F},\widehat{H})} \biggr) \Biggr) \\ &\hphantom{\triangle \mathcal{M}_{n} =}{} +\gamma \varLambda ( \widehat{F},\widehat{H} ) \Biggl( \sum _{j=n-m_{3}+1}^{n}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \widehat{S} ) } \biggr) -\sum_{j=n-m_{3}} ^{n-1}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \widehat{S} ) } \biggr) \Biggr) . \end{aligned}$$
Using inequalities (45) and (46) by replacing \(F^{\ast }\), \(H^{\ast }\), \(\rho ^{\ast }\) with F̂, Ĥ, ρ̂, we get
$$\begin{aligned} \Delta \mathcal{M}_{n} \leq{}& \frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( 1-\frac{\varLambda (\widehat{F},\widehat{H})}{\varLambda (F _{n+1},\widehat{H})} \biggr) ( F_{n+1}-F_{n} ) \\ &{}+\frac{me ^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( 1-\frac{\digamma _{1}( \widehat{K})}{\digamma _{1}(K_{n+1})} \biggr) ( K_{n+1}-K_{n} ) \\ &{} +e^{-\mu _{3}\tau _{3}} \biggl( 1-\frac{\digamma _{2}(\widehat{S})}{ \digamma _{2}(S_{n+1})} \biggr) ( S_{n+1}-S_{n} ) \\ &{} +\frac{ ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{\theta } \biggl( 1- \frac{\digamma _{3}(\widehat{H})}{\digamma _{3}(H_{n+1})} \biggr) ( H_{n+1}-H_{n} ) \\ &{} +\frac{d ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{q\theta } ( Y_{n+1}-Y_{n} ) + \frac{\lambda e^{-\mu _{3} \tau _{3}}}{g} \biggl( 1-\frac{\digamma _{5}(\widehat{Z})}{\digamma _{5}(Z _{n+1})} \biggr) ( Z_{n+1}-Z_{n} ) \biggr] \\ &{} +\frac{c}{\theta } \bigl( a+\lambda \digamma _{5}( \widehat{Z}) \bigr) \digamma _{3} ( \widehat{H} ) \biggl[ \frac{\digamma _{3} ( H_{n+1} ) }{\digamma _{3}(\widehat{H})}-\frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(\widehat{H})}+\ln \biggl( \frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widehat{F},\widehat{H}) \biggl[ \frac{ \varLambda ( F_{n+1},H_{n} ) }{\varLambda (\widehat{F}, \widehat{H})}-\frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda (\widehat{F},\widehat{H})} \\ &{} +\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m _{1}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr] \\ & {}+\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widehat{F},\widehat{H}) \biggl[ \frac{\varLambda ( F_{n+1},H_{n} ) }{\varLambda (\widehat{F},\widehat{H})}-\frac{ \varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) }{\varLambda ( \widehat{F},\widehat{H})} \\ &{} +\ln \biggl( \frac{\varLambda ( F_{n-m _{2}+1},H_{n-m_{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr] \\ & {}+\gamma \varLambda ( \widehat{F},\widehat{H} ) \biggl[ \frac{ \digamma _{2} ( S_{n+1} ) }{\digamma _{2} ( \widehat{S} ) }- \frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( \widehat{S} ) }+\ln \biggl( \frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) \biggr] . \end{aligned}$$
From Eqs. (11)–(16) we have
$$\begin{aligned} \Delta \mathcal{M}_{n} \leq{}& \gamma \biggl( 1- \frac{\varLambda ( \widehat{F},\widehat{H})}{\varLambda (F_{n+1},\widehat{H})} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varLambda ( F_{n+1},H_{n} ) \bigr) \\ &{} +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( 1-\frac{\digamma _{1}( \widehat{K})}{\digamma _{1}(K_{n+1})} \biggr) \bigl[ ( 1- \varepsilon ) e^{-\mu _{1}\tau _{1}}\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \\ &{} - ( \alpha +m ) \digamma _{1} ( K_{n+1} ) \bigr] \\ &{} +e^{-\mu _{3}\tau _{3}} \biggl( 1-\frac{\digamma _{2}(\widehat{S})}{ \digamma _{2}(S_{n+1})} \biggr) \bigl[ \varepsilon e^{-\mu _{2}\tau _{2}}\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) +m\digamma _{1} ( K_{n+1} ) \\ &{}-a\digamma _{2} ( S_{n+1} ) -\lambda \digamma _{2} ( S_{n+1} ) \digamma _{5} ( Z_{n+1} ) \bigr] \\ & {}+\frac{ ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{ \theta } \biggl( 1- \frac{\digamma _{3}(\widehat{H})}{\digamma _{3}(H_{n+1})} \biggr) \bigl[ \theta e^{-\mu _{3}\tau _{3}}\digamma _{2} ( S_{n-m_{3}+1} ) -c \digamma _{3} ( H_{n+1} ) \\ &{} -d\digamma _{3} ( H_{n+1} ) \digamma _{4} ( Y_{n+1} ) \bigr] \\ &{} +\frac{d ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{q \theta } \bigl[ q\digamma _{3} ( H_{n+1} ) \digamma _{4} ( Y_{n+1} ) -\eta \digamma _{4} ( Y_{n+1} ) \bigr] \\ &{} +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} \biggl( 1-\frac{\digamma _{5}( \widehat{Z})}{\digamma _{5}(Z_{n+1})} \biggr) \bigl[ g \digamma _{2} ( S_{n+1} ) \digamma _{5} ( Z_{n+1} ) -\xi \digamma _{5} ( Z_{n+1} ) \bigr] \\ &{} +\frac{c}{\theta } \bigl( a+\lambda \digamma _{5}( \widehat{Z}) \bigr) \digamma _{3} ( \widehat{H} ) \biggl[ \frac{\digamma _{3} ( H_{n+1} ) }{\digamma _{3}(\widehat{H})}-\frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(\widehat{H})}+\ln \biggl( \frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widehat{F},\widehat{H}) \biggl[ \frac{ \varLambda ( F_{n+1},H_{n} ) }{\varLambda (\widehat{F}, \widehat{H})}-\frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda (\widehat{F},\widehat{H})} \\ &{} +\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m _{1}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr] \\ & +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widehat{F},\widehat{H}) \biggl[ \frac{\varLambda ( F_{n+1},H_{n} ) }{\varLambda (\widehat{F},\widehat{H})}-\frac{ \varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) }{\varLambda ( \widehat{F},\widehat{H})} \\ &{}+\ln \biggl( \frac{\varLambda ( F_{n-m _{2}+1},H_{n-m_{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr] \\ &{} +\gamma \varLambda ( \widehat{F},\widehat{H} ) \biggl[ \frac{ \digamma _{2} ( S_{n+1} ) }{\digamma _{2} ( \widehat{S} ) }- \frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( \widehat{S} ) }+\ln \biggl( \frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) \biggr] . \end{aligned}$$
(55)
Collecting terms of Eq. (55), we get
$$\begin{aligned} \Delta \mathcal{M}_{n} \leq {}&\gamma \biggl( 1- \frac{\varLambda ( \widehat{F},\widehat{H})}{\varLambda (F_{n+1},\widehat{H})} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta ( \widehat{F} ) \bigr) +\gamma \varTheta ( \widehat{F} ) \biggl( 1- \frac{\varLambda (\widehat{F},\widehat{H})}{\varLambda (F_{n+1}, \widehat{H})} \biggr) \\ &{} +\gamma \frac{\varLambda (\widehat{F},\widehat{H})}{\varLambda (F_{n+1}, \widehat{H})}\varLambda ( F_{n+1},H_{n} ) \\ &{} -\frac{m ( 1- \varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{\alpha +m} \varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \frac{\digamma _{1}( \widehat{K})}{\digamma _{1}(K_{n+1})} \\ &{} +me^{-\mu _{3}\tau _{3}}\digamma _{1}(\widehat{K})-\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) \frac{\digamma _{2}(\widehat{S})}{ \digamma _{2}(S_{n+1})} \\ &{} -me^{-\mu _{3}\tau _{3}}\digamma _{1} ( K_{n+1} ) \frac{\digamma _{2}(\widehat{S})}{\digamma _{2}(S_{n+1})} \\ &{} -ae^{-\mu _{3}\tau _{3}}\digamma _{2} ( S_{n+1} ) +\lambda e^{-\mu _{3}\tau _{3}}\digamma _{2}(\widehat{S})\digamma _{5} ( Z _{n+1} ) +ae^{-\mu _{3}\tau _{3}}\digamma _{2}(\widehat{S}) \\ & {}+ \bigl( a+\lambda \digamma _{5}(\widehat{Z}) \bigr) e^{-\mu _{3} \tau _{3}}\digamma _{2}(S_{n-m_{3}+1}) \\ &{} - \bigl( a+ \lambda \digamma _{5}( \widehat{Z}) \bigr) e^{-\mu _{3}\tau _{3}}\digamma _{2}(S_{n-m_{3}+1})\frac{ \digamma _{3}(\widehat{H})}{\digamma _{3}(H_{n+1})} \\ &{} +\frac{c ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{ \theta }\digamma _{3} ( \widehat{H} ) + \frac{d ( a+ \lambda \digamma _{5}(\widehat{Z}) ) }{\theta }\digamma _{3} ( \widehat{H} ) \digamma _{4} ( Y_{n+1} ) \\ &{} -\frac{d ( a+\lambda \digamma _{5}(\widehat{Z}) ) \eta }{q\theta }\digamma _{4} ( Y_{n+1} ) \\ & {}-\lambda e^{-\mu _{3}\tau _{3}}\digamma _{5} ( \widehat{Z} ) \digamma _{2} ( S_{n+1} ) -\frac{\lambda e^{-\mu _{3}\tau _{3}}\xi }{g}\digamma _{5} ( Z_{n+1} ) +\frac{\lambda e^{- \mu _{3}\tau _{3}}\xi }{g}\digamma _{5} ( \widehat{Z} ) \\ &{} -\frac{c}{\theta } \bigl( a+\lambda \digamma _{5}( \widehat{Z}) \bigr) \digamma _{3} ( H_{n} ) + \frac{c}{\theta } \bigl( a+\lambda \digamma _{5}(\widehat{Z}) \bigr) \digamma _{3} ( \widehat{H} ) \ln \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widehat{F},\widehat{H})\ln \biggl( \frac{ \varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda ( F _{n+1},H_{n} ) } \biggr) \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widehat{F},\widehat{H})\ln \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H _{n-m_{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \\ & {}+\gamma \varLambda ( \widehat{F},\widehat{H} ) \biggl( \frac{ \digamma _{2} ( S_{n+1} ) }{\digamma _{2} ( \widehat{S} ) }- \frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( \widehat{S} ) }+\ln \frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) . \end{aligned}$$
Using the conditions of Q̂
$$\begin{aligned} &\varTheta ( \widehat{F} ) =\varLambda ( \widehat{F}, \widehat{H} ) , \\ &( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}}\varLambda ( \widehat{F},\widehat{H} ) = ( \alpha +m ) \digamma _{1} ( \widehat{K} ) , \\ &\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widehat{F},\widehat{H} ) +me^{-\mu _{3}\tau _{3}}\digamma _{1} ( \widehat{K} ) =\gamma \varLambda ( \widehat{F}, \widehat{H} ) = \bigl( a+\lambda \digamma _{5} ( \widehat{Z} ) \bigr) e^{-\mu _{3}\tau _{3}}\digamma _{2} ( \widehat{S} ) , \\ &\theta e^{-\mu _{3}\tau _{3}}\digamma _{2} ( \widehat{S} ) =c\digamma _{3} ( \widehat{H} ) , \\ &\digamma _{2} ( \widehat{S} ) =\frac{\xi }{g}, \end{aligned}$$
we get
$$\begin{aligned} &\Delta \mathcal{M}_{n} \leq \gamma \biggl( 1- \frac{\varLambda ( \widehat{F},\widehat{H})}{\varLambda (F_{n+1},\widehat{H})} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta ( \widehat{F} ) \bigr)+ \gamma \varLambda (\widehat{F},\widehat{H}) \biggl( 1- \frac{\varLambda ( \widehat{F},\widehat{H})}{\varLambda (F_{n+1},\widehat{H})} \biggr) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +\gamma \varLambda (\widehat{F},\widehat{H})\frac{\varLambda (F_{n+1},H _{n})}{\varLambda (F_{n+1},\widehat{H})} \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{}- \frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{\alpha +m} \varLambda (\widehat{F},\widehat{H})\frac{\varLambda ( F_{n-m_{1}+1},H _{n-m_{1}} ) \digamma _{1}(\widehat{K})}{\varLambda (\widehat{F}, \widehat{H})\digamma _{1}(K_{n+1})} \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widehat{F},\widehat{H}) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{}-\varepsilon e ^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda (\widehat{F},\widehat{H})\frac{ \varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) \digamma _{2}( \widehat{S})}{\varLambda (\widehat{F},\widehat{H})\digamma _{2}(S_{n+1})} \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} -\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widehat{F},\widehat{H}) \frac{\digamma _{1}(K_{n+1})\digamma _{2}(\widehat{S})}{\digamma _{1}(\widehat{K}) \digamma _{2}(S_{n+1})}+\gamma \varLambda (\widehat{F},\widehat{H}) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{}- \gamma \varLambda (\widehat{F},\widehat{H})\frac{\digamma _{3}( \widehat{H})\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H _{n+1})\digamma _{2}(\widehat{S})} \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +\gamma \varLambda (\widehat{F},\widehat{H})-\gamma \varLambda ( \widehat{F}, \widehat{H})\frac{\digamma _{3}(H_{n})}{\digamma _{3}( \widehat{H})}+\frac{d ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{\theta } \biggl( \digamma _{3} ( \widehat{H} ) -\frac{ \eta }{q} \biggr) \digamma _{4} ( Y_{n+1} ) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +\gamma \varLambda (\widehat{F},\widehat{H})\ln \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{}+\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}- \mu _{3}\tau _{3}}}{\alpha +m}\varLambda (\widehat{F},\widehat{H})\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{ \varLambda ( F_{n+1},H_{n} ) } \biggr) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widehat{F},\widehat{H})\ln \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H _{n-m_{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} + \gamma \varLambda (\widehat{F}, \widehat{H})\ln \biggl( \frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) \\ &\hphantom{\Delta \mathcal{M}_{n} }{} =\gamma \biggl( 1-\frac{\varLambda (\widehat{F},\widehat{H})}{\varLambda (F_{n+1},\widehat{H})} \biggr) \bigl(\varTheta ( F_{n+1} ) - \varTheta ( \widehat{F} ) \bigr) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{}+\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{\alpha +m} \varLambda (\widehat{F},\widehat{H}) \biggl[ 5-\frac{\varLambda (\widehat{F}, \widehat{H})}{\varLambda (F_{n+1},\widehat{H})} \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} -\frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \digamma _{1}( \widehat{K})}{\varLambda (\widehat{F},\widehat{H})\digamma _{1}(K_{n+1})}-\frac{ \digamma _{1}(K_{n+1})\digamma _{2}(\widehat{S})}{\digamma _{1}( \widehat{K})\digamma _{2}(S_{n+1})} \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{}-\frac{\digamma _{3}(\widehat{H}) \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H_{n+1}) \digamma _{2}(\widehat{S})}- \frac{\varLambda ( F_{n+1},\widehat{H} ) \digamma _{3}(H_{n})}{\varLambda ( F _{n+1},H_{n} ) \digamma _{3}(\widehat{H})} \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m _{1}} ) \digamma _{2} ( S_{n-m_{3}+1} ) \digamma _{3} ( H_{n} ) }{\varLambda ( F_{n+1},H_{n} ) \digamma _{2}(S_{n+1})\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ & \hphantom{\Delta \mathcal{M}_{n} \leq}{}+\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widehat{F},\widehat{H}) \biggl[ 4-\frac{\varLambda (\widehat{F}, \widehat{H})}{\varLambda (F_{n+1},\widehat{H})}-\frac{\varLambda ( F _{n-m_{2}+1},H_{n-m_{2}} ) \digamma _{2}(\widehat{S})}{\varLambda ( \widehat{F},\widehat{H})\digamma _{2}(S_{n+1})} \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{}-\frac{\digamma _{3}( \widehat{H})\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H _{n+1})\digamma _{2}(\widehat{S})} \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} -\frac{\varLambda ( F_{n+1},\widehat{H} ) \digamma _{3}(H _{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}(\widehat{H})}+ \ln \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) \digamma _{2} ( S_{n-m_{3}+1} ) \digamma _{3} ( H_{n} ) }{\varLambda ( F_{n+1},H_{n} ) \digamma _{2}(S _{n+1})\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +\frac{d ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{ \theta } \bigl( \digamma _{3} ( \widehat{H} ) - \digamma _{3} ( \widetilde{H} ) \bigr) \digamma _{4} ( Y_{n+1} ) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +\gamma \varLambda (\widehat{F},\widehat{H}) \biggl[ -1+\frac{\varLambda ( F_{n+1},\widehat{H} ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}(\widehat{H})}+ \frac{\varLambda (F_{n+1},H_{n})}{\varLambda (F_{n+1},\widehat{H})}-\frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(\widehat{H})} \biggr] , \\ &\Delta \mathcal{M}_{n} \leq \gamma \biggl( 1- \frac{\varLambda ( \widehat{F},\widehat{H})}{\varLambda (F_{n+1},\widehat{H})} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta ( \widehat{F} ) \bigr) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{}-\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{ \alpha +m}\varLambda (\widehat{F},\widehat{H}) \biggl[ G \biggl( \frac{ \varLambda (\widehat{F},\widehat{H})}{\varLambda (F_{n+1},\widehat{H})} \biggr) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +G \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \digamma _{1}(\widehat{K})}{\varLambda (\widehat{F},\widehat{H})\digamma _{1}(K _{n+1})} \biggr) +G \biggl( \frac{\digamma _{1}(K_{n+1})\digamma _{2}( \widehat{S})}{\digamma _{1}(\widehat{K})\digamma _{2}(S_{n+1})} \biggr) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +G \biggl( \frac{\digamma _{3}(\widehat{H})\digamma _{2} ( S_{n-m _{3}+1} ) }{\digamma _{3}(H_{n+1})\digamma _{2}(\widehat{S})} \biggr) \\ & \hphantom{\Delta \mathcal{M}_{n} \leq}{}+G \biggl( \frac{\varLambda ( F_{n+1},\widehat{H} ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}( \widehat{H})} \biggr) \biggr] -\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3} \tau _{3}}\varLambda (\widehat{F},\widehat{H}) \biggl[ G \biggl( \frac{ \varLambda (\widehat{F},\widehat{H})}{\varLambda (F_{n+1},\widehat{H})} \biggr) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +G \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) \digamma _{2}(\widehat{S})}{\varLambda (\widehat{F},\widehat{H})\digamma _{2}(S_{n+1})} \biggr) +G \biggl( \frac{ \digamma _{3}(\widehat{H})\digamma _{2} ( S_{n-m_{3}+1} ) }{ \digamma _{3}(H_{n+1})\digamma _{2}(\widehat{S})} \biggr) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +G \biggl( \frac{ \varLambda ( F_{n+1},\widehat{H} ) \digamma _{3}(H_{n})}{ \varLambda ( F_{n+1},H_{n} ) \digamma _{3}(\widehat{H})} \biggr) \biggr] \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +\frac{d ( a+\lambda \digamma _{5}(\widehat{Z}) ) }{ \theta }\digamma _{3} ( \widetilde{H} ) \bigl( \mathcal{R} _{1}^{Y}/\mathcal{R}_{2}^{Z}-1 \bigr) \digamma _{4} ( Y_{n+1} ) \\ &\hphantom{\Delta \mathcal{M}_{n} \leq}{} +\gamma \varLambda (\widehat{F},\widehat{H}) \biggl( 1-\frac{\varLambda (F _{n+1},\widehat{H})}{\varLambda (F_{n+1},H_{n})} \biggr) \biggl( \frac{ \varLambda (F_{n+1},H_{n})}{\varLambda (F_{n+1},\widehat{H})}-\frac{\digamma _{3} ( H_{n} ) }{\digamma _{3} ( \widehat{H} ) } \biggr) . \end{aligned}$$
(56)
Using Conditions C1–C4, we get that the first and last terms of Eq. (56) are less than or equal to zero. Moreover, if \(\mathcal{R}_{1}^{Y}/\mathcal{R}_{2}^{Z}\leq 1\), we get \(\Delta \mathcal{M}_{n}\leq 0\), and thus \(\mathcal{M}_{n}\) is a monotone decreasing sequence. Since \(\mathcal{M}_{n}\geq 0\), then there is a limit \(\lim_{n\rightarrow \infty }\mathcal{M}_{n}\geq 0\). Therefore, \(\lim_{n\rightarrow \infty }\Delta \mathcal{M} _{n}=0\), which implies that \(\lim_{n\rightarrow \infty }F _{n}=\widehat{F}\), \(\lim_{n\rightarrow \infty }K_{n}= \widehat{K}\), \(\lim_{n\rightarrow \infty }S_{n}= \widehat{S}\), \(\lim_{n\rightarrow \infty }H_{n}= \widehat{H}\), and \(\lim_{n\rightarrow \infty } ( \mathcal{R}_{1}^{Y}/\mathcal{R}_{2}^{Z}-1 ) Y_{n+1}=0\). We have two cases as follows:
\(\mathcal{R}_{1}^{Y}/\mathcal{R}_{2}^{Z}=1\), from Eq. (13)
$$ 0=\varepsilon e^{-\mu _{2}\tau _{2}}\varLambda ( \widehat{F}, \widehat{H} ) +m \digamma _{1} ( \widehat{K} ) -a \digamma _{2} ( \widehat{S} ) -\lambda \digamma _{2} ( \widehat{S} ) \lim _{n\rightarrow \infty }\digamma _{5} ( Z_{n+1} ), $$
(57)
and this gives \(\lim_{n\rightarrow \infty }Z_{n}= \widehat{Z}\). Moreover, from Eq. (14) we have
$$ 0=\theta e^{-\mu _{3}\tau _{3}}\digamma _{2} ( \widehat{S} ) -c \digamma _{3} ( \widehat{H} ) -d\digamma _{3} ( \widehat{H} ) \lim_{n\rightarrow \infty }\digamma _{4} ( Y_{n+1} ) , $$
(58)
then we get \(\lim_{n\rightarrow \infty }Y_{n}=0\).
\(\mathcal{R}_{1}^{Y}/\mathcal{R}_{2}^{Z}<1\), \(\lim_{n \rightarrow \infty }Y_{n}=0\). From Eq. (57) we get \(\lim_{n\rightarrow \infty }Z_{n}=\widehat{Z}\). Then we get that Q̂ is G.A.S.
□
Proof of Theorem 5
Define \(\mathcal{V}_{n}(F_{n},K_{n},S _{n},H_{n},Y_{n},Z_{n})\):
$$\begin{aligned} \mathcal{V}_{n} ={}&\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n}-\widetilde{F}- \int _{\widetilde{F}}^{F_{n}}\frac{\varLambda (\widetilde{F},\widetilde{H})}{\varLambda (\varsigma ,\widetilde{H})}\,d\varsigma \biggr) +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K _{n}-\widetilde{K}- \int _{\widetilde{K}}^{K_{n}}\frac{\digamma _{1}( \widetilde{K})}{\digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n}-\widetilde{S}- \int _{ \widetilde{S}}^{S_{n}}\frac{\digamma _{2}(\widetilde{S})}{\digamma _{2}( \varsigma )}\,d\varsigma \biggr) \\ &{} +\frac{ ( a+\lambda \digamma _{5}( \widetilde{Z}) ) }{\theta } \biggl( H_{n}-\widetilde{H}- \int _{\widetilde{H}}^{H_{n}}\frac{\digamma _{3}(\widetilde{H})}{ \digamma _{3}(\varsigma )}\,d\varsigma \biggr) \\ &{} +\frac{d ( a+\lambda \digamma _{5}(\widetilde{Z}) ) }{q\theta } \biggl( Y_{n}-\widetilde{Y}- \int _{\widetilde{Y}}^{Y_{n}}\frac{ \digamma _{4}(\widetilde{Y})}{\digamma _{4}(\varsigma )}\,d\varsigma \biggr) \\ &{} +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} \biggl( Z_{n}-\widetilde{Z}- \int _{ \widetilde{Z}}^{Z_{n}}\frac{\digamma _{5}(\widetilde{Z})}{\digamma _{5}( \varsigma )}\,d\varsigma \biggr) \biggr] \\ &{} +\frac{ ( c+d\digamma _{4}(\widetilde{Y}) ) }{\theta } \bigl( a+\lambda \digamma _{5}( \widetilde{Z}) \bigr) \digamma _{3} ( \widetilde{H} ) G \biggl( \frac{\digamma _{3} ( H _{n} ) }{\digamma _{3}(\widetilde{H})} \biggr) \\ &{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widetilde{F},\widetilde{H})\sum _{j=n-m _{1}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{ \varLambda (\widetilde{F},\widetilde{H})} \biggr) \\ &{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widetilde{F},\widetilde{H})\sum _{j=n-m_{2}}^{n-1}G \biggl( \frac{ \varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widetilde{F}, \widetilde{H})} \biggr) +\gamma \varLambda ( \widetilde{F}, \widetilde{H} ) \sum_{j=n-m_{3}}^{n-1}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \widetilde{S} ) } \biggr) . \end{aligned}$$
Clearly, \(\mathcal{V}_{n}(F_{n},K_{n},S_{n},H_{n},Y_{n},Z_{n})>0\) for all \(F_{n},K_{n},S_{n},H_{n},Y_{n},Z_{n}>0\) and \(\mathcal{V}_{n}( \widetilde{F},\widetilde{K},\widetilde{S},\widetilde{H},\widetilde{Y}, \widetilde{Z})=0\). We compute \(\Delta \mathcal{V}_{n}=\mathcal{V}_{n+1}- \mathcal{V}_{n}\) as follows:
$$\begin{aligned} &\triangle \mathcal{V}_{n} =\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n+1}-\widetilde{F}- \int _{\widetilde{F}}^{F_{n+1}}\frac{ \varLambda (\widetilde{F},\widetilde{H})}{\varLambda (\varsigma , \widetilde{H})}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{}+\frac{me^{-\mu _{3}\tau _{3}}}{ \alpha +m} \biggl( K_{n+1}-\widetilde{K}- \int _{\widetilde{K}}^{K_{n+1}}\frac{ \digamma _{1}(\widetilde{K})}{\digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n+1}-\widetilde{S}- \int _{ \widetilde{S}}^{S_{n+1}}\frac{\digamma _{2}(\widetilde{S})}{\digamma _{2}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{}+\frac{ ( a+\lambda \digamma _{5}(\widetilde{Z}) ) }{\theta } \biggl( H_{n+1}-\widetilde{H}- \int _{\widetilde{H}}^{H_{n+1}}\frac{\digamma _{3}(\widetilde{H})}{ \digamma _{3}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\frac{d ( a+\lambda \digamma _{5}(\widetilde{Z}) ) }{q\theta } \biggl( Y_{n+1}-\widetilde{Y}- \int _{\widetilde{Y}}^{Y_{n+1}}\frac{ \digamma _{4}(\widetilde{Y})}{\digamma _{4}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} \biggl( Z_{n+1}-\widetilde{Z}- \int _{ \widetilde{Z}}^{Z_{n+1}}\frac{\digamma _{5}(\widetilde{Z})}{\digamma _{5}( \varsigma )}\,d\varsigma \biggr) \biggr] \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\frac{ ( c+d\digamma _{4}(\widetilde{Y}) ) }{\theta } \bigl( a+\lambda \digamma _{5}( \widetilde{Z}) \bigr) \digamma _{3} ( \widetilde{H} ) G \biggl( \frac{\digamma _{3} ( H _{n+1} ) }{\digamma _{3}(\widetilde{H})} \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widetilde{F},\widetilde{H}) \sum _{j=n-m_{1}+1}^{n}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widetilde{F},\widetilde{H})} \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widetilde{F},\widetilde{H})\sum _{j=n-m_{2}+1}^{n}G \biggl( \frac{ \varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widetilde{F}, \widetilde{H})} \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{}+\gamma \varLambda ( \widetilde{F}, \widetilde{H} ) \sum_{j=n-m_{3}+1}^{n}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \widetilde{S} ) } \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} -\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n}- \widetilde{F}- \int _{\widetilde{F}}^{F_{n}}\frac{\varLambda ( \widetilde{F},\widetilde{H})}{\varLambda (\varsigma ,\widetilde{H})}\,d\varsigma \biggr) +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K _{n}-\widetilde{K}- \int _{\widetilde{K}}^{K_{n}}\frac{\digamma _{1}( \widetilde{K})}{\digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n}-\widetilde{S}- \int _{ \widetilde{S}}^{S_{n}}\frac{\digamma _{2}(\widetilde{S})}{\digamma _{2}( \varsigma )}\,d\varsigma \biggr) +\frac{ ( a+\lambda \digamma _{5}( \widetilde{Z}) ) }{\theta } \biggl( H_{n}-\widetilde{H}- \int _{\widetilde{H}}^{H_{n}}\frac{\digamma _{3}(\widetilde{H})}{ \digamma _{3}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\frac{d ( a+\lambda \digamma _{5}(\widetilde{Z}) ) }{q\theta } \biggl( Y_{n}-\widetilde{Y}- \int _{\widetilde{Y}}^{Y_{n}}\frac{ \digamma _{4}(\widetilde{Y})}{\digamma _{4}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{}+\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} \biggl( Z_{n}-\widetilde{Z}- \int _{ \widetilde{Z}}^{Z_{n}}\frac{\digamma _{5}(\widetilde{Z})}{\digamma _{5}( \varsigma )}\,d\varsigma \biggr) \biggr] \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} -\frac{ ( c+d\digamma _{4}(\widetilde{Y}) ) }{\theta } \bigl( a+\lambda \digamma _{5}( \widetilde{Z}) \bigr) \digamma _{3} ( \widetilde{H} ) G \biggl( \frac{\digamma _{3} ( H _{n} ) }{\digamma _{3}(\widetilde{H})} \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} -\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widetilde{F},\widetilde{H})\sum _{j=n-m _{1}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{ \varLambda (\widetilde{F},\widetilde{H})} \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} -\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widetilde{F},\widetilde{H})\sum _{j=n-m_{2}}^{n-1}G \biggl( \frac{ \varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widetilde{F}, \widetilde{H})} \biggr) -\gamma \varLambda ( \widetilde{F}, \widetilde{H} ) \sum_{j=n-m_{3}}^{n-1}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \widetilde{S} ) } \biggr) , \\ &\Delta \mathcal{V}_{n} =\frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( F_{n+1}-F_{n}- \int _{F_{n}}^{F_{n+1}}\frac{\varLambda ( \widetilde{F},\widetilde{H})}{\varLambda (\varsigma ,\widetilde{H})}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( K _{n+1}-K_{n}- \int _{K_{n}}^{K_{n+1}}\frac{\digamma _{1}(\widetilde{K})}{ \digamma _{1}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +e^{-\mu _{3}\tau _{3}} \biggl( S_{n+1}-S_{n}- \int _{S_{n}}^{S_{n+1}}\frac{ \digamma _{2}(\widetilde{S})}{\digamma _{2}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\frac{ ( a+\lambda \digamma _{5}(\widetilde{Z}) ) }{\theta } \biggl( H_{n+1}-H_{n}- \int _{H_{n}}^{H_{n}+1}\frac{\digamma _{3}(\widetilde{H})}{\digamma _{3}( \varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\frac{d ( a+\lambda \digamma _{5}(\widetilde{Z}) ) }{q\theta } \biggl( Y_{n+1}-Y_{n}- \int _{Y_{n}}^{Y_{n+1}}\frac{\digamma _{4}( \widetilde{Y})}{\digamma _{4}(\varsigma )}\,d\varsigma \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\frac{ \lambda e^{-\mu _{3}\tau _{3}}}{g} \biggl( Z_{n+1}-Z_{n}- \int _{Z_{n}} ^{Z_{n+1}} \frac{\digamma _{5}(\widetilde{Z})}{\digamma _{5}(\varsigma )}\,d \varsigma \biggr) \biggr] \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\frac{ ( c+d\digamma _{4}(\widetilde{Y}) ) }{\theta } \bigl( a+\lambda \digamma _{5}( \widetilde{Z}) \bigr) \digamma _{3} ( \widetilde{H} ) \biggl[ G \biggl( \frac{\digamma _{3} ( H_{n+1} ) }{\digamma _{3}(\widetilde{H})} \biggr) -G \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}( \widetilde{H})} \biggr) \biggr] \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widetilde{F},\widetilde{H}) \Biggl( \sum _{j=n-m_{1}+1}^{n}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widetilde{F},\widetilde{H})} \biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} -\sum_{j=n-m_{1}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widetilde{F},\widetilde{H})} \biggr) \Biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widetilde{F},\widetilde{H}) \Biggl( \sum_{j=n-m_{2}+1}^{n}G \biggl( \frac{ \varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widetilde{F}, \widetilde{H})} \biggr) -\sum_{j=n-m_{2}}^{n-1}G \biggl( \frac{\varLambda ( F_{j+1},H_{j} ) }{\varLambda (\widetilde{F},\widetilde{H})} \biggr) \Biggr) \\ &\hphantom{\triangle \mathcal{V}_{n} =}{} +\gamma \varLambda ( \widetilde{F},\widetilde{H} ) \Biggl( \sum _{j=n-m_{3}+1}^{n}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{\digamma _{2} ( \widetilde{S} ) } \biggr) -\sum_{j=n-m _{3}}^{n-1}G \biggl( \frac{\digamma _{2} ( S_{j+1} ) }{ \digamma _{2} ( \widetilde{S} ) } \biggr) \Biggr) . \end{aligned}$$
Using inequalities (45) and (46) by replacing \(F^{\ast }\), \(H^{\ast }\), \(\rho ^{\ast }\) with F̃, H̃, ρ̃, we obtain
$$\begin{aligned} \Delta \mathcal{V}_{n} \leq{}& \frac{1}{\phi ( h ) } \biggl[ \gamma \biggl( 1-\frac{\varLambda (\widetilde{F},\widetilde{H})}{\varLambda (F_{n+1},\widetilde{H})} \biggr) ( F_{n+1}-F_{n} ) +\frac{me ^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( 1-\frac{\digamma _{1}( \widetilde{K})}{\digamma _{1}(K_{n+1})} \biggr) ( K_{n+1}-K_{n} ) \\ &{} +e^{-\mu _{3}\tau _{3}} \biggl( 1-\frac{\digamma _{2}(\widetilde{S})}{ \digamma _{2}(S_{n+1})} \biggr) ( S_{n+1}-S_{n} ) \\ &{} +\frac{ ( a+\lambda \digamma _{5}(\widetilde{Z}) ) }{\theta } \biggl( 1- \frac{\digamma _{3}(\widetilde{H})}{\digamma _{3}(H_{n+1})} \biggr) ( H_{n+1}-H_{n} ) \\ &{} +\frac{d ( a+\lambda \digamma _{5}(\widetilde{Z}) ) }{q\theta } \biggl( 1-\frac{\digamma _{4}(\widetilde{Y})}{\digamma _{4}(Y _{n+1})} \biggr) ( Y_{n+1}-Y_{n} ) \\ &{} +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} \biggl( 1- \frac{\digamma _{5}(\widetilde{Z})}{\digamma _{5}(Z_{n+1})} \biggr) ( Z_{n+1}-Z_{n} ) \biggr] \\ &{} +\frac{ ( c+d\digamma _{4}(\widetilde{Y}) ) }{\theta } \bigl( a+\lambda \digamma _{5}( \widetilde{Z}) \bigr) \digamma _{3} ( \widetilde{H} ) \biggl[ \frac{\digamma _{3} ( H_{n+1} ) }{\digamma _{3}(\widetilde{H})}-\frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(\widetilde{H})}+\ln \biggl( \frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ & {}+\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widetilde{F},\widetilde{H}) \biggl[ \frac{ \varLambda ( F_{n+1},H_{n} ) }{\varLambda (\widetilde{F}, \widetilde{H})}-\frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda (\widetilde{F},\widetilde{H})} \\ &{} +\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m _{1}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr] \\ & {}+\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widetilde{F},\widetilde{H}) \biggl[ \frac{\varLambda ( F_{n+1},H _{n} ) }{\varLambda (\widetilde{F},\widetilde{H})}-\frac{\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) }{\varLambda (\widetilde{F}, \widetilde{H})} \\ &{} +\ln \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H_{n-m _{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr] \\ &{} +\gamma \varLambda ( \widetilde{F},\widetilde{H} ) \biggl[ \frac{\digamma _{2} ( S_{n+1} ) }{\digamma _{2} ( \widetilde{S} ) }-\frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( \widetilde{S} ) }+\ln \biggl( \frac{ \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) \biggr] . \end{aligned}$$
From Eqs. (11)–(16) we have
$$\begin{aligned} &\Delta \mathcal{V}_{n} \leq \gamma \biggl( 1- \frac{\varLambda ( \widetilde{F},\widetilde{H})}{\varLambda (F_{n+1},\widetilde{H})} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varLambda ( F_{n+1},H_{n} ) \bigr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\frac{me^{-\mu _{3}\tau _{3}}}{\alpha +m} \biggl( 1-\frac{\digamma _{1}( \widetilde{K})}{\digamma _{1}(K_{n+1})} \biggr) \bigl[ ( 1- \varepsilon ) e^{-\mu _{1}\tau _{1}}\varLambda ( F_{n-m_{1}+1},H _{n-m_{1}} ) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} - ( \alpha +m ) \digamma _{1} ( K _{n+1} ) \bigr] \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +e^{-\mu _{3}\tau _{3}} \biggl( 1-\frac{\digamma _{2}(\widetilde{S})}{ \digamma _{2}(S_{n+1})} \biggr) \bigl[ \varepsilon e^{-\mu _{2}\tau _{2}}\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) +m\digamma _{1} ( K_{n+1} ) -a\digamma _{2} ( S_{n+1} ) \\ & \hphantom{\Delta \mathcal{V}_{n} \leq}{}-\lambda \digamma _{2} ( S_{n+1} ) \digamma _{5} ( Z_{n+1} ) \bigr] \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\frac{ ( a+\lambda \digamma _{5}(\widetilde{Z}) ) }{ \theta } \biggl( 1-\frac{\digamma _{3}(\widetilde{H})}{\digamma _{3}(H _{n+1})} \biggr) \bigl[ \theta e^{-\mu _{3}\tau _{3}}\digamma _{2} ( S_{n-m_{3}+1} ) -c\digamma _{3} ( H_{n+1} ) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} -d\digamma _{3} ( H_{n+1} ) \digamma _{4} ( Y_{n+1} ) \bigr] \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\frac{d ( a+\lambda \digamma _{5}(\widetilde{Z}) ) }{q \theta } \biggl( 1-\frac{\digamma _{4}(\widetilde{Y})}{\digamma _{4}(Y _{n+1})} \biggr) \bigl[ q \digamma _{3} ( H_{n+1} ) \digamma _{4} ( Y_{n+1} ) -\eta \digamma _{4} ( Y_{n+1} ) \bigr] \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\frac{\lambda e^{-\mu _{3}\tau _{3}}}{g} \biggl( 1-\frac{\digamma _{5}( \widetilde{Z})}{\digamma _{5}(Z_{n+1})} \biggr) \bigl[ g \digamma _{2} ( S_{n+1} ) \digamma _{5} ( Z_{n+1} ) -\xi \digamma _{5} ( Z_{n+1} ) \bigr] \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\frac{ ( c+d\digamma _{4}(\widetilde{Y}) ) }{\theta } \bigl( a+\lambda \digamma _{5}( \widetilde{Z}) \bigr) \digamma _{3} ( \widetilde{H} ) \biggl[ \frac{\digamma _{3} ( H_{n+1} ) }{\digamma _{3}(\widetilde{H})}-\frac{\digamma _{3} ( H_{n} ) }{\digamma _{3}(\widetilde{H})}+\ln \biggl( \frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widetilde{F},\widetilde{H}) \biggl[ \frac{ \varLambda ( F_{n+1},H_{n} ) }{\varLambda (\widetilde{F}, \widetilde{H})}-\frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda (\widetilde{F},\widetilde{H})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m _{1}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr] \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widetilde{F},\widetilde{H}) \biggl[ \frac{\varLambda ( F_{n+1},H _{n} ) }{\varLambda (\widetilde{F},\widetilde{H})}-\frac{\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) }{\varLambda (\widetilde{F}, \widetilde{H})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{}+\ln \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H_{n-m _{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \biggr] \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\gamma \varLambda ( \widetilde{F},\widetilde{H} ) \biggl[ \frac{\digamma _{2} ( S_{n+1} ) }{\digamma _{2} ( \widetilde{S} ) }-\frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( \widetilde{S} ) }+\ln \biggl( \frac{ \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) \biggr], \\ &\Delta \mathcal{V}_{n} \leq \gamma \biggl( 1- \frac{\varLambda ( \widetilde{F},\widetilde{H})}{\varLambda (F_{n+1},\widetilde{H})} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta ( \widetilde{F} ) \bigr) +\gamma \varTheta ( \widetilde{F} ) \biggl( 1- \frac{\varLambda (\widetilde{F}, \widetilde{H})}{\varLambda (F_{n+1},\widetilde{H})} \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\gamma \frac{\varLambda (\widetilde{F},\widetilde{H})}{\varLambda (F _{n+1},\widetilde{H})}\varLambda ( F_{n+1},H_{n} ) -\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{ \alpha +m}\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \frac{ \digamma _{1}(\widetilde{K})}{\digamma _{1}(K_{n+1})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +me^{-\mu _{3}\tau _{3}}\digamma _{1}(\widetilde{K})-\varepsilon e ^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( F_{n-m_{2}+1},H _{n-m_{2}} ) \frac{\digamma _{2}(\widetilde{S})}{\digamma _{2}(S _{n+1})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{}-me^{-\mu _{3}\tau _{3}}\digamma _{1} ( K_{n+1} ) \frac{ \digamma _{2}(\widetilde{S})}{\digamma _{2}(S_{n+1})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} -ae^{-\mu _{3}\tau _{3}}\digamma _{2} ( S_{n+1} ) +\lambda e^{-\mu _{3}\tau _{3}}\digamma _{2}(\widetilde{S})\digamma _{5} ( Z _{n+1} ) +ae^{-\mu _{3}\tau _{3}}\digamma _{2}(\widetilde{S}) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{}+ \bigl( a+\lambda \digamma _{5}( \widetilde{Z}) \bigr) e^{-\mu _{3}\tau _{3}}\digamma _{2}(S_{n-m_{3}+1}) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} - \bigl( a+\lambda \digamma _{5}(\widetilde{Z}) \bigr) e^{-\mu _{3} \tau _{3}}\digamma _{2}(S_{n-m_{3}+1}) \frac{\digamma _{3}(\widetilde{H})}{ \digamma _{3}(H_{n+1})}+\frac{ ( a+\lambda \digamma _{5}( \widetilde{Z}) ) c}{\theta }\digamma _{3} ( \widetilde{H} ) \\ & \hphantom{\Delta \mathcal{V}_{n} \leq}{}+\frac{d ( a+\lambda \digamma _{5}(\widetilde{Z}) ) }{q \theta }\digamma _{3} ( \widetilde{H} ) \digamma _{4} ( Y_{n+1} ) -\frac{d ( a+\lambda \digamma _{5}(\widetilde{Z}) ) \eta }{q\theta }\digamma _{4} ( Y_{n+1} ) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\frac{d ( a+\lambda \digamma _{5}(\widetilde{Z}) ) \eta }{q\theta }\digamma _{4}(\widetilde{Y})-\lambda e^{-\mu _{3}\tau _{3}}\digamma _{5} ( \widetilde{Z} ) \digamma _{2} ( S _{n+1} ) -\frac{\lambda e^{-\mu _{3}\tau _{3}}\xi }{g}\digamma _{5} ( Z_{n+1} ) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} + \frac{\lambda e^{-\mu _{3}\tau _{3}}\xi }{g}\digamma _{5} ( \widetilde{Z} ) \\ & \hphantom{\Delta \mathcal{V}_{n} \leq}{}-\frac{ ( c+d\digamma _{4}(\widetilde{Y}) ) }{\theta } \bigl( a+\lambda \digamma _{5}( \widetilde{Z}) \bigr) \digamma _{3} ( H_{n} ) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} + \frac{ ( c+d\digamma _{4}(\widetilde{Y}) ) }{\theta } \bigl( a+\lambda \digamma _{5}(\widetilde{Z}) \bigr) \digamma _{3} ( \widetilde{H} ) \ln \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \\ & \hphantom{\Delta \mathcal{V}_{n} \leq}{}+\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widetilde{F},\widetilde{H})\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widetilde{F},\widetilde{H})\ln \biggl( \frac{\varLambda ( F_{n-m _{2}+1},H_{n-m_{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\gamma \varLambda ( \widetilde{F},\widetilde{H} ) \biggl( \frac{\digamma _{2} ( S_{n+1} ) }{\digamma _{2} ( \widetilde{S} ) }-\frac{\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( \widetilde{S} ) }+\ln \biggl( \frac{ \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) \biggr) . \end{aligned}$$
Using the conditions of Q̃
$$\begin{aligned} &\varTheta ( \widetilde{F} ) =\varLambda ( \widetilde{F}, \widetilde{H} ) , \\ &( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}}\varLambda ( \widetilde{F},\widetilde{H} ) = ( \alpha +m ) \digamma _{1} ( \widetilde{K} ) , \\ &\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widetilde{F},\widetilde{H} ) +me^{-\mu _{3}\tau _{3}}\digamma _{1} ( \widetilde{K} ) =\gamma \varLambda ( \widetilde{F},\widetilde{H} ) = \bigl( a+\lambda \digamma _{5} ( \widetilde{Z} ) \bigr) e^{-\mu _{3}\tau _{3}}\digamma _{2} ( \widetilde{S} ) , \\ &\theta e^{-\mu _{3}\tau _{3}}\digamma _{2} ( \widetilde{S} ) = \bigl( c+d\digamma _{4}(\widetilde{Y}) \bigr) \digamma _{3} ( \widetilde{H} ) , \\ &\digamma _{2} ( \widetilde{S} ) =\frac{\xi }{g},\qquad \digamma _{3} ( \widetilde{H} ) =\frac{\eta }{q}, \end{aligned}$$
we get
$$\begin{aligned} &\Delta \mathcal{V}_{n} \leq \gamma \biggl( 1- \frac{\varLambda ( \widetilde{F},\widetilde{H})}{\varLambda (F_{n+1},\widetilde{H})} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta ( \widetilde{F} ) \bigr) +\gamma \varLambda (\widetilde{F}, \widetilde{H}) \biggl( 1-\frac{\varLambda (\widetilde{F},\widetilde{H})}{ \varLambda (F_{n+1},\widetilde{H})} \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\gamma \varLambda (\widetilde{F},\widetilde{H})\frac{\varLambda (F_{n+1},H _{n})}{\varLambda (F_{n+1},\widetilde{H})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{}- \frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{\alpha +m}\varLambda ( \widetilde{F},\widetilde{H}) \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m _{1}} ) \digamma _{1}(\widetilde{K})}{\varLambda (\widetilde{F}, \widetilde{H})\digamma _{1}(K_{n+1})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widetilde{F},\widetilde{H}) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{}-\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda (\widetilde{F}, \widetilde{H})\frac{\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) \digamma _{2}(\widetilde{S})}{\varLambda (\widetilde{F},\widetilde{H}) \digamma _{2}(S_{n+1})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} -\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widetilde{F},\widetilde{H}) \frac{ \digamma _{1}(K_{n+1})\digamma _{2}(\widetilde{S})}{\digamma _{1}( \widetilde{K})\digamma _{2}(S_{n+1})}+\gamma \varLambda (\widetilde{F}, \widetilde{H}) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{}-\gamma \varLambda (\widetilde{F},\widetilde{H})\frac{ \digamma _{3}(\widetilde{H})\digamma _{2} ( S_{n-m_{3}+1} ) }{ \digamma _{3}(H_{n+1})\digamma _{2}(\widetilde{S})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\gamma \varLambda (\widetilde{F},\widetilde{H})-\gamma \varLambda ( \widetilde{F},\widetilde{H})\frac{\digamma _{3}(H_{n})}{\digamma _{3}( \widetilde{H})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{}+\gamma \varLambda ( \widetilde{F},\widetilde{H})\ln \biggl( \frac{\digamma _{3} ( H_{n} ) }{\digamma _{3} ( H_{n+1} ) } \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3} \tau _{3}}}{\alpha +m}\varLambda (\widetilde{F},\widetilde{H})\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widetilde{F},\widetilde{H})\ln \biggl( \frac{\varLambda ( F_{n-m _{2}+1},H_{n-m_{2}} ) }{\varLambda ( F_{n+1},H_{n} ) } \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{}+\gamma \varLambda (\widetilde{F}, \widetilde{H})\ln \biggl( \frac{ \digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{2} ( S_{n+1} ) } \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} }{} =\gamma \biggl( 1-\frac{\varLambda (\widetilde{F},\widetilde{H})}{ \varLambda (F_{n+1},\widetilde{H})} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta ( \widetilde{F} ) \bigr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{ \alpha +m} \varLambda (\widetilde{F},\widetilde{H}) \biggl[ 5-\frac{\varLambda (\widetilde{F},\widetilde{H})}{\varLambda (F_{n+1},\widetilde{H})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} -\frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \digamma _{1}( \widetilde{K})}{\varLambda (\widetilde{F},\widetilde{H})\digamma _{1}(K _{n+1})}-\frac{\digamma _{1}(K_{n+1})\digamma _{2}(\widetilde{S})}{ \digamma _{1}(\widetilde{K})\digamma _{2}(S_{n+1})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{}-\frac{\digamma _{3}( \widetilde{H})\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H _{n+1})\digamma _{2}(\widetilde{S})}- \frac{\varLambda ( F_{n+1}, \widetilde{H} ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H _{n} ) \digamma _{3}(\widetilde{H})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\ln \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m _{1}} ) \digamma _{2} ( S_{n-m_{3}+1} ) \digamma _{3} ( H_{n} ) }{\varLambda ( F_{n+1},H_{n} ) \digamma _{2}(S_{n+1})\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3}\tau _{3}}\varLambda ( \widetilde{F},\widetilde{H}) \biggl[ 4-\frac{\varLambda (\widetilde{F}, \widetilde{H})}{\varLambda (F_{n+1},\widetilde{H})}-\frac{\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) \digamma _{2}(\widetilde{S})}{\varLambda (\widetilde{F},\widetilde{H})\digamma _{2}(S_{n+1})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{}-\frac{\digamma _{3}(\widetilde{H})\digamma _{2} ( S_{n-m_{3}+1} ) }{ \digamma _{3}(H_{n+1})\digamma _{2}(\widetilde{S})} \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} -\frac{\varLambda ( F_{n+1},\widetilde{H} ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}(\widetilde{H})}+\ln \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H _{n-m_{2}} ) \digamma _{2} ( S_{n-m_{3}+1} ) \digamma _{3} ( H_{n} ) }{\varLambda ( F_{n+1},H_{n} ) \digamma _{2}(S_{n+1})\digamma _{3} ( H_{n+1} ) } \biggr) \biggr] \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\gamma \varLambda (\widetilde{F},\widetilde{H}) \biggl[ -1+ \frac{ \varLambda ( F_{n+1},\widetilde{H} ) \digamma _{3}(H_{n})}{ \varLambda ( F_{n+1},H_{n} ) \digamma _{3}(\widetilde{H})}+\frac{ \varLambda (F_{n+1},H_{n})}{\varLambda (F_{n+1},\widetilde{H})}-\frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3}(\widetilde{H})} \biggr] , \\ &\Delta \mathcal{V}_{n} \leq \gamma \biggl( 1- \frac{\varLambda ( \widetilde{F},\widetilde{H})}{\varLambda (F_{n+1},\widetilde{H})} \biggr) \bigl( \varTheta ( F_{n+1} ) -\varTheta ( \widetilde{F} ) \bigr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{}-\frac{m ( 1-\varepsilon ) e^{-\mu _{1}\tau _{1}-\mu _{3}\tau _{3}}}{\alpha +m}\varLambda ( \widetilde{F}, \widetilde{H}) \biggl[ G \biggl( \frac{\varLambda ( \widetilde{F},\widetilde{H})}{\varLambda (F_{n+1},\widetilde{H})} \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +G \biggl( \frac{\varLambda ( F_{n-m_{1}+1},H_{n-m_{1}} ) \digamma _{1}(\widetilde{K})}{\varLambda (\widetilde{F},\widetilde{H}) \digamma _{1}(K_{n+1})} \biggr) +G \biggl( \frac{\digamma _{1}(K_{n+1}) \digamma _{2}(\widetilde{S})}{\digamma _{1}(\widetilde{K})\digamma _{2}(S _{n+1})} \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +G \biggl( \frac{\digamma _{3}(\widetilde{H})\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H_{n+1})\digamma _{2}( \widetilde{S})} \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +G \biggl( \frac{\varLambda ( F_{n+1},\widetilde{H} ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H_{n} ) \digamma _{3}( \widetilde{H})} \biggr) \biggr] -\varepsilon e^{-\mu _{2}\tau _{2}-\mu _{3} \tau _{3}}\varLambda (\widetilde{F},\widetilde{H}) \biggl[ G \biggl( \frac{ \varLambda (\widetilde{F},\widetilde{H})}{\varLambda (F_{n+1},\widetilde{H})} \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +G \biggl( \frac{\varLambda ( F_{n-m_{2}+1},H_{n-m_{2}} ) \digamma _{2}(\widetilde{S})}{\varLambda (\widetilde{F},\widetilde{H})\digamma _{2}(S_{n+1})} \biggr) +G \biggl( \frac{\digamma _{3}(\widetilde{H})\digamma _{2} ( S_{n-m_{3}+1} ) }{\digamma _{3}(H_{n+1})\digamma _{2}( \widetilde{S})} \biggr) \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{}+G \biggl( \frac{\varLambda ( F_{n+1}, \widetilde{H} ) \digamma _{3}(H_{n})}{\varLambda ( F_{n+1},H _{n} ) \digamma _{3}(\widetilde{H})} \biggr) \biggr] \\ &\hphantom{\Delta \mathcal{V}_{n} \leq}{} +\gamma \varLambda (\widetilde{F},\widetilde{H}) \biggl( 1- \frac{ \varLambda (F_{n+1},\widetilde{H})}{\varLambda (F_{n+1},H_{n})} \biggr) \biggl( \frac{\varLambda (F_{n+1},H_{n})}{\varLambda (F_{n+1},\widetilde{H})}-\frac{ \digamma _{3} ( H_{n} ) }{\digamma _{3} ( \widetilde{H} ) } \biggr) . \end{aligned}$$
(59)
Using Conditions C1–C4, we get that the first and last terms of Eq. (59) are less than or equal to zero. Thus, \(\mathcal{V}_{n}\) is a monotone decreasing sequence. Since \(\mathcal{V}_{n}\geq 0\), then there is a limit \(\lim_{n\rightarrow \infty }\mathcal{V}_{n}\geq 0\). Therefore, \(\lim_{n\rightarrow \infty }\Delta \mathcal{V}_{n}=0\), which implies that \(\lim_{n\rightarrow \infty }F_{n}=\widetilde{F}\), \(\lim_{n\rightarrow \infty }K_{n}=\widetilde{K}\), \(\lim_{n\rightarrow \infty }S_{n}=\widetilde{S}\), \(\lim_{n\rightarrow \infty }H_{n}=\widetilde{H}\). From Eqs. (13) and (14) we have
$$\begin{aligned} &0 =\varepsilon e^{-\mu _{2}\tau _{2}}\varLambda ( \widetilde{F}, \widetilde{H} ) +m \digamma _{1} ( \widetilde{K} ) -a \digamma _{2} ( \widetilde{S} ) -\lambda \digamma _{2} ( \widetilde{S} ) \lim _{n\rightarrow \infty }\digamma _{5} ( Z_{n+1} ) , \\ &0 =\theta e^{-\mu _{3}\tau _{3}}\digamma _{2} ( \widetilde{S} ) -c \digamma _{3} ( \widetilde{H} ) -d \digamma _{2} ( \widetilde{H} ) \lim_{n\rightarrow \infty }\digamma _{4} ( Y_{n+1} ) , \end{aligned}$$
then \(\lim_{n\rightarrow \infty }Y_{n}=\widetilde{Y}\) and \(\lim_{n\rightarrow \infty }Z_{n}=\widetilde{Z}\). Then we get Q̃ is G.A.S. □