Some equilibrium, stability, instability and oscillatory results for an extended discrete epidemic model with evolution memory

Abstract

This paper investigates the existence and potential uniqueness of equilibrium points and some stability, instability and oscillatory properties of a discrete nonlinear epidemic model, which generalises previous Stevic’s model, whose solutions possess memory from a finite chain of preceding samples. An application example is provided where the proposed model is ‘ad hoc’ adapted to a class of SIS models widely used in epidemiology.

1 Introduction

In this paper, some properties of equilibrium points as well as some stability and instability properties of the following nonlinear discrete epidemic model are investigated:

$$x_{n+1}=max(0,(1-\sum _{j=0}^{k-1}F(x_{n-j}))(1-M({\overline{x}}_n)e^{-AG({\overline{x}}_n)})),n\in {\mathbf{N}}_0,$$

(1.1)

with ${\mathbf{N}}_0=\mathbf{N}\cup \{0\}$, where ${\overline{x}}_n=(x_n,x_{n-1},\dots ,x_{n-q+1})$ is a $q(\in \mathbf{N})$-tuple of values of the solution sequence previous to its $(n+1)$th value and $F:{\mathbf{R}}_{0+}\to \mathbf{R}$ and $M,G:{\mathbf{R}}_{0+}^q\to \mathbf{R}$ under any set of initial conditions $x_i\ge 0$; $i=1-p,2-p,\dots ,0$, where $p=max(k,q)$, and $A\in {\mathbf{R}}_{+}$ is a weighting forgetting factor in the model. Note that such sets of initial conditions guarantee that the associated solution sequence $\{x_n\}$, $n\in {\mathbf{N}}_0\cup (-\overline{p})$, where $\overline{p}=\{1,2,\dots ,p\}$, is nonnegative by construction. It has to be pointed out that Stevic studied negative solutions of the epidemic model

$$x_{n+1}=max(0,(1-\sum _{j=0}^{k-1}{x}_{n-j})(1-e^{-A{x}_n}))$$

(1.2)

for $A\in (0,\mathrm{\infty })$ and also described and interpreted prior work on nonnegative solutions of the same epidemic model and their stability, instability and oscillation properties [1]. See also [2–5] for some related work. Later on, the discrete epidemic model was extended to include two coupled extended difference equations [6]. It turns out that both continuous-type and discrete-type epidemic models are of great importance in research nowadays because of its intrinsic interest in medical applications and because of their rich dynamics which make them also attractive to mathematicians involved in the investigations of nonlinear differential and difference equations. See, for instance, [7–11]. Note that discrete modelling techniques are very relevant in the study of ecology and biology problems as well, like, for instance, the discretization of logistic equations [12] leading to discrete models such as those related to the well-known Riecker, Beverton-Holt and Hassel equations (see, for instance, [13–21] and references therein). A way of establishing a direct generalisation of Zhang-Shi’s [5], Stevic’s [1], and Papaschinopoulos et al.’s [6] analysis for a related epidemic model is to restrict the codomains of the various functions to be nonnegative by defining them as $F:{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$; $M,G:{\mathbf{R}}_{0+}^q\to {\mathbf{R}}_{0+}$, where ${\mathbf{R}}_{0+}=\{z\in \mathbf{R}:z\ge 0\}={\mathbf{R}}_{+}\cup \{0\}$ with ${\mathbf{R}}_{+}=\{z\in \mathbf{R}:z>0\}$. However, under a more general discussion, it can be allowed for the functions to have ranges in R. The function $F:{\mathbf{R}}_{0+}\to \mathbf{R}$ is assumed to be upper-bounded and lower-bounded by known polynomials in x of degree q. The constraint $max({\sum }_{j=0}^{k-1}F(x_{n-j}),M({\overline{x}}_n)e^{-AG({\overline{x}}_n)})\le 1$ for $j\in {\mathbf{N}}_{0+}$ is not assumed so that both sequences $(1-{\sum }_{j=0}^{k-1}F(x_{n-j}))$, $(1-M({\overline{x}}_n)e^{-AG({\overline{x}}_n)})$ may be eventually negative in (1.1) while generating a nonnegative solution sequence. Note that the solution evolution of the proposed model can be interpreted as the propagation of the infection (say, roughly speaking, the infected population or a normalised value for it) from certain initial conditions and subject to weighting factor parameterization and a number of discrete delays. A considerable freedom of implementation of the proposed model in terms of choices of the parameterization structures F, M, A and the number of terms in the parameterization sequences is allowed. The above model remembers, in a much more general context and discretized version, the original Bernoulli proposal to introduce a simple epidemic model with just a single variable being the solution of a scalar equation. However, this model can be useful for situations involving two (or even three) variables as, for instance, the SI-epidemic model when the total population remains constant for all time during the illness cycle, i.e. for the case when mortality caused by the disease is not expected, or even for models of three variables. Related interpretations and practical use are addressed in the simulated examples, and it has to be pointed out that simple structures for epidemic models are often preferred in medical structures compared to complex alternative structures. A simulated example is also provided with a detailed study combining the proposed epidemic models with a class of SIS epidemic models that have been previously known in the background literature [16, 17]. Finally, it has to be pointed out that the properties of stability, instability and oscillatory behaviour of the solutions are very relevant issues in the study of epidemic models. See, for instance, [16–21] and references therein. Thus, the study and associated discussion provided concerning the proposed model pay a special attention to them.

2 Equilibrium points

To study the existence of potential equilibrium points, we set the values of solution sequence (1.1) to a constant one x, which yields

$$x=max(0,(1-kF(x))(1-M(\overline{x})e^{-AG(\overline{x})})),$$

(2.1)

where $\overline{x}=(x,x,\dots ,x)$ is in ${\mathbf{R}}^q$. Define $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ as follows:

$$g(x)=max(0,(1-kF(x))(1-M(\overline{x})e^{-AG(\overline{x})}))-x.$$

(2.2)

Note that $x\in {\mathbf{R}}_{0+}$ is an equilibrium point of (1.1) if and only if $g(x)=0$. The existence of equilibrium points of (1.1) are subject to the following direct result.

Theorem 2.1 The following properties hold.

• (i) $x=0$ is an equilibrium point of (1.1) if and only if $g(0)=0$, equivalently if and only if $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})\le 0$, where $0\in \mathbf{R}$ and $\overline{0}=(0,0,\dots ,0)\in {\mathbf{R}}^q$, equivalently if and only if

  $$\begin{array}{rcl}(F(0)=1/k)& \vee & (M(\overline{0})=e^{AG(\overline{0})})\\ \vee & ([(F(0)\ne 1/k)\wedge M(\overline{0})\ne e^{AG(\overline{0})}]\iff [(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})<0]),\end{array}$$

where the symbols ‘∨’ and ‘∧’ stand for ‘or’ and ‘and’ (i.e. for logic disjunction and conjunction, respectively).

• (ii) $x\in {\mathbf{R}}_{+}$ (respectively, $x\in {\mathbf{R}}_{0+}$) is a positive (respectively, a nonnegative) equilibrium point of (1.1) if and only $g(x)=0$ for such $x\in {\mathbf{R}}_{+}$ (respectively, for such $x\in {\mathbf{R}}_{0+}$). If $g(x)\ne 0\iff (1-kF(x))(1-M(\overline{x})e^{-AG(\overline{x})})\ne x>0$; $\mathrm{\forall }x\in [x_1,x_2]\subset {\mathbf{R}}_{0+}$, then there is no equilibrium point of (1.1) within the real interval $[x_1,x_2]$. Finally, a necessary condition for $x\in {\mathbf{R}}_{0+}$ (respectively, for $x\in {\mathbf{R}}_{+}$) to be an equilibrium point of (1.1) is that $(1-kF(x))(1-M(\overline{x})e^{-AG(\overline{x})})\ge 0$ (respectively, $(1-kF(x))(1-M(\overline{x})e^{-AG(\overline{x})})>0$).

• (iii) There is no positive (respectively, no nonnegative) equilibrium point of (1.1) if and only if $g(x)\ne 0$; $\mathrm{\forall }x\in {\mathbf{R}}_{+}$ (respectively, if and only if $g(x)\ne 0$; $\mathrm{\forall }x\in {\mathbf{R}}_{0+}$).

• (iv) $x=0$ is the unique nonnegative equilibrium point of (1.1) if and only if $(1-kF(x))(1-M(\overline{x})e^{-AG(\overline{x})})\le 0$; $\mathrm{\forall }x\in {\mathbf{R}}_{0+}$.

Proof It follows that $x=0$ is an equilibrium point of (1.1) if and only if $g(0)=0\iff (1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})\le 0$ from (2.1)-(2.2). This proves directly Property (i). To prove Property (ii), note that the logic proposition $g(x)\ne 0\iff (1-kF(x))(1-M(\overline{x})e^{-AG(\overline{x})})\ne x>0$ for any $x\in [x_1,x_2]\subset {\mathbf{R}}_{+}$ is equivalent to its contrapositive logic proposition $\mathrm{\neg }\mathrm{\exists }x\in [x_1,x_2]$ such that $(1-kF(x))(1-M(\overline{x})e^{-AG(\overline{x})})\le 0\iff g(x)=0$ (‘¬’ stands for logic negation) so that there is no equilibrium point of (1.1) in $[x_1,x_2]$. This proves the first part of Property (ii). The last part of Property (ii) follows by contradiction since $(1-kF(x))(1-M(\overline{x})e^{-AG(\overline{x})})\le 0$ implies $g(x)=-x<0$; $\mathrm{\forall }x\in {\mathbf{R}}_{+}$, so that there is no positive equilibrium point of (1.1), and, on the other hand, $g(x)=0$ if and only if $x=0$ is an equilibrium point, which is impossible since $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})>0$. Hence, Property (ii). Property (iii) is direct since any equilibrium point of (1.1) in ${\mathbf{R}}_{0+}$ implies and is implied by the condition $g(x)=0$. The sufficiency part of Property (iv) follows since the given condition implies that $g(0)=0$, then $x=0$ is an equilibrium point of (1.1) and $g(x)=-x<0$ for $x\in {\mathbf{R}}_{+}$, so that there is no positive equilibrium point of (1.1). The necessity part of Property (iv) follows since $x=0$ is an equilibrium point of (1.1) only if $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})\le 0$, which is unique in ${\mathbf{R}}_{0+}$, since any $x\in {\mathbf{R}}_{+}$ is an equilibrium point of (1.1) only if $(1-kF(x))(1-M(\overline{x})e^{-AG(\overline{x})})>0$ from Property (ii). □

Some parts of the subsequent analysis are simplified subject to the following assumption.

Assumption 2.2 $F:{\mathbf{R}}_{0+}\to \mathbf{R}$ is differentiable on ${\mathbf{R}}_{0+}$ and $M,G:{\mathbf{R}}_{0+}^q\to \mathbf{R}$ are differentiable in ${\mathbf{R}}_{0+}^q$.

Note that if Assumption 2.2 holds, then $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ is also everywhere differentiable. The following result is a direct conclusion of Theorem 2.1.

Theorem 2.3 A sufficient condition for $x_1\in {\mathbf{R}}_{0+}$ to be the unique equilibrium point of (1.1) is that $g(x_1)=0$ and $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ fulfils one of the conditions below:

• (a) strictly monotone in ${\mathbf{R}}_{0+}$;

• (b) non-decreasing (or, respectively, decreasing) in $[x_1,\mathrm{\infty })$ and, furthermore, strictly increasing (or, respectively, strictly decreasing) in some real subinterval $[x_1,x_3)\subseteq [x_1,\mathrm{\infty })$ of nonzero measure and, in addition if $x_1\in {\mathbf{R}}_{+}$, either strictly decreasing (or, respectively, strictly increasing) in some interval $[x_2,x_1]\subset {\mathbf{R}}_{0+}$ of nonzero measure and, correspondingly, either decreasing (or, respectively, non-decreasing) in some (eventually being of zero measure) interval $[0,x_2]$;

• (c) non-decreasing (or, respectively, decreasing) in $[0,\mathrm{\infty })$ and, furthermore, strictly increasing (or, respectively, strictly decreasing) in some real subinterval $[x_2,x_1)\subseteq [0,x_1)$ of nonzero measure.

The result also holds under Assumption  2.2 if $g^{\prime }:{\mathbf{R}}_{+}\to \mathbf{R}$ is continuous, nonzero, and has a constant sign in some real intervals $[x_2,x_1)\subseteq [0,x_1)$ and $(x_1,x_3)\subseteq [x_1,\mathrm{\infty })$ of nonzero measure while being identically zero in $[0,x_2)\cup [x_3,\mathrm{\infty })$.

Proof If $g(x_1)=0$ for some $x_1\in {\mathbf{R}}_{0+}$ and $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ is strictly monotone, then $g(x)\ne 0$; $\mathrm{\forall }x(\ne x_1)\in {\mathbf{R}}_{0+}$ so that there is no equilibrium point of (1.1) other than $x_1$ on ${\mathbf{R}}_{0+}$. The result has been proven under Condition (a). Now, assume that $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ is strictly increasing in $[x_1,x_3)\subseteq [x_1,\mathrm{\infty })$ and non-decreasing in $[x_1,\mathrm{\infty })$. Then, $g(x)\ge g(x_3)>g(y)>g(x_1)=0$; $\mathrm{\forall }x\in (x_3,\mathrm{\infty })$, $\mathrm{\forall }y\in (x_1,x_3)$. If $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ is strictly decreasing in $[x_1,x_3)\subseteq [x_1,\mathrm{\infty })$ and decreasing in $[x_1,\mathrm{\infty })$, then $0=g(x_1)>g(y)>g(x_3)\ge g(x)$; $\mathrm{\forall }y\in (x_1,x_3)$, $\mathrm{\forall }x\in (x_3,\mathrm{\infty })$. Thus, $x_1$ is the unique equilibrium point of (1.1) in $[x_1,\mathrm{\infty })$. The result under Condition (b) is already proven for $x_1=0$ but not yet for $x_1\in {\mathbf{R}}_{+}$. Thus, assume now that $x_1\in {\mathbf{R}}_{+}$, that the above conditions hold and that, in addition, $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ is either strictly decreasing (or, respectively, strictly increasing) in some interval $[x_2,x_1]\subset {\mathbf{R}}_{0+}$ of nonzero measure and, correspondingly, is either decreasing (or, respectively, non-decreasing) within an interval $[0,x_2]$ being of zero or nonzero measure. Then, there is no $x(\ne x_1)\in {\mathbf{R}}_{0+}$ being an equilibrium point of (1.1). Hence, the result fully follows under Condition (b). A particular case occurs when the function is continuously differentiable and either strictly increasing or strictly decreasing in each of the real intervals $[0,x_1)$ and $(x_1,\mathrm{\infty })$. Hence, the theorem. □

Theorem 2.4 Assume that $F:{\mathbf{R}}_{0+}\to \mathbf{R}$ satisfies $\{x\in {\mathbf{R}}_{+}:F(x)=1/k\}=\mathrm{\varnothing }$ and that Assumption  2.2 holds with all the derivatives referred to as being everywhere continuous in their definition domains. Define $m^{\prime }:{\mathbf{R}}_{0+}\to \mathbf{R}$ as

$$m^{\prime }(x)=(A{G}^{\prime }(x)+\frac{k{F}^{\prime }(x)}{1-kF(x)})M(\overline{x})-\frac{1+k{F}^{\prime }(x)}{1-kF(x)}{e}^{AG(\overline{x})};\mathrm{\forall }x\in {\mathbf{R}}_{0+}.$$

(2.3)

Then the following properties hold.

• (i) Assume that $F:{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ and $M,G:{\mathbf{R}}_{0+}^q\to {\mathbf{R}}_{0+}$ satisfy $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})\le 0$, and

  $$M^{\prime }(x)=m^{\prime }(x)+\epsilon (x);\mathrm{\forall }x\in {\mathbf{R}}_{0+}$$

  (2.4)

for some given everywhere continuous $\epsilon :{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ if $g(0)<0$, where

$$g^{\prime }(0)=\{\begin{array}{cc}-1\hfill & \mathit{\text{if}}(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})<0,\hfill \\ [M(\overline{0})A{G}^{\prime }(\overline{0})-M^{\prime }(\overline{0})](1-kF(0))e^{-AG(\overline{0})}\hfill \\ +k{F}^{\prime }(0)(M(\overline{0})e^{-AG(\overline{0})}-1)-1\hfill & \mathit{\text{if}}(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})=0.\hfill \end{array}$$

(2.5)

Then $x=0$ is the unique nonnegative equilibrium point of (1.1) in ${\mathbf{R}}_{0+}$.

• (ii) Property (i) also holds if $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})=0$ and if $M^{\prime }(x)=m^{\prime }(x)-\epsilon (x)$; $\mathrm{\forall }x\in {\mathbf{R}}_{0+}$ with $g^{\prime }(0)>0$, where

  $$g^{\prime }(0)=[M(\overline{0})A{G}^{\prime }(\overline{0})-M^{\prime }(\overline{0})](1-kF(0))e^{-AG(\overline{0})}+k{F}^{\prime }(0)(M(\overline{0})e^{-AG(\overline{0})}-1)-1$$

  (2.6)

and $\epsilon :{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ is everywhere continuous.

• (iii) Assume that $F:{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ and $M,G:{\mathbf{R}}_{0+}^q\to {\mathbf{R}}_{0+}$ satisfy $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})>0$. Then $\mathrm{\exists }{x}_1\in {\mathbf{R}}_{+}$, which satisfies $g(x_1)=0$, is a unique nonnegative equilibrium point of (1.1) if $M^{\prime }(x)=m^{\prime }(x)+\epsilon (x)$ subject to (2.5) with $\epsilon :{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ being everywhere continuous, (strictly) positive in $[0,x_1+{\epsilon }_0]$ for some sufficiently large${\epsilon }_0\in {\mathbf{R}}_{0+}$, and decreasing in ${\mathbf{R}}_{0+}$.

• (iv) Assume that $F:{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ and $M,G:{\mathbf{R}}_{0+}^q\to {\mathbf{R}}_{0+}$ satisfy $g(0)=(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})>0$. Then, $\mathrm{\neg }\mathrm{\exists }{x}_1\in {\mathbf{R}}_{0+}$ is a nonnegative equilibrium point of (1.1) if $M^{\prime }(x)=m^{\prime }(x)-\epsilon (x)$ with $\epsilon :{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ is everywhere continuous.

Proof Property (i) is first proven. Direct calculation from (2.2) under Assumption 2.2 yields

$$\begin{array}{rcl}{g}^{\prime }(x)& =& (1-kF(x))(A{G}^{\prime }(\overline{x})M^{\prime }(\overline{x})-M(\overline{x}))e^{-AG(\overline{x})}-k{F}^{\prime }(x)(1-M(\overline{x})e^{-AG(\overline{x})})-1\\ =& [M(\overline{x})(A{G}^{\prime }(\overline{x})(1-kF(x))+k{F}^{\prime }(x))-M^{\prime }(\overline{x})(1-kF(x))]e^{-AG(\overline{x})}-k{F}^{\prime }(x)-1\\ =& [M(\overline{x})A{G}^{\prime }(\overline{x})-M^{\prime }(\overline{x})](1-kF(x))e^{-AG(\overline{x})}+k{F}^{\prime }(x)(M(\overline{x})e^{-AG(\overline{x})}-1)-1\end{array}$$

(2.7)

for any $x\in {\mathbf{R}}_{0+}$ such that $(1-kF(x))(1-M(\overline{x})e^{-AG(\overline{x})})\ge 0$, and $g^{\prime }(x)=-1$ for any $x\in {\mathbf{R}}_{0+}$ such that $(1-kF(x))(1-M(\overline{x})e^{-AG(\overline{x})})\le 0$. It is assumed that $g^{\prime }(0)=a-1<g(0)=0$, obtained from (2.5), where $a=a(0)$ is defined by

$$a=\{\begin{array}{cc}0\hfill & \text{if }(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})<0,\hfill \\ [M(\overline{0})A{G}^{\prime }(\overline{0})-M^{\prime }(\overline{0})](1-kF(0))e^{-AG(\overline{0})}\hfill \\ +k{F}^{\prime }(0)(M(\overline{0})e^{-AG(\overline{0})}-1)\hfill & \text{otherwise}.\hfill \end{array}$$

(2.8)

Define $m^{\prime }:{\mathbf{R}}_{0+}\to \mathbf{R}$ as

$$m^{\prime }(x)=(A{G}^{\prime }(x)+\frac{k{F}^{\prime }(x)}{1-kF(x)})M(\overline{x})-\frac{1+k{F}^{\prime }(x)}{1-kF(x)}{e}^{AG(\overline{x})};\mathrm{\forall }x\in {\mathbf{R}}_{0+},$$

(2.9)

which is everywhere continuous in ${\mathbf{R}}_{+}$ since the subset of its second-class discontinuity points is empty by hypothesis. Then, from (2.4), one has for any $x\in {\mathbf{R}}_{+}$ such that $F(x)\ne 1/k$, then for any $x\in {\mathbf{R}}_{+}$ since $\{x\in {\mathbf{R}}_{+}:F(x)=1/k\}=\mathrm{\varnothing }$, that

$$\begin{array}{r}{g}^{\prime }(x)\le 0\iff M^{\prime }(x)\ge m^{\prime }(x);g^{\prime }(x)<0\iff M^{\prime }(x)>m^{\prime }(x);\\ g^{\prime }(x)=0\iff M^{\prime }(x)=m^{\prime }(x);\mathrm{\forall }x\in {\mathbf{R}}_{+},\end{array}$$

(2.10a)

$$\begin{array}{r}{g}^{\prime }(x)\ge 0\iff M^{\prime }(x)\le m^{\prime }(x);\\ g^{\prime }(x)>0\iff M^{\prime }(x)<m^{\prime }(x);\mathrm{\forall }x\in {\mathbf{R}}_{+}.\end{array}$$

(2.10b)

Now, $x=0$ is an equilibrium point of (1.1), equivalently, $g(0)=0$, with $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})\le 0$ and $g^{\prime }(0)=a-1<0$ with $a=0$ if $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})<0$ and $a=a(0)=(M(\overline{0})A{G}^{\prime }(\overline{0})-M^{\prime }(\overline{0}))(1-kF(0))e^{-AG(\overline{0})}+k{F}^{\prime }(0)(M(\overline{0})e^{-AG(\overline{0})}-1)<1$ if $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})=0$. In both cases, $g^{\prime }(0)<0$ and, since $g^{\prime }:{\mathbf{R}}_{0+}\to \mathbf{R}$ is continuous from (2.3)-(2.4), since $\{x\in {\mathbf{R}}_{+}:F(x)=1/k\}=\mathrm{\varnothing }$ implies that $m^{\prime }:{\mathbf{R}}_{0+}\to \mathbf{R}$ is continuous, there is some $x_1\in {\mathbf{R}}_{+}$ such that $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ is strictly decreasing on $[0,x_1)$ from (2.4)-(2.5). Thus, $0\ne g(x)<g(0)=0$; $\mathrm{\forall }x\in [0,x_1)$ and $g(x)\le g(x_1)<g(0)=0$; $\mathrm{\forall }x\in {\mathbf{R}}_{+}$ since $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ is decreasing in ${\mathbf{R}}_{0+}$ if (2.10a) holds. Hence, Property (i). Property (ii) is a dual property of (i) with $g^{\prime }(0)=a-1>g(0)=0$ and $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ being non-decreasing in ${\mathbf{R}}_{0+}$ under (2.10b).

Property (iii) holds since $g(0)=(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})>0$ (so that $x=0$ is not an equilibrium point of (1.1)) with $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ being strictly decreasing in $[0,x_1+{\epsilon }_0]$ and decreasing on $[0,\mathrm{\infty })$ from (2.10a) for some $x_1\in {\mathbf{R}}_{+}$ (since $\epsilon :{\mathbf{R}}_{0+}\mathrm{\setminus }[0,x_1+{\epsilon }_0]\to {\mathbf{R}}_{+}$) which exists so that $g(x)\le g(x_1+\epsilon )<g(x_1)=0<g(y)$; $\mathrm{\forall }x(>x_1),y(<x_1)\in {\mathbf{R}}_{+}$. Then $x_1$ is the unique equilibrium point of (1.1) in ${\mathbf{R}}_{0+}$ if ${\epsilon }_0$ is large enough. Property (iv) holds since $g(0)>0$ (so that $x=0$ is not an equilibrium point of (1.1)) with $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ being non-decreasing in $[0,\mathrm{\infty })$ from (2.10b). Then there is no equilibrium point of (1.1) in ${\mathbf{R}}_{0+}$. □

The existence of the zero equilibrium and another positive equilibrium point of (1.1) can be given by a direct extension of Theorem 2.4(i)-(ii) as follows.

Theorem 2.5 Assume that $F:{\mathbf{R}}_{0+}\to \mathbf{R}$ satisfies $\{x\in {\mathbf{R}}_{+}:F(x)=1/k\}=\mathrm{\varnothing }$ and that Assumption  2.2 holds with all the derivatives referred to as being everywhere continuous in their definition domains. Then the following properties hold.

• (i) Assume, in addition, that $F:{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ and $M,G:{\mathbf{R}}_{0+}^q\to {\mathbf{R}}_{0+}$ satisfy $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})\le 0$, $M^{\prime }(x)=m^{\prime }(x)+\epsilon (x)$; $\mathrm{\forall }x\in [0,x_1)$ and $M^{\prime }(x)=m^{\prime }(x)-\epsilon (x)$; $\mathrm{\forall }x\in [x_1,\mathrm{\infty })$ for some $x_1\in {\mathbf{R}}_{+}$, any continuous $\epsilon :{\mathbf{R}}_{0+}\to {\mathbf{R}}_{+}$ with $g^{\prime }(0)<0$ satisfying (2.5) (i.e. $g^{\prime }(0)=a-1<0$ if $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})=0$ and $g^{\prime }(0)=-1<0$ if $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})<0$). Then, $x=0$ and $x=x_2>x_1$ for some $x_2\in {\mathbf{R}}_{+}$ are the only two nonnegative equilibrium points of (1.1) in ${\mathbf{R}}_{0+}$.

• (ii) Property (i) also holds if $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})=0$ and if $M^{\prime }(x)=m^{\prime }(x)-\epsilon (x)$; $\mathrm{\forall }x\in [0,x_1)$ and $M^{\prime }(x)=m^{\prime }(x)+\epsilon (x)$; $\mathrm{\forall }x\in [x_1,\mathrm{\infty })$ for some $x_1\in {\mathbf{R}}_{+}$ and $g^{\prime }(0)>0$, according to (2.6), for any continuous $\epsilon :{\mathbf{R}}_{0+}\to {\mathbf{R}}_{+}$.

Outline of proof The proof of Property (i) follows with $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ being strictly decreasing on some interval $[0,x_1)$ with $g^{\prime }(0)<0$ and strictly increasing on $[x_1,\mathrm{\infty })$ since $g^{\prime }(0)=-1<0$ if $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})<0$ and $g^{\prime }(0)=a-1<0$ if $g(0)=(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})=0$. Then there is $x_2(>x_1)\in {\mathbf{R}}_{+}$ such that $0=g(0)>g(x)>g(x_1)<g(y)<g(x_2)=0<g(z)$; $\mathrm{\forall }x\in (0,x_1)$, $x\in (x_1,x_2)$, $\mathrm{\forall }z\in (x_2,\mathrm{\infty })$. Then there is no $x\in {\mathbf{R}}_{0+}$ other than $x=0$ and $x=x_2$ such that $g(x)=0$. Hence, Property (i). The proof of Property (ii) is similar by noting that $0=g(0)<g(x)<g(x_1)>g(y)>g(x_2)=0>g(z)$ with $g(x)$ being strictly increasing in $[0,x_1)$ and strictly decreasing in $(x_1,\mathrm{\infty })$. Hence, the result. □

Note that the conditions of Theorem 2.5 can be relaxed in the sense that it is not necessary for $g:{\mathbf{R}}_{0+}\to \mathbf{R}$ to be strictly monotone in $(x_2,\mathrm{\infty })$ but non-decreasing for Part (i) and decreasing for Part (ii).

Examples 2.6 (1) If we consider the particular case $p=1$, $x=\overline{x}$, $F(x)=G(x)=x$, then $F(0)=G(0)=0$, $M(x)=M(0)=1$ so that $F^{\prime }(x)=G^{\prime }(x)=F^{\prime }(0)=G^{\prime }(0)=1$, $M^{\prime }(x)=M^{\prime }(0)=0$, one gets from Theorem 2.4(i) that the condition $M^{\prime }(0)=0\ge A-1$, equivalently, $0<A\le 1$, leads to $x=0$ being an equilibrium point of (1.1). Also, one has again from Theorem 2.4(i) that $g(0)=0$ and the following condition holds for $x\in [0,1/k)$ and $0<A\le 1$:

$$0=M^{\prime }(x)\ge m^{\prime }(x)=A-\frac{1}{1-kx}(e^{Ax}+k(e^{Ax}-1)).$$

Then $x=0$ is the only equilibrium point of (1.1) if $0<A\le 1$ for any $k\in \mathbf{N}$, since $g:{\mathbf{R}}_{0+}\to {\mathbf{R}}_{+}$ is decreasing for $x\in [0,1/k)$ from the above inequality and $g^{\prime }(0)<0$ implies that $\mathrm{\neg }\mathrm{\exists }{x}_1\in {\mathbf{R}}_{+}$ such that $g(x_1)=0$. This coincides with former results obtained in [1, 2] for $k=1$.

• (2) Consider the equilibrium equation $x=(1+kx)(e^{Ax}+1)$ of $x_{n+1}=(1+{\sum }_{j=0}^{k-1}{x}_{n-j})(1+e^{A{x}_n})$ with $min(x_{1-i}:i\in \overline{p})\ge 0$. Thus, $F(x)=G(x)=-x$, $F^{\prime }(x)=G^{\prime }(x)=-1$, $M(x)=-1$, $M^{\prime }(x)=0$, $g(0)=2$, $g^{\prime }(0)=A+2k-1>0$, and $g:{\mathbf{R}}_{0+}\to {\mathbf{R}}_{+}$ is not decreasing for $A\ge 0$ since

  $$0=M^{\prime }(x)\ge m^{\prime }(x)=A+\frac{1}{1+kx}(k(e^{-Ax}+1)-e^{-Ax})$$

cannot hold for $x\in {\mathbf{R}}_{0+}$. Thus, $\mathrm{\neg }\mathrm{\exists }x\in {\mathbf{R}}_{0+}$ is an equilibrium point for any $A\ge 0$.

Theorems 2.3, 2.4 and 2.5 may be extended by removing the condition $\{x\in {\mathbf{R}}_{+}:F(x)=1/k\}=\mathrm{\varnothing }$ and the continuity of the derivatives of the vector functions in Assumption 2.2 by allowing such derivatives to be impulsive at the points of $x\in {\mathbf{R}}_{+}$ such that $F(x)=1/k$, if any. The sign of eventual impulses should be such that they do not change the needed non-decreasing, decreasing or strictly monotone properties of $g:{\mathbf{R}}_{0+}\to \mathbf{R}$. We denote in the following by $f(x^{-})$ the left limits and by $f(x)$ the right limits of functions at points of the functions $f:{\mathbf{R}}_{0+}\to \mathbf{R}$ which are distinct if such functions are discontinuous at x. Such an extension is as follows.

Theorem 2.7 Let Assumption  2.2 hold with $F:{\mathbf{R}}_{0+}\to \mathbf{R}$ being continuous on ${\mathbf{R}}_{0+}\mathrm{\setminus }{S}_{\mathrm{imp}}$ and impulsive on $S_{\mathrm{imp}}$, and $M^{\prime },G^{\prime }:{\mathbf{R}}_{0+}^p\to \mathbf{R}$ being continuous on ${\mathbf{R}}_{0+}^p\mathrm{\setminus }{\overline{S}}_{\mathrm{imp}}$ and impulsive on ${\overline{S}}_{\mathrm{imp}}$, where $S_{\mathrm{imp}}=\{x\in {\mathbf{R}}_{+}:F(x)=1/k\}$ and ${\overline{S}}_{\mathrm{imp}}=\{\overline{x}=(x,x,\dots ,x)\in {\mathbf{R}}_{0+}^p:x\in S_{\mathrm{imp}}\}$ which can be empty or nonempty (note that $\overline{x}\in {\overline{S}}_{\mathrm{imp}}$ if and only if $x\in S_{\mathrm{imp}}$). Let the function $m^{\prime }:{\mathbf{R}}_{0+}\to \mathbf{R}$ be defined in (2.3). Then the following properties hold.

• (i) Assume that $F:{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ and $M,G:{\mathbf{R}}_{0+}^q\to {\mathbf{R}}_{0+}$ satisfy $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})\le 0$, and $M^{\prime }(x)=m^{\prime }(x)+\epsilon (x)$; $\mathrm{\forall }x\in {\mathbf{R}}_{0+}$ for some continuous $\epsilon :{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ being everywhere continuous and satisfying (2.5). Then $x=0$ is the unique nonnegative equilibrium point of (1.1) in ${\mathbf{R}}_{0+}$ provided that either $F^{\prime }(x)M(\overline{x})-(1+k{F}^{\prime }(x))e^{AG(\overline{x})}=0$ or $sgn(F^{\prime }(x)M(\overline{x})-(1+k{F}^{\prime }(x))e^{AG(\overline{x})})<0$; $\mathrm{\forall }x\in S_{\mathrm{imp}}$.

• (ii) Property (i) also holds if $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})=0$ and if $M^{\prime }(x)=m^{\prime }(x)-\epsilon (x)$; $\mathrm{\forall }x\in {\mathbf{R}}_{0+}$ with $\epsilon :{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ being everywhere continuous and satisfying (2.6) and $g^{\prime }(0)>0$ provided that either $F^{\prime }(x)M(\overline{x})-(1+k{F}^{\prime }(x))e^{AG(\overline{x})}=0$ or $sgn(F^{\prime }(x)M(\overline{x})-(1+k{F}^{\prime }(x))e^{AG(\overline{x})})>0$; $\mathrm{\forall }x\in S_{\mathrm{imp}}$.

• (iii) Assume that $F:{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ and $M,G:{\mathbf{R}}_{0+}^q\to {\mathbf{R}}_{0+}$ satisfy $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})>0$. Then, $\mathrm{\exists }{x}_1\in {\mathbf{R}}_{+}$, which satisfies $g(x_1)=0$, is then a unique nonnegative equilibrium point of (1.1) if $M^{\prime }(x)=m^{\prime }(x)+\epsilon (x)$, with $\epsilon :{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ being everywhere continuous and strictly decreasing in $[0,x_1]$ and decreasing in $(x_1,\mathrm{\infty })$, provided that $g^{\prime }(0)$ fulfils (2.5), $F^{\prime }(x)M(\overline{x})-(1+k{F}^{\prime }(x))e^{AG(\overline{x})}=0$, or $sgn(F^{\prime }(x)M(\overline{x})-(1+k{F}^{\prime }(x))e^{AG(\overline{x})})<0$ with $\frac{F^{\prime }(x)M(\overline{x})-(1+k{F}^{\prime }(x))e^{AG(\overline{x})}}{1-KF(x)}\le -g(x^{-})$; $\mathrm{\forall }x\in S_{\mathrm{imp}}$. If $\frac{F^{\prime }(x)M(\overline{x})-(1+k{F}^{\prime }(x))e^{AG(\overline{x})}}{1-KF(x)}=-g(x^{-})$, then $x_1=x$.

• (iv) Assume that $F:{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ and $M,G:{\mathbf{R}}_{0+}^q\to {\mathbf{R}}_{0+}$ satisfy $(1-kF(0))(1-M(\overline{0})e^{-AG(\overline{0})})>0$. Then $\mathrm{\neg }\mathrm{\exists }{x}_1\in {\mathbf{R}}_{0+}$ is a nonnegative equilibrium point of (1.1) if $M^{\prime }(x)=m^{\prime }(x)-\epsilon (x)$ with $\epsilon :{\mathbf{R}}_{0+}\to {\mathbf{R}}_{0+}$ being everywhere continuous, and $g^{\prime }(0)$ subject to (2.6), provided that either $F^{\prime }(x)M(\overline{x})-(1+k{F}^{\prime }(x))e^{AG(\overline{x})}=0$ or $sgn(F^{\prime }(x)M(\overline{x})-{(1+k{F}^{\prime }(x)e)}^{AG(\overline{x})})>0$; $\mathrm{\forall }x\in S_{\mathrm{imp}}$.

Outline of proof Note that the various extended conditions in Theorem 2.7(i)-(iv) with respect to those of Theorem 2.4 imply that $g(x)\le g(x^{-})$ (respectively, $g(x)\ge g(x^{-})$) if $x\in S_{\mathrm{imp}}$ and $g:(x-\epsilon ,x)\to \mathbf{R}$ is decreasing (respectively, non-decreasing) for some $\epsilon \in {\mathbf{R}}_{+}$ since the constraints $M^{\prime }(x)=m^{\prime }(x)\pm \epsilon (x)$ of Theorem 2.4 also hold at the discontinuities of $m^{\prime }(x)$ at $x=-1/k$. □

3 Stability results

A result on boundedness of the solutions of (1.1) and then its stability under certain parametrical constraints is now given as follows.

Theorem 3.1 Assume the following:

• (1) The real sequences $\{F_n\}$ and $\{M_n\}$ of respective general terms $F_n=F(x_n)$ and $M_n=M({\overline{x}}_n)$ satisfy the constraints ${\nu }_n{x}_n\le F_n\le {\mu }_n{x}_n$ and ${\delta }_n{max}_{n-p+1\le i\le n}{x}_i\le M_n\le {\omega }_n{max}_{n-p+1\le i\le n}{x}_i$ for some nonnegative real sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$, $\{{\delta }_n\}$ and $\{{\omega }_n\}$ with ${\mu }_n\ge {\nu }_n$ and ${\omega }_n\ge {\delta }_n$; $\mathrm{\forall }n\in {\mathbf{N}}_0$.

• (2) There are $a,b(\ge a)\in {\mathbf{R}}_{0+}$ such that $b\ge x_i\ge a$; $i=1-p,2-p,\dots ,0$.

• (3) $G:{\mathbf{R}}_{0+}^q\to \mathbf{R}$, which generates the real sequence $\{G_n\}$ of general terms $G_n=G(x_n)$ satisfies the following constraints on ${\mathbf{R}}_{0+}$:

  $$a\le G_n(x)\le b\mathit{\text{if}}A\ge 0;a\ge G_n(x)\ge b\mathit{\text{if}}A<0.$$

Then the following properties hold:

• (i) $b\ge x_i\ge a$; $\mathrm{\forall }i\in \{1-p,2-p,\dots ,0\}\cup \mathbf{N}$ if the sequences $\{{\nu }_n\}$, $\{{\mu }_n\}$ satisfy

  $$\begin{array}{l}\frac{e^{Ab}}{a^2({\sum }_{j=0}^{k-1}{\nu }_{n-j})}[a+b({\omega }_n{e}^{-Aa}+\sum _{j=0}^{k-1}{\mu }_{n-j})-1]\\ \le {\delta }_n\le {\omega }_n\le \frac{e^{Aa}}{b^2({\sum }_{j=0}^{k-1}{\mu }_{n-j})}[b+a({\delta }_n{e}^{-Ab}+\sum _{j=0}^{k-1}{\nu }_{n-j})-1];\end{array}$$

  (3.1)

$\mathrm{\forall }n\in {\mathbf{N}}_0$ for given a, b, $\{{\nu }_n\}$ and $\{{\mu }_n\}$.

• (ii) A sufficient condition for (3.1) to hold is

  $$\begin{array}{c}{(1-\frac{1}{ab({\sum }_{j=0}^{k-1}{\nu }_{n-j})({\sum }_{j=0}^{k-1}{\mu }_{n-j})})}^{-1}\frac{e^{Ab}}{a^2({\sum }_{j=0}^{k-1}{\nu }_{n-j})}\hfill \\ \times [a-1+b\left(\sum _{j=0}^{k-1}{\mu }_{n-j}\right)+\frac{1}{{\sum }_{j=0}^{k-1}{\nu }_{n-j}}+\frac{b({\sum }_{j=0}^{k-1}{\mu }_{n-j}-1)}{a({\sum }_{j=0}^{k-1}{\nu }_{n-j})}]\hfill \\ \le {\delta }_n\le {\omega }_n\hfill \\ \le {(1-\frac{1}{ab({\sum }_{j=0}^{k-1}{\nu }_{n-j})({\sum }_{j=0}^{k-1}{\mu }_{n-j})})}^{-1}\frac{e^{Aa}}{b^2({\sum }_{j=0}^{k-1}{\mu }_{n-j})}\hfill \\ \times [b-1+a\left(\sum _{j=0}^{k-1}{\nu }_{n-j}\right)+\frac{1}{{\sum }_{j=0}^{k-1}{\mu }_{n-j}}+\frac{a({\sum }_{j=0}^{k-1}{\nu }_{n-j}-1)}{b({\sum }_{j=0}^{k-1}{\mu }_{n-j})}];\hfill \end{array}$$

$\mathrm{\forall }n\in {\mathbf{N}}_0$ provided that $({\sum }_{j=0}^{k-1}{\nu }_{n-j})({\sum }_{j=0}^{k-1}{\mu }_{n-j})\ne \frac{1}{ab}$.

Proof Note that $1-{\omega }_n{\overline{x}}_n{e}^{-Aa}\le 1-M_n{e}^{-A{G}_n}\le 1-{\delta }_n{\overline{x}}_n{e}^{-Ab}$; $\mathrm{\forall }n\in {\mathbf{N}}_0$ since

$$\begin{array}{c}[(a\le G_n(x)\le b\text{ if }A\ge 0)\wedge (a\ge G_n(x)\ge b\text{ if }A<0)]\hfill \\ \Rightarrow [e^{Aa}\le e^{A{G}_n}\le e^{Ab}];\mathrm{\forall }n\in {\mathbf{N}}_0.\hfill \end{array}$$

Assume that $0\le a\le x_n\le b<\mathrm{\infty }$ for some $a,b(\ge a)\in {\mathbf{R}}_{+}$ and $p(\ge n)\in {\mathbf{N}}_0$. Proceed now by complete induction by assuming that if there is some $n(\ge p)\in \mathbf{N}$ such that $b\ge x_i\ge a$; $\mathrm{\forall }i\in \overline{n}$, then $b\ge x_{n+1}\ge a$. If $x_{n+1}=0$, this holds trivially for any $a,b(\ge a)\in {\mathbf{R}}_{0+}$. Note that for $i\in \overline{n}$,

$${\nu }_{i}a\le F_i\le {\mu }_{i}b;e^{Aa}\le e^{A{G}_i}\le e^{Ab},b{\omega }_i{e}^{-Aa}\ge M_i{e}^{-A{G}_i}\ge a{\delta }_i{e}^{-Ab},$$

(3.2)

$$1-b{\omega }_i{e}^{-Aa}\le 1-{\omega }_i{\overline{x}}_i{e}^{-Aa}\le 1-M_i{e}^{-A{G}_i}\le 1-{\delta }_i{\overline{x}}_i{e}^{-Ab}\le 1-a{\delta }_i{e}^{-Ab}.$$

(3.3)

If $x_{n+1}\ne 0$ then it satisfies from (1.1) by using (3.2)-(3.3) that

$$\begin{array}{rcl}a& \le & a^2{\delta }_n\left(\sum _{j=0}^{k-1}{\nu }_{n-j}\right)e^{-Ab}-b({\omega }_n{e}^{-Aa}+\sum _{j=0}^{k-1}{\mu }_{n-j})+1\\ \le & x_{n+1}=(1-\sum _{j=0}^{k-1}{F}_{n-j})(1-M_n{e}^{-A{G}_n})\\ =& -F_n+F_n{M}_n{e}^{-A{G}_n}+\left(\sum _{j=1}^{k-1}{F}_{n-j}\right)M_n{e}^{-A{G}_n}-\left(\sum _{j=1}^{k-1}{F}_{n-j}\right)-M_n{e}^{-A{G}_n}+1\\ \le & b^2{\omega }_n\left(\sum _{j=0}^{k-1}{\mu }_{n-j}\right)e^{-Aa}-a({\delta }_n{e}^{-Ab}+\sum _{j=0}^{k-1}{\nu }_{n-j})+1\le b;\end{array}$$

(3.4)

$\mathrm{\forall }n\in {\mathbf{N}}_0$ provided that (3.1) holds. Thus, if $b\ge x_i\ge a$; $i=1-p,2-p,\dots ,0$, then $b\ge x_i\ge a$; $\mathrm{\forall }i\in \{1-p,2-p,\dots ,0\}\cup \mathbf{N}$. Hence, Property (i).

Property (ii) follows by replacing the lower-bound of ${\delta }_n$ of (3.1) in the upper-bound of ${\omega }_n$ to get a more stringent upper-bound ${\overline{\omega }}_n$ of ${\omega }_n$, which does not depend on ${\delta }_n$ since

$$\begin{array}{rcl}{\omega }_n& \le & {\overline{\omega }}_n\\ =& \frac{e^{Aa}}{b^2({\sum }_{j=0}^{k-1}{\mu }_{n-j})}[b-1+a\left(\sum _{j=0}^{k-1}{\nu }_{n-j}\right)\\ +\frac{1}{a({\sum }_{j=0}^{k-1}{\nu }_{n-j})}(a+b{\omega }_n{e}^{-Aa}+b\sum _{j=0}^{k-1}{\mu }_{n-j}-1)];\mathrm{\forall }n\in {\mathbf{N}}_0.\end{array}$$

In a close way, we can get a more stringent lower-bound ${\underline{\delta }}_n$ of ${\delta }_n$, which does not depend on ${\omega }_n$ since

$$\begin{array}{rcl}{\delta }_n& \ge & {\underline{\delta }}_n\\ =& \frac{e^{Ab}}{a^2({\sum }_{j=0}^{k-1}{\nu }_{n-j})}[a-1+b\left(\sum _{j=0}^{k-1}{\mu }_{n-j}\right)\\ +\frac{1}{b({\sum }_{j=0}^{k-1}{\mu }_{n-j})}(b+a{\delta }_n{e}^{-Ab}+a\sum _{j=0}^{k-1}{\nu }_{n-j}-1)];\mathrm{\forall }n\in {\mathbf{N}}_0.\end{array}$$

 □

Remark 3.2 (Brief historical note)

It can be pointed out that assumptions of the type in the assumption in Theorem 3.1 are very relevant to some classical control problems for both continuous-time or discrete-time descriptions of dynamic systems, like, for instance, those of absolute stability in the Lure and Popov (following Vasile Mihai Popov - Galati, Romania, 1928) senses or Popovian hyperstability [22]. Basically, if there are uncertainties in parameterization, which is a very common drawback from fabrication dispersion of components for devices constructed for applications, a robust regulator or controller, in general, has to be able to stabilise all the particular elements of the whole series within some error margin, not just the theoretically nominal one. The lack in appropriately formulating that problem implied during Second World War II a lack of well-regulated equilibrium positioning of guns of some Soviet military tanks with the associate lack of effectiveness in military operations. This was the initial point of the theory of Popov’s absolute stability later on being generalised to hyperstability after including the phenomena of existence of unmodelled dynamics. This was apparently one of the reasons for Popov’s decision of to investigate the simultaneous stabilization of devices of the same family subject to parametrical dispersion of components related to a theoretical nominal ideal device (source: old private communication by PhD supervisor ID Landau, a former Popov’s collaborator and later on a relevant researcher in the field, to the first author of this paper). See also [23] for relevant content honouring Popov’s work. In the context of this paper, we can attribute the unmodelled or parametrical errors to a non-exact parameterization of the epidemic model for each possible situation.

A set of stability and instability properties, implying, furthermore, that any nontrivial solution of (1.1) is strictly monotone for $n\in {\mathbf{N}}_0$, are presented in the next two results. Some of the properties depend on parametrical conditions of lower- and upper-bounding sequences of $\{F_n\}$.

Theorem 3.3 Assume that the real sequence $\{F_n\}$ satisfies the constraints ${\sum }_{i=0}^m{\nu }_{ni}{x}_n^i\le F_n\le {\sum }_{i=0}^m{\mu }_{ni}{x}_n^i$ for some real sequences $\{{\nu }_{ni}\}$, $\{{\mu }_{ni}\}$ with ${\mu }_{ni}\ge {\nu }_{ni}$; $i\in \overline{m}\cup \{0\}$, $\mathrm{\forall }n\in {\mathbf{N}}_0$. The following properties hold.

• (i) Any nontrivial nonnegative solution $\{x_n\}$ of (1.1) is uniformly bounded and strictly decreasing for $n\in {\mathbf{N}}_0$ and initial conditions $1\ge min(1,{min}_{n\in {\mathbf{N}}_{0+}}(\frac{1-{\sum }_{j=n-k+1}^n{\mu }_{j0}}{{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\mu }_{ji}}))\ge x_{-p+1}\ge \cdots \ge x_{-1}\ge x_0$, and then it converges to the zero equilibrium point under the following condition:

  $$\begin{array}{l}([(\sum _{j=n-k+1}^n{\mu }_{j0}<1)\wedge (\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}\ge 0)]\\ \vee [(\sum _{j=n-k+1}^n{\mu }_{j0}>1)\wedge (\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}\le 0)])\\ \wedge (M_n\ge e^{A{G}_n});\mathrm{\forall }n\in {\mathbf{N}}_0.\end{array}$$

  (3.5)

• (ii) Any nontrivial nonnegative solution $\{x_n\}$ of (1.1) is uniformly bounded and strictly decreasing for $n\in {\mathbf{N}}_0$ and initial conditions $min(1,{min}_{n\in {\mathbf{N}}_{0+}}(\frac{{\sum }_{j=n-k+1}^n{\nu }_{j0}-1}{|{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\nu }_{ji}|}))>x_{-p+1}\ge \cdots \ge x_{-1}\ge x_0$, and then it converges to the zero equilibrium point under the following condition:

  $$[(\sum _{j=n-k+1}^n{\nu }_{j0}<1)\wedge ({\nu }_{ji}\le 0;\mathrm{\forall }j\in {\mathbf{N}}_{0+},\mathrm{\forall }i\in \overline{m})]\wedge (M_n\le e^{A{G}_n});\mathrm{\forall }n\in {\mathbf{N}}_0.$$

  (3.6)

Proof Define ${\lambda }_n=1-M_n{e}^{-A{G}_n}$. Thus, one gets

$$x_{n+1}-x_n=(1-\sum _{j=n-k+1}^n{F}_j){\lambda }_n-x_n<0;\mathrm{\forall }n\in {\mathbf{N}}_0$$

(3.7)

from (1.1) if $x_n>0$, ${\sum }_{j=n-k+1}^n{F}_j\le 1$ and ${\lambda }_n\le 0$, which is guaranteed with $M_n\ge e^{A{G}_n}$; $\mathrm{\forall }n\in {\mathbf{N}}_0$, if for any $n\in {\mathbf{N}}_0$, the constraints $0<x_n<\cdots <x_0\le x_{-1}\le \cdots \le x_{-p+1}\le min(1,{min}_{n\in {\mathbf{N}}_{0+}}(\frac{1-{\sum }_{j=n-k+1}^n{\mu }_{j0}}{{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\mu }_{ji}}))$ hold for any given $n\in {\mathbf{N}}_{0+}$, together with

$$\begin{array}{rcl}\sum _{j=n-k+1}^n{F}_j& \le & \sum _{j=n-k+1}^n{\mu }_{j0}+\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}{x}_j^i\le \sum _{j=n-k+1}^n{\mu }_{j0}+\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}max(x_j,x_j^m)\\ \le & \sum _{j=n-k+1}^n{\mu }_{j0}+\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}{x}_j\le \sum _{j=n-k+1}^n{\mu }_{j0}+\left(\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}\right)x_{-p+1}\\ \le & 1\end{array}$$

(3.8a)

so that

$$\begin{array}{rcl}0& <& x_{n+1}<x_n<\cdots <x_0\le x_{-p+1}\\ \le & min(1,\underset{n\in {\mathbf{N}}_{0+}}{min}\left(\frac{1-{\sum }_{j=n-k+1}^n{\mu }_{j0}}{{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\mu }_{ji}}\right));\mathrm{\forall }n\in {\mathbf{N}}_0\end{array}$$

(3.8b)

provided that $min(1,{min}_{n\in {\mathbf{N}}_{0+}}(\frac{1-{\sum }_{j=n-k+1}^n{\mu }_{j0}}{{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\mu }_{ji}}))\ge x_{-p+1}\ge \cdots \ge x_{-1}\ge x_0$. It has been proven by complete induction that for any given $n\in {\mathbf{N}}_0$,

$$\begin{array}{l}0<x_n<\cdots <x_0\le x_{-1}\le \cdots \le x_{-p+1}\le min(1,\underset{n\in {\mathbf{N}}_{0+}}{min}\left(\frac{1-{\sum }_{j=n-k+1}^n{\mu }_{j0}}{{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\mu }_{ji}}\right))\le 1\\ \Rightarrow 0<x_{n+1}<x_n<\cdots <x_0\le x_{-p+1}\\ \phantom{\Rightarrow 0}\le min(1,\underset{n\in {\mathbf{N}}_{0+}}{min}\left(\frac{1-{\sum }_{j=n-k+1}^n{\mu }_{j0}}{{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\mu }_{ji}}\right))\le 1.\end{array}$$

(3.9)

A close result also holds involving non-strict inequalities if $x_n=0$, ${\sum }_{j=n-k+1}^n{F}_j\le 1$ and ${\lambda }_n\le 0$ or if $x_n\ge 0$, ${\sum }_{j=n-k+1}^n{F}_j>1$ and ${\lambda }_n\ge 0$, then $x_j=0$; $\mathrm{\forall }j(\ge n+1)\in {\mathbf{N}}_0$ from (1.1). Then (3.5) guarantees the convergence to zero of the sequence $\{x_n\}$; $n\in {\mathbf{N}}_0$ which is, furthermore, strictly decreasing. Hence, Property (i).

To prove Property (ii), note that (3.7) also holds from (1.1) for $x_n>0$ if ${\sum }_{j=n-k+1}^n{F}_j\ge 1$ and ${\lambda }_n\ge 0$, which is guaranteed with $M_n\le e^{A{G}_n}$; $\mathrm{\forall }n\in {\mathbf{N}}_{0+}$, if ${\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\nu }_{ji}\le 0$, and then the complete induction method gives

$$\begin{array}{c}min(1,\underset{n\in {\mathbf{N}}_{0+}}{min}\left(\frac{{\sum }_{j=n-k+1}^n{\nu }_{j0}-1}{|{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\nu }_{ji}|}\right))\hfill \\ >x_{-p+1}\ge \cdots \ge x_{-1}\ge x_0>x_n\text{for any given }n\in {\mathbf{N}}_0,\hfill \end{array}$$

together with

$$\begin{array}{l}\sum _{j=n-k+1}^n{F}_j\ge \sum _{j=n-k+1}^n{\nu }_{j0}-\left|\sum _{j=n-k+1}^n\sum _{i=1}^m{\nu }_{ji}\right|max(x_j,x_j^m)\\ \phantom{\sum _{j=n-k+1}^n{F}_j}\ge \sum _{j=n-k+1}^n{\nu }_{j0}-\left|\sum _{j=n-k+1}^n\sum _{i=1}^m{\nu }_{ji}\right|x_{-p+1}\ge 1\\ \Rightarrow min(1,\underset{n\in {\mathbf{N}}_{0+}}{min}\left(\frac{{\sum }_{j=n-k+1}^n{\nu }_{j0}-1}{|{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\nu }_{ji}|}\right))\\ \phantom{\Rightarrow }>x_{-p+1}\ge \cdots \ge x_{-1}\ge x_0>x_n>x_{n+1};\mathrm{\forall }n\in {\mathbf{N}}_0\end{array}$$

(3.10)

provided that $min(1,{min}_{n\in {\mathbf{N}}_{0+}}(\frac{{\sum }_{j=n-k+1}^n{\nu }_{j0}-1}{|{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\nu }_{ji}|}))>x_{-p+1}\ge \cdots \ge x_{-1}\ge x_0$. On the other hand, if $x_n\ge 0$ with ${\sum }_{j=n-k+1}^n{F}_j\ge 1$ and ${\lambda }_n\ge 0$ or if $x_n\ge 0$, ${\sum }_{j=n-k+1}^n{F}_j\le 1$ and ${\lambda }_n\le 0$, then $x_j=0$; $\mathrm{\forall }j(\ge n+1)\in {\mathbf{N}}_0$ from (1.1). Then (3.6) guarantees the convergence to zero of the sequence $\{x_n\}$; $n\in {\mathbf{N}}_0$ which is, furthermore, strictly decreasing. Hence, Property (ii). □

A dual result to Theorem 3.3 concerned with the instability situations follows.

Theorem 3.4 The following properties hold.

• (i) Any nontrivial nonnegative solution $\{x_n\}$ of (1.1) is strictly increasing for $n\in {\mathbf{N}}_0$ and initial conditions $x_{-p+1}\le \cdots \le x_{-1}\le x_0>0$, and then it tends to +∞ under the following condition:

  $$\begin{array}{r}(e^{A{G}_n}(1+\frac{1}{{\sum }_{j=n-k+1}^n{\nu }_{j1}})>M_n\ge e^{A{G}_n})\\ \wedge (\sum _{j=n-k+1}^n{\nu }_{j0}\ge 1)\wedge ({\nu }_{ji}\ge 0;\mathrm{\forall }j\in {\mathbf{N}}_{0+},\mathrm{\forall }i\in \overline{m});\mathrm{\forall }n\in {\mathbf{N}}_0\end{array}$$

  (3.11)

with $M_n>e^{A{G}_n}$ if $x_n\ne 0$ and ${\sum }_{j=n-k+1}^n{\mu }_{j1}>0$ with $\{F_n\}$ satisfying the same constraints as those of Theorem  3.3.

• (ii) Any nontrivial nonnegative solution $\{x_n\}$ of (1.1) is strictly increasing for $n\in {\mathbf{N}}_0$ and initial conditions $x_{-p+1}\le \cdots \le x_{-1}\le x_0>0$, and then it tends to +∞ under the following condition:

  $$(M_n\le e^{A{G}_n})\wedge (\sum _{j=n-k+1}^n{\mu }_{j0}\le 1)\wedge ({\mu }_{ji}\le 0;\mathrm{\forall }j\in {\mathbf{N}}_{0+},\mathrm{\forall }i\in \overline{m});\mathrm{\forall }n\in {\mathbf{N}}_0$$

  (3.12)

with $M_n>e^{A{G}_n}$ if $x_n\ne 0$ and ${\sum }_{j=n-k+1}^n{\mu }_{j1}<0$.

Proof Assume that $x_j>x_{j-1}>\cdots >x_0\ge x_{-1}\ge \cdots \ge x_{-p+1}$ for any given $j(\le n)\in {\mathbf{N}}_0$. If $x_{n+1}\ne 0$, then it follows that

$$\begin{array}{rcl}{x}_{n+1}& =& (1-\sum _{j=n-k+1}^n{F}_j)(1-M_n{e}^{-A{G}_n})=(\sum _{j=n-k+1}^n{F}_j-1)(M_n{e}^{-A{G}_n}-1)\\ \ge & (\sum _{j=n-k+1}^n{\nu }_{j0}-1+\sum _{j=n-k+1}^n{\nu }_{j1}{x}_j+\sum _{j=n-k+1}^n\sum _{i=2}^m{\nu }_{ji}{x}_j^i)(M_n{e}^{-A{G}_n}-1)>x_n\\ \ge & (\sum _{j=n-k+1}^n{\nu }_{j0}-1+\sum _{j=n-k+1}^n{\nu }_{j1}{x}_j+\sum _{j=n-k+1}^n\sum _{i=2}^m{\nu }_{ji}min(x_n,x_n^m))(M_n{e}^{-A{G}_n}-1)>x_n\end{array}$$

if either

$$\begin{array}{r}{M}_n\ge e^{A{G}_n};\sum _{j=n-k+1}^n{\nu }_{j1}>\frac{1}{1-M_n{e}^{-A{G}_n}};\\ \sum _{j=n-k+1}^n{\nu }_{j0}\ge 1;{\nu }_{ji}\ge 0;\mathrm{\forall }j,n\in {\mathbf{N}}_0,\mathrm{\forall }i\in \overline{m}\end{array}$$

(3.13)

with the first inequality being strict for any $x_n\ne 0$, or

$$\begin{array}{r}{M}_n\le e^{A{G}_n};\sum _{j=n-k+1}^n|{\nu }_{j1}|>\frac{1}{1-M_n{e}^{-A{G}_n}};\\ \sum _{j=n-k+1}^n{\nu }_{j0}\le 1;{\nu }_{ji}\le 0;\mathrm{\forall }j,n\in {\mathbf{N}}_0,\mathrm{\forall }i\in \overline{m}\end{array}$$

(3.14)

with the first inequality being strict for any $x_n\ne 0$, and the proof of Property (i) follows by complete induction after combining the first two conditions of (3.13). The proof of Property (ii) follows from (3.14) in the same way after combining the first two conditions which reduce to the first one. □

The constraints ${\sum }_{i=0}^m{\nu }_{ni}{x}_n^i\le F_n\le {\sum }_{i=0}^m{\mu }_{ni}{x}_n^i$; $\mathrm{\forall }n\in {\mathbf{N}}_0$ have been used in order to simplify the discussion. More general constraints can be used instead as, for instance, ${\sum }_{i=0}^{m_n}{\nu }_{ni}{x}_n^i\le F_n\le {\sum }_{i=0}^{m_n}{\mu }_{ni}{x}_n^i$ or ${\sum }_{i\in I_n}{\nu }_{ni}{x}_n^i\le F_n\le {\sum }_{i\in I_n}{\mu }_{ni}{x}_n^i$, where $\{m_n\}$ is a nonnegative sequence of polynomial degrees subject to $\underline{m}\le m_n\le \overline{m}$; $\mathrm{\forall }n\in {\mathbf{N}}_0$ and $\{I_n\}$ is a nonnegative real sequence of sets of cardinal $(m_n+1)$; $\mathrm{\forall }n\in {\mathbf{N}}_0$. The above second constraints are not necessarily of polynomial type. Theorems 3.3-3.4 have the following direct extensions such that the stability and instability conditions depend not only of the parameters but on the solution sequence as well.

Theorem 3.5 The following results hold.

• (i) Assume that the sequence $\{M_n\}$ satisfies the constraint

  $$\begin{array}{r}{M}_n>e^{A{G}_n}(1+\frac{x_n}{({\sum }_{j=n-k+1}^n{\nu }_{j0}-1)+({\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\nu }_{ji})min(x_n,x_n^m)});\\ \mathrm{\forall }n\in {\mathbf{N}}_0.\end{array}$$

  (3.15)

Thus, any nontrivial nonnegative solution $\{x_n\}$ of (1.1) under initial conditions $x_{-p+1}\ge \cdots \ge x_{-1}\ge x_0>0$ is uniformly bounded and strictly decreasing and then converges to the zero equilibrium point. If the above inequality is non-strict, then any nontrivial solution still is uniformly bounded satisfying $x_n\le x_0\le x_{-p+1}<+\mathrm{\infty }$, $\mathrm{\forall }n\in {\mathbf{N}}_0$.

• (ii) Assume that the sequence $\{M_n\}$ satisfies the constraint

  $$\begin{array}{r}{M}_n<e^{A{G}_n}(1-\frac{x_n}{({\sum }_{j=n-k+1}^n{\mu }_{j0}-1)+({\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\mu }_{ji})max(x_n,x_n^m)});\\ \mathrm{\forall }n\in {\mathbf{N}}_0.\end{array}$$

  (3.16)

Thus, any nontrivial nonnegative solution $\{x_n\}$ of (1.1) under initial conditions $x_{-p+1}\le \cdots \le x_{-1}\le x_0>0$ is strictly increasing and then tends to +∞. If the above inequality is non-strict, then any nontrivial solution is bounded from below satisfying $x_{n+1}\ge x_n\ge x_0\ge x_{-p+1}>0$, $\mathrm{\forall }n\in {\mathbf{N}}_0$.

Proof Note that for $x_n>0$ and $x_{n+1}\ge 0$, one has

$$\begin{array}{r}{x}_{n+1}=(1-\sum _{j=n-k+1}^n{F}_j)(1-M_n{e}^{-A{G}_n})\le x_n\\ \iff M_n\ge e^{A{G}_n}(1+\frac{x_n}{{\sum }_{j=n-k+1}^n{F}_j-1});\mathrm{\forall }n\in {\mathbf{N}}_0.\end{array}$$

(3.17)

If $x_n=0$ then $x_j=0$; $\mathrm{\forall }j(\ge n)\in {\mathbf{N}}_0$ from (1.1). Thus, Eq. (3.15) is a sufficient condition for (3.17) to hold. This proves the first part of Property (i) and the solution sequence is strictly decreasing. If the inequality in (3.15) is non-strict, then $x_{n+1}\le x_n\le x_0\le x_{-p+1}$; $\mathrm{\forall }n\in {\mathbf{N}}_0$ leading to the second part of Property (i). Property (ii) follows by proving that (3.16) guarantees that

$$x_{n+1}=max(0,(1-\sum _{j=n-k+1}^n{F}_j)(1-M_n{e}^{-A{G}_n}))>(\ge )x_n;\mathrm{\forall }n\in {\mathbf{N}}_0.$$

 □

Theorems 3.3-3.4 may be combined to give mixed conditions for the solution to be oscillatory while being uniformly bounded as follows.

Theorem 3.6 Assume that the real sequence $\{F_n\}$ satisfies the constraints: ${\sum }_{i=0}^m{\nu }_{ni}{x}_n^i\le F_n\le {\sum }_{i=0}^m{\mu }_{ni}{x}_n^i$ for some nonnegative real sequences $\{{\nu }_{ni}\}$, $\{{\mu }_{ni}\}$ with ${\mu }_{ni}\ge {\nu }_{ni}$; $i\in \overline{m}\cup \{0\}$, $\mathrm{\forall }n\in {\mathbf{N}}_0$. Then the following properties hold.

• (i) Any nontrivial nonnegative solution $\{x_n\}$ of (1.1) for initial conditions $min(1,{min}_{n\in {\mathbf{N}}_{0+}}(\frac{1-{\sum }_{j=n-k+1}^n{\mu }_{j0}}{{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\mu }_{ji}}))\ge x_{-p+1}\ge \cdots \ge x_{-1}\ge x_0$ is uniformly bounded and oscillatory if it satisfies, for each two consecutive intervals on nonnegative integers, the following constraints:

  $$\begin{array}{l}([(\sum _{j=n-k+1}^n{\mu }_{j0}<1)\wedge (\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}\ge 0)]\\ \vee [(\sum _{j=n-k+1}^n{\mu }_{j0}>1)\wedge (\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}\le 0)])\\ \wedge (M_n\ge e^{A{G}_n});\mathrm{\forall }n\in [k\left(\sum _{i=1}^j{n}_i\right),k\left(\sum _{i=1}^{j+1}{n}_i\right)-1),\end{array}$$

  (3.18)

$$\begin{array}{l}(e^{A{G}_n}(1+\frac{1}{{\sum }_{j=n-k+1}^n{\nu }_{j1}})>M_n\ge e^{A{G}_n})\\ \wedge (\sum _{j=n-k+1}^n{\nu }_{j0}\ge 1)\wedge ({\nu }_{ji}\ge 0;\mathrm{\forall }j\in {\mathbf{N}}_0,\mathrm{\forall }i\in \overline{m});\\ \mathrm{\forall }n\in [k\left(\sum _{i=1}^{j+1}{n}_i\right),k\left(\sum _{i=1}^{j+2}{n}_i\right)-1)\end{array}$$

(3.19)

with $M_n>e^{A{G}_n}$ if $x_n\ne 0$ in (3.19), for any given finite $j\in {\mathbf{N}}_0$ for any set of finite numbers $n_i\in \mathbf{N}$ for $i\ge 1$, $i\in {\mathbf{N}}_0$ with $n_0=0$, which satisfy $x_{k({\sum }_{i=1}^{j+2}{n}_i)-1}\le x_{k({\sum }_{i=1}^{j+1}{n}_i)-1}$. The solution is alternately strictly decreasing and strictly increasing for each two consecutive such intervals.

• (ii) Any nontrivial nonnegative solution $\{x_n\}$ of (1.1) for initial conditions $min(1,{min}_{n\in {\mathbf{N}}_{0+}}(\frac{1-{\sum }_{j=n-k+1}^n{\mu }_{j0}}{{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\mu }_{ji}}))\ge x_{-p+1}\ge \cdots \ge x_{-1}\ge x_0$ is uniformly bounded and oscillatory if it satisfies the following constraints for each two consecutive intervals on nonnegative integers:

  $$\begin{array}{l}([(\sum _{j=n-k+1}^n{\mu }_{j0}<1)\wedge (\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}\ge 0)]\\ \vee [(\sum _{j=n-k+1}^n{\mu }_{j0}>1)\wedge (\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}\le 0)])\\ \wedge (M_n\ge e^{A{G}_n});\mathrm{\forall }n\in [k\left(\sum _{i=1}^j{n}_i\right),k\left(\sum _{i=1}^{j+1}{n}_i\right)-1),\end{array}$$

  (3.20)

$$\begin{array}{l}(M_n\le e^{A{G}_n})\wedge (\sum _{j=n-k+1}^n{\mu }_{j0}\le 1)\wedge ({\mu }_{ji}\le 0;\mathrm{\forall }j\in {\mathbf{N}}_{0+},\mathrm{\forall }i\in \overline{m});\\ \mathrm{\forall }n\in [k\left(\sum _{i=1}^{j+1}{n}_i\right),k\left(\sum _{i=1}^{j+2}{n}_i\right)-1)\end{array}$$

(3.21)

with $M_n<e^{A{G}_n}$ if $x_n\ne 0$ in (3.21), for any given finite $j\in {\mathbf{N}}_0$ and for any set of finite numbers $n_i\in \mathbf{N}$ for $i\ge 1$, $i\in {\mathbf{N}}_{0+}$ with $n_0=0$, which satisfy $x_{k({\sum }_{i=1}^{j+2}{n}_i)-1}\le x_{k({\sum }_{i=1}^{j+1}{n}_i)-1}$. The solution is alternately strictly decreasing and strictly increasing for each two consecutive such intervals.

• (iii) Any nontrivial nonnegative solution $\{x_n\}$ of (1.1) for initial conditions $min(1,{min}_{n\in {\mathbf{N}}_{0+}}(\frac{{\sum }_{j=n-k+1}^n{\nu }_{j0}-1}{|{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\nu }_{ji}|}))>x_{-p+1}\ge \cdots \ge x_{-1}\ge x_0$ is uniformly bounded and oscillatory if it satisfies the following constraints for each two consecutive intervals on nonnegative integers:

  $$\begin{array}{l}[(\sum _{j=n-k+1}^n{\nu }_{j0}<1)\wedge ({\nu }_{ji}\le 0;\mathrm{\forall }j\in {\mathbf{N}}_0,\mathrm{\forall }i\in \overline{m})]\\ \wedge (M_n\le e^{A{G}_n});\mathrm{\forall }n\in [k\left(\sum _{i=1}^j{n}_i\right),k\left(\sum _{i=1}^{j+1}{n}_i\right)-1),\end{array}$$

  (3.22)

$$\begin{array}{l}(e^{A{G}_n}(1+\frac{1}{{\sum }_{j=n-k+1}^n{\nu }_{j1}})>M_n\ge e^{A{G}_n})\\ \wedge (\sum _{j=n-k+1}^n{\nu }_{j0}\ge 1)\wedge ({\nu }_{ji}\ge 0;\mathrm{\forall }j\in {\mathbf{N}}_0,\mathrm{\forall }i\in \overline{m});\\ \mathrm{\forall }n\in [k\left(\sum _{i=1}^{j+1}{n}_i\right),k\left(\sum _{i=1}^{j+2}{n}_i\right)-1)\end{array}$$

(3.23)

with $M_n>e^{A{G}_n}$ if $x_n\ne 0$ in (3.23), for any given finite $j\in {\mathbf{N}}_0$ for any set of finite numbers $n_i\in \mathbf{N}$ for $i\ge 1$, $i\in {\mathbf{N}}_0$ with $n_0=0$, which satisfy $x_{k({\sum }_{i=1}^{j+2}{n}_i)-1}\le x_{k({\sum }_{i=1}^{j+1}{n}_i)-1}$. The solution is alternately strictly decreasing and strictly increasing for each two consecutive such intervals.

• (iv) Any nontrivial nonnegative solution $\{x_n\}$ of (1.1) for initial conditions $min(1,{min}_{n\in {\mathbf{N}}_{0+}}(\frac{{\sum }_{j=n-k+1}^n{\nu }_{j0}-1}{|{\sum }_{j=n-k+1}^n{\sum }_{i=1}^m{\nu }_{ji}|}))>x_{-p+1}\ge \cdots \ge x_{-1}\ge x_0$ is uniformly bounded and oscillatory if it satisfies the following constraints for each two consecutive intervals on nonnegative integers:

  $$\begin{array}{l}[(\sum _{j=n-k+1}^n{\nu }_{j0}<1)\wedge ({\nu }_{ji}\le 0;\mathrm{\forall }j\in {\mathbf{N}}_{0+},\mathrm{\forall }i\in \overline{m})]\wedge (M_n\le e^{A{G}_n});\\ \mathrm{\forall }n\in [k\left(\sum _{i=1}^j{n}_i\right),k\left(\sum _{i=1}^{j+1}{n}_i\right)-1),\end{array}$$

  (3.24)

$$\begin{array}{l}(M_n\le e^{A{G}_n})\wedge (\sum _{j=n-k+1}^n{\mu }_{j0}\le 1)\wedge ({\mu }_{ji}\le 0;\mathrm{\forall }j\in {\mathbf{N}}_{0+},\mathrm{\forall }i\in \overline{m});\\ \mathrm{\forall }n\in [k\left(\sum _{i=1}^{j+1}{n}_i\right),k\left(\sum _{i=1}^{j+2}{n}_i\right)-1)\end{array}$$

(3.25)

with $M_n<e^{A{G}_n}$ if $x_n\ne 0$ in (3.25), for any given finite $j\in {\mathbf{N}}_0$ for any set of finite numbers $n_i\in \mathbf{N}$ for $i\ge 1$, $i\in {\mathbf{N}}_0$, with $n_0=0$, which satisfy $x_{k({\sum }_{i=1}^{j+2}{n}_i)-1}\le x_{k({\sum }_{i=1}^{j+1}{n}_i)-1}$; $\mathrm{\forall }j\in {\mathbf{N}}_0$. The solution is alternately strictly decreasing and strictly increasing for each two consecutive such intervals.

• (v) Any nontrivial nonnegative solution $\{x_n\}$ of (1.1) for initial conditions $x_{-p+1}\le \cdots \le x_{-1}\le x_0>0$ is uniformly bounded and oscillatory if it satisfies the following constraints for each two consecutive intervals on nonnegative integers:

  $$\begin{array}{l}(e^{A{G}_n}(1+\frac{1}{{\sum }_{j=n-k+1}^n{\nu }_{j1}})>M_n\ge e^{A{G}_n})\\ \wedge (\sum _{j=n-k+1}^n{\nu }_{j0}\ge 1)\wedge ({\nu }_{ji}\ge 0;\mathrm{\forall }j\in {\mathbf{N}}_0,\mathrm{\forall }i\in \overline{m});\\ \mathrm{\forall }n\in [k\left(\sum _{i=1}^j{n}_i\right),k\left(\sum _{i=1}^{j+1}{n}_i\right)-1),\end{array}$$

  (3.26)

$$\begin{array}{l}([(\sum _{j=n-k+1}^n{\mu }_{j0}<1)\wedge (\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}\ge 0)]\\ \vee [(\sum _{j=n-k+1}^n{\mu }_{j0}>1)\wedge (\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}\le 0)]);\\ \mathrm{\forall }n\in [k\left(\sum _{i=1}^{j+1}{n}_i\right),k\left(\sum _{i=1}^{j+2}{n}_i\right)-1)\end{array}$$

(3.27)

with $M_n>e^{A{G}_n}$ if $x_n\ne 0$ in (3.26), for any given finite $j\in {\mathbf{N}}_0$ for any set of finite numbers $n_i\in \mathbf{N}$ for $i\ge 1$, $i\in {\mathbf{N}}_0$ with $n_0=0$, which satisfy $x_{k({\sum }_{i=1}^{j+2}{n}_i)-1}\le x_{k({\sum }_{i=1}^{j+1}{n}_i)-1}$. The solution is alternately strictly increasing and strictly decreasing for each two consecutive such intervals. If the above inequality is strict, then the solution converges asymptotically to the zero equilibrium point.

• (vi) Any nontrivial nonnegative solution $\{x_n\}$ of (1.1) for initial conditions $x_{-p+1}\le \cdots \le x_{-1}\le x_0>0$ is uniformly bounded and oscillatory if it satisfies the following constraints for each two consecutive intervals on nonnegative integers:

  $$\begin{array}{l}(M_n\le e^{A{G}_n})\wedge (\sum _{j=n-k+1}^n{\mu }_{j0}\le 1)\wedge ({\mu }_{ji}\le 0;\mathrm{\forall }j\in {\mathbf{N}}_{0+},\mathrm{\forall }i\in \overline{m});\\ \mathrm{\forall }n\in [k\left(\sum _{i=1}^j{n}_i\right),k\left(\sum _{i=1}^{j+1}{n}_i\right)-1),\end{array}$$

  (3.28)

$$\begin{array}{l}([(\sum _{j=n-k+1}^n{\mu }_{j0}<1)\wedge (\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}\ge 0)]\\ \vee [(\sum _{j=n-k+1}^n{\mu }_{j0}>1)\wedge (\sum _{j=n-k+1}^n\sum _{i=1}^m{\mu }_{ji}\le 0)]);\\ \mathrm{\forall }n\in [k\left(\sum _{i=1}^{j+1}{n}_i\right),k\left(\sum _{i=1}^{j+2}{n}_i\right)-1)\end{array}$$

(3.29)

with $M_n<e^{A{G}_n}$ if $x_n\ne 0$ in (3.28), for any given finite $j\in {\mathbf{N}}_0$ for any set of finite numbers $n_i\in \mathbf{N}$ for $i\ge 1$, $i\in {\mathbf{N}}_0$ with $n_0=0$, which satisfy $x_{k({\sum }_{i=1}^{j+2}{n}_i)-1}\le x_{k({\sum }_{i=1}^{j+1}{n}_i)-1}$; $\mathrm{\forall }j\in {\mathbf{N}}_0$. The solution is alternately strictly increasing and strictly decreasing for each two consecutive such intervals. If the above inequality is strict, then the solution converges asymptotically to the zero equilibrium point.

• (vii) Any nontrivial nonnegative solution $\{x_n\}$ of (1.1) for initial conditions $x_{-p+1}\le \cdots \le x_{-1}\le x_0>0$ is uniformly bounded and oscillatory if it satisfies the following constraints for each two consecutive intervals on nonnegative integers:

  $$\begin{array}{l}(e^{A{G}_n}(1+\frac{1}{{\sum }_{j=n-k+1}^n{\nu }_{j1}})>M_n\ge e^{A{G}_n})\\ \wedge (\sum _{j=n-k+1}^n{\nu }_{j0}\ge 1)\wedge ({\nu }_{ji}\ge 0;\mathrm{\forall }j\in {\mathbf{N}}_{0+},\mathrm{\forall }i\in \overline{m});\\ \mathrm{\forall }n\in [k\left(\sum _{i=1}^j{n}_i\right),k\left(\sum _{i=1}^{j+1}{n}_i\right)\end{array}$$