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Positive solutions for a system of Riemann–Liouville fractional boundary value problems with p-Laplacian operators

Abstract

We study the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with p-Laplacian operators, nonnegative nonlinearities and positive parameters, subject to coupled nonlocal boundary conditions which contain Riemann–Stieltjes integrals and various fractional derivatives. We use the Guo–Krasnosel’skii fixed point theorem in the proof of the main existence results.

Introduction

In this paper we consider the system of nonlinear ordinary fractional differential equations with \(\varrho _{1}\)-Laplacian and \(\varrho _{2}\)-Laplacian operators

$$ \textstyle\begin{cases} D_{0+}^{\alpha _{1}}(\varphi _{\varrho _{1}} (D_{0+}^{\beta _{1}} x(t)))+ \lambda f(t,x(t),y(t))=0,\quad t\in (0,1), \\ D_{0+}^{\alpha _{2}}(\varphi _{\varrho _{2}}(D_{0+}^{\beta _{2}}y(t)))+ \mu g(t,x(t),y(t))=0,\quad t\in (0,1), \end{cases} $$
(1)

with the coupled nonlocal boundary conditions

$$ \textstyle\begin{cases} x^{(j)}(0)=0, \quad j=0,\ldots ,n-2; \qquad D_{0+}^{\beta _{1}}x(0)=0, \\ D_{0+}^{\gamma _{0}}x(1)= \sum_{i=1}^{p} \int _{0}^{1} D_{0+}^{\gamma _{i}}y(t) \,dH_{i}(t), \\ y^{(j)}(0)=0, \quad j=0,\ldots ,m-2;\qquad D_{0+}^{\beta _{2}}y(0)=0, \\ D_{0+}^{\delta _{0}}y(1)= \sum_{i=1}^{q} \int _{0}^{1} D_{0+}^{\delta _{i}}x(t) \,dK_{i}(t), \end{cases} $$
(2)

where \(\alpha _{1}, \alpha _{2}\in (0,1]\), \(\beta _{1}\in (n-1,n]\), \(\beta _{2}\in (m-1,m]\), \(n, m\in \mathbb{N}\), \(n, m\ge 3\), \(p,q\in \mathbb{N}\), \(\gamma _{i}\in \mathbb{R}\) for all \(i=0,1,\ldots ,p\), \(0\le \gamma _{1}<\gamma _{2}<\cdots <\gamma _{p}\le \delta _{0}< \beta _{2}-1\), \(\delta _{0}\ge 1\), \(\delta _{i}\in \mathbb{R}\) for all \(i=0,1,\ldots ,q\), \(0\le \delta _{1}<\delta _{2}<\cdots <\delta _{q}\le \gamma _{0}< \beta _{1}-1\), \(\gamma _{0}\ge 1\), \(\varrho _{1}, \varrho _{2}>1\), \(\varphi _{\varrho _{i}}(s)=|s|^{\varrho _{i}-2}s\), \(\varphi _{\varrho _{i}}^{-1}=\varphi _{\rho _{i}}\), \(\frac{1}{\varrho _{i}}+\frac{1}{\rho _{i}}=1\), \(i=1,2\), \(\lambda , \mu >0\), the functions \(f, g\in C([0,1]\times \mathbb{R}_{+}\times \mathbb{R}_{+}, \mathbb{R}_{+})\) (\(\mathbb{R}_{+}=[0,\infty )\)), the integrals from (2) are Riemann–Stieltjes integrals with \(H_{i}\), \(i=1,\ldots ,p\), and \(K_{i}\), \(i=1,\ldots ,q\), functions of bounded variation, and \(D_{0+}^{k}\) denotes the Riemann–Liouville derivative of order k (for \(k=\alpha _{1}, \beta _{1}, \alpha _{2}, \beta _{2}, \gamma _{i}\) for \(i=0,1,\ldots ,p\), \(\delta _{i}\) for \(i=0,1,\ldots ,q\)).

We present sufficient conditions on the functions f and g, and intervals for the parameters λ and μ such that problem (1), (2) has positive solutions. A positive solution of problem (1), (2) is a pair of functions \((x,y)\in (C([0,1],\mathbb{R}_{+}))^{2}\), satisfying (1) and (2) with \(x(t)>0\) for all \(t\in (0,1]\) or \(y(t)>0\) for all \(t\in (0,1]\). We also investigate the nonexistence of positive solutions for this problem. The problem (1), (2) is a generalization of the problem studied in [30]. Indeed, if \(p=1\), \(q=1\), \(\gamma _{0}=p_{1}\), \(\gamma _{1}=q_{1}\), \(\delta _{0}=p_{2}\), \(\delta _{1}=q_{2}\), \(H_{1}\) is a step function given by \(H_{1}(t)=\{0, t\in [0,\xi _{1}); a_{1}, t\in [\xi _{1},\xi _{2}); a_{1}+a_{2}, t\in [\xi _{2},\xi _{3});\ldots ;\sum_{i=1}^{N}a_{i}, t\in [\xi _{N},1]\}\), and \(K_{1}\) is a step function given by \(K_{1}(t)=\{0, t\in [0,\eta _{1}); b_{1}, t\in [\eta _{1}, \eta _{2}); b_{1}+b_{2}, t\in [\eta _{2},\eta _{3});\ldots ; \sum_{i=1}^{M} b_{i}, t\in [\eta _{M},1]\}\), then the boundary conditions (2) become the multi-point boundary conditions \((BC)\) from [30]. The system (1) with uncoupled multi-point boundary conditions was studied in [29]. Fractional differential equations and systems of fractional differential equations, subject to various multi-point or Riemann–Stieltjes integral boundary conditions, were studied in the last years in [18, 1725, 28, 31, 37, 3941]. For various applications of the fractional calculus in several scientific and engineering fields, the reader may consult the books [10, 11, 26, 27, 3436], and the papers [9, 1215, 32, 33, 38].

In Sect. 2, we study a nonlocal boundary value problem for fractional differential equations with p-Laplacian operators, and we present some properties of the associated Green functions. Section 3 is devoted to the main existence theorems for the positive solutions with respect to a cone for our problem (1), (2), which are based on the Guo–Krasnosel’skii fixed point theorem (see [16]). In Sect. 4, we present some nonexistence results for the positive solutions of (1), (2), and in Sect. 5, we give an example which illustrates our results.

Preliminary results

We consider the system of fractional differential equations

$$ \textstyle\begin{cases} D_{0+}^{\alpha _{1}}(\varphi _{\varrho _{1}} (D_{0+}^{\beta _{1}} x(t)))+ \psi (t)=0, \quad t\in (0,1), \\ D_{0+}^{\alpha _{2}}(\varphi _{\varrho _{2}}(D_{0+}^{\beta _{2}}y(t)))+ \chi (t)=0, \quad t\in (0,1), \end{cases} $$
(3)

with the coupled boundary conditions (2), where \(\psi , \chi \in C(0,1)\cap L^{1}(0,1)\).

We denote \(\varphi _{\varrho _{1}}(D_{0^{+}}^{\beta _{1}}x(t))=u(t)\) and \(\varphi _{\varrho _{2}}(D_{0^{+}}^{\beta _{2}}y(t))=v(t)\). Then problem (3), (2) is equivalent to the following three problems:

$$\begin{aligned}& \textstyle\begin{cases} D_{0+}^{\alpha _{1}}u(t)+\psi (t)=0, \quad 0< t< 1, \\ u(0)=0, \end{cases}\displaystyle \end{aligned}$$
(4)
$$\begin{aligned}& \textstyle\begin{cases} D_{0+}^{\alpha _{2}}v(t)+\chi (t)=0,\quad 0< t< 1, \\ v(0)=0, \end{cases}\displaystyle \end{aligned}$$
(5)

and

$$ \textstyle\begin{cases} D_{0+}^{\beta _{1}}x(t)=\varphi _{\rho _{1}}(u(t)),\quad t\in (0,1), \\ D_{0+}^{\beta _{2}}y(t)=\varphi _{\rho _{2}}(v(t)),\quad t\in (0,1), \end{cases} $$
(6)

with the boundary conditions

$$ \textstyle\begin{cases} x^{(j)}(0)=0, \quad j=0,\ldots ,n-2;\qquad D_{0+}^{\gamma _{0}}x(1)= \sum_{i=1}^{p} \int _{0}^{1}D_{0+}^{\gamma _{i}} y(t) \,dH_{i}(t), \\ y^{(j)}(0)=0, \quad j=0,\ldots ,m-2;\qquad D_{0+}^{\delta _{0}}y(1)= \sum_{i=1}^{q} \int _{0}^{1} D_{0+}^{\delta _{i}}x(t) \,dK_{i}(t). \end{cases} $$
(7)

The first two problems (4) and (5) have the unique solutions \(u\in C[0,1]\) and \(v\in C[0,1]\), respectively, of the form

$$ u(t)=-I_{0+}^{\alpha _{1}}\psi (t)=- \frac{1}{\varGamma (\alpha _{1})} \int _{0}^{t} (t-\nu )^{ \alpha _{1}-1}\psi ( \nu ) \,d\nu ,\quad t\in [0,1], $$
(8)

and

$$ v(t)=-I_{0+}^{\alpha _{2}}\chi (t)=- \frac{1}{\varGamma (\alpha _{2})} \int _{0}^{t} (t-\nu )^{ \alpha _{2}-1}\chi ( \nu ) \,d\nu ,\quad t\in [0,1], $$
(9)

respectively.

For the third problem (6), (7), we denote

$$ \begin{aligned}[b] \Delta ={}& \frac{\varGamma (\beta _{1})\varGamma (\beta _{2})}{\varGamma (\beta _{1}-\gamma _{0})\varGamma (\beta _{2}-\delta _{0})}- \Biggl( \sum_{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1}\tau ^{\beta _{2}-\gamma _{i}-1} \,dH_{i}( \tau ) \Biggr) \\ &{}\times \Biggl( \sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1}\tau ^{\beta _{1}-\delta _{i}-1} \,dK_{i}( \tau ) \Biggr). \end{aligned} $$
(10)

Lemma 2.1

If\(\Delta \neq0\), then the unique solution\((x,y)\in C[0,1]\times C[0,1]\)of problem (6), (7) is given by

$$\begin{aligned}& x(t)= \frac{1}{\varGamma (\beta _{1})} \int _{0}^{t} (t-\nu )^{\beta _{1}-1}\varphi _{\rho _{1}} \bigl(u(\nu ) \bigr) \,d\nu \\& \hphantom{x(t)= {}}{}+ \frac{t^{\beta _{1}-1}}{\Delta } \Biggl[- \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{1}-\gamma _{0})\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-\nu )^{\beta _{1}-\gamma _{0}-1} \varphi _{\rho _{1}} \bigl(u(\nu ) \bigr) \,d\nu \\& \hphantom{x(t)= {}}{}+ \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\delta _{0})} \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{0}^{\nu } (\nu -\tau )^{\beta _{2}- \gamma _{i}-1} \varphi _{\rho _{2}} \bigl(v(\tau ) \bigr) \,d\tau \biggr) \,dH_{i}( \nu ) \\& \hphantom{x(t)= {}}{}- \Biggl( \sum_{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} {\nu }^{\beta _{2}-\gamma _{i}-1} \,dH_{i}( \nu ) \Biggr) \\& \hphantom{x(t)= {}}{}\times \biggl( \frac{1}{\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1- \nu )^{\beta _{2}-\delta _{0}-1}\varphi _{\rho _{2}} \bigl(v(\nu ) \bigr) \,d\nu \biggr) \\& \hphantom{x(t)= {}}{}+ \Biggl( \sum_{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} {\nu }^{\beta _{2}-\gamma _{i}-1} \,dH_{i}( \nu ) \Biggr) \\& \hphantom{x(t)= {}}{}\times \Biggl( \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} \biggl( \int _{0}^{\nu } (\nu -\tau )^{ \beta _{1}-\delta _{i}-1} \varphi _{\rho _{1}} \bigl(u(\tau ) \bigr) \,d\tau \biggr) \,dK_{i}( \nu ) \Biggr) \Biggr], \\& \hphantom{x(t)= {}}{} \forall t\in [0,1], \\& y(t)= \frac{1}{\varGamma (\beta _{2})} \int _{0}^{t} (t-\nu )^{\beta _{2}-1}\varphi _{\rho _{2}} \bigl(v(\nu ) \bigr) \,d\nu \\& \hphantom{y(t)={}}{}+ \frac{t^{\beta _{2}-1}}{\Delta } \Biggl[- \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\gamma _{0})\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-\nu )^{\beta _{2}-\delta _{0}-1} \varphi _{\rho _{2}} \bigl(v(\nu ) \bigr) \,d\nu \\& \hphantom{y(t)={}}{}+ \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\gamma _{0})} \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} \biggl( \int _{0}^{\nu } (\nu -\tau )^{\beta _{1}- \delta _{i}-1} \varphi _{\rho _{1}} \bigl(u(\tau ) \bigr) \,d\tau \biggr) \,dK_{i}( \nu ) \\& \hphantom{y(t)={}}{}- \Biggl( \sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} {\nu }^{\beta _{1}-\delta _{i}-1} \,dK_{i}( \nu ) \Biggr) \\& \hphantom{y(t)={}}{}\times \biggl( \frac{1}{\varGamma (\beta _{1}-\gamma _{0})} \int _{0}^{1}(1- \nu )^{\beta _{1}-\gamma _{0}-1}\varphi _{\rho _{1}} \bigl(u(\nu ) \bigr) \,d\nu \biggr) \\& \hphantom{y(t)={}}{}+ \Biggl( \sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} {\nu }^{\beta _{1}-\delta _{i}-1} \,dK_{i}( \nu ) \Biggr) \\& \hphantom{y(t)={}}{}\times \Biggl( \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{0}^{\nu } (\nu -\tau )^{ \beta _{2}-\gamma _{i}-1} \varphi _{\rho _{2}} \bigl(v(\tau ) \bigr) \,d\tau \biggr) \,dH_{i}( \nu ) \Biggr) \Biggr], \\& \hphantom{y(t)={}}{} \forall t\in [0,1]. \end{aligned}$$
(11)

Proof

The solutions \((x,y)\in (C(0,1)\cap L^{1}(0,1))^{2}\) of system (6) are

$$ \begin{aligned} x(t)=I_{0+}^{\beta _{1}}\varphi _{\rho _{1}} \bigl(u(t) \bigr)+a_{1}t^{\beta _{1}-1}+a_{2} t^{\beta _{1}-2}+\cdots +a_{n} t^{\beta _{1}-n}, \\ y(t)=I_{0+}^{\beta _{2}}\varphi _{\rho _{2}} \bigl(v(t) \bigr)+b_{1}t^{\beta _{2}-1}+b_{2}t^{ \beta _{2}-2}+ \cdots +b_{m} t^{\beta _{2}-m}, \end{aligned} $$

with \(a_{1},a_{2},\ldots ,a_{n},b_{1},b_{2},\ldots ,b_{m}\in \mathbb{R}\). By using the boundary conditions \(x(0)=x'(0)=\cdots =x^{(n-2)}(0)=0\) and \(y(0)=y'(0)=\cdots =y^{(m-2)}=0\), we obtain \(a_{2}=\cdots =a_{n}=0\) and \(b_{2}=\cdots =b_{m}=0\). So the above solutions become

$$ \begin{aligned} &x(t)=I_{0+}^{\beta _{1}} \varphi _{\rho _{1}} \bigl(u(t) \bigr)+a_{1}t^{\beta _{1}-1}= \frac{1}{\varGamma (\beta _{1})} \int _{0}^{t} (t-\nu )^{\beta _{1}-1} \varphi _{\rho _{1}} \bigl(u(\nu ) \bigr) \,d\nu +a_{1} t^{\beta _{1}-1}, \\ &y(t)=I_{0+}^{\beta _{2}}\varphi _{\rho _{2}} \bigl(v(t) \bigr)+b_{1}t^{\beta _{2}-1}= \frac{1}{\varGamma (\beta _{2})} \int _{0}^{t}(t-\nu )^{\beta _{2}-1} \varphi _{\rho _{2}} \bigl(v(\nu ) \bigr) \,d\nu +b_{1} t^{\beta _{2}-1}. \end{aligned} $$
(12)

For the obtained functions x and y we have

$$\begin{aligned}& D_{0+}^{\gamma _{i}}y(t)=b_{1} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})}t^{\beta _{2}- \gamma _{i}-1}+I_{0+}^{\beta _{1}-\gamma _{i}} \varphi _{\rho _{2}} \bigl(v(t) \bigr),\quad i=1,\ldots ,p, \\& D_{0+}^{\gamma _{0}}x(t)=a_{1} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\gamma _{0})}t^{\beta _{1}- \gamma _{0}-1}+I_{0+}^{\beta _{1}-\gamma _{0}} \varphi _{\rho _{1}} \bigl(u(t) \bigr), \\& D_{0+}^{\delta _{i}}x(t)=a_{1} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})}t^{\beta _{1}- \delta _{i}-1}+I_{0+}^{\beta _{1}-\delta _{i}} \varphi _{\rho _{1}} \bigl(u(t) \bigr),\quad i=1,\ldots ,q, \\& D_{0+}^{\delta _{0}}y(t)=b_{1} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\delta _{0})}t^{\beta _{2}- \delta _{0}-1}+I_{0+}^{\beta _{2}-\delta _{0}} \varphi _{\rho _{2}} \bigl(v(t) \bigr), \\& \begin{aligned} D_{0+}^{\gamma _{0}}x(1)&=a_{1} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\gamma _{0})}+I_{0+}^{ \beta _{1}-\gamma _{0}} \varphi _{\rho _{1}} \bigl(u(t) \bigr)|_{t=1} \\ &=a_{1} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\gamma _{0})}+ \frac{1}{\varGamma (\beta _{1}-\gamma _{0})} \int _{0}^{1}(1-\nu )^{\beta _{1}-\gamma _{0}-1}\varphi _{\rho _{1}} \bigl(u( \nu ) \bigr) \,d\nu, \end{aligned} \\& \begin{aligned} D_{0+}^{\delta _{0}}y(1)&=b_{1} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\delta _{0})}+I_{0+}^{ \beta _{2}-\delta _{0}} \varphi _{\rho _{2}} \bigl(v(t) \bigr)|_{t=1} \\ &=b_{1} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\delta _{0})}+ \frac{1}{\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-\nu )^{\beta _{2}-\delta _{0}-1}\varphi _{\rho _{2}} \bigl(v( \nu ) \bigr) \,d\nu . \end{aligned} \end{aligned}$$

By imposing the boundary conditions \(D_{0+}^{\gamma _{0}}x(1)=\sum_{i=1}^{p}\int _{0}^{1} D_{0+}^{ \gamma _{i}} y(s) \,dH_{i}(s)\) and \(D_{0+}^{\delta _{0}}y(1)=\sum_{i=1}^{q}\int _{0}^{1} D_{0+}^{ \delta _{i}} x(s) \,dK_{i}(s)\), we deduce the following system in \(a_{1}\) and \(b_{1}\):

$$ \textstyle\begin{cases} a_{1} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\gamma _{0})}-b_{1} \sum_{i=1}^{p} \int _{0}^{1} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})}t^{\beta _{2}- \gamma _{i}-1}\,dH_{i}(t) \\ \quad = - \frac{1}{\varGamma (\beta _{1}-\gamma _{0})} \int _{0}^{1}(1-\nu )^{\beta _{1}-\gamma _{0}-1} \varphi _{\rho _{1}}(u(\nu )) \,d\nu \\ \qquad {} + \sum_{i=1}^{p} \int _{0}^{1} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} ( \int _{0}^{t} (t-\nu )^{\beta _{2}-\gamma _{i}-1} \varphi _{\rho _{2}}(v(\nu )) \,d\nu )\,dH_{i}(t), \\ b_{1} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\delta _{0})}-a_{1} \sum_{i=1}^{q} \int _{0}^{1} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})}t^{\beta _{1} -\delta _{i}-1}\,dK_{i}(t) \\ \quad = - \frac{1}{\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-\nu )^{\beta _{2}-\delta _{0}-1} \varphi _{\rho _{2}}(v(\nu )) \,d\nu \\ \qquad {} + \sum_{i=1}^{q} \int _{0}^{1} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} ( \int _{0}^{t} (t-\nu )^{\beta _{1}-\delta _{i}-1} \varphi _{\rho _{1}}(u(\nu )) \,d\nu )\,dK_{i}(t). \end{cases} $$
(13)

The above system in the unknowns \(a_{1}\) and \(b_{1}\) has the determinant Δ, which, by the assumption of the lemma, is nonzero. So the system (13) has the unique solution

$$\begin{aligned}& a_{1}=- \frac{\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{1}-\gamma _{0})\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-\nu )^{\beta _{1}-\gamma _{0}-1} \varphi _{\rho _{1}} \bigl(u(\nu ) \bigr) \,d\nu \\& \hphantom{a_{1}={}}{}+ \frac{\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{2}-\delta _{0})} \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{0}^{t} (t-\nu )^{\beta _{2}-\gamma _{i}-1} \varphi _{\rho _{2}} \bigl(v(\nu ) \bigr) \,d\nu \biggr) \,dH_{i}(t) \\& \hphantom{a_{1}={}}{}- \frac{1}{\Delta } \Biggl( \sum _{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} t^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(t) \Biggr) \\& \hphantom{a_{1}={}}{}\times \biggl( \frac{1}{\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1- \nu )^{\beta _{2}-\delta _{0}-1}\varphi _{\rho _{2}} \bigl(v(\nu ) \bigr) \,d\nu \biggr) \\& \hphantom{a_{1}={}}{}+ \frac{1}{\Delta } \Biggl( \sum _{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} t^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(t) \Biggr) \\& \hphantom{a_{1}={}}{}\times \Biggl( \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} \biggl( \int _{0}^{t}(t-\nu )^{\beta _{1}-\delta _{i}-1} \varphi _{\rho _{1}} \bigl(u(\nu ) \bigr) \,d\nu \biggr)\,dK_{i}(t) \Biggr), \\& b_{1}=- \frac{\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-\nu )^{\beta _{2}-\delta _{0}-1} \varphi _{\rho _{2}} \bigl(v(\nu ) \bigr) \,d\nu \\& \hphantom{b_{1}={}}{}+ \frac{\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})} \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} \biggl( \int _{0}^{t} (t-\nu )^{\beta _{1}-\delta _{i}-1} \varphi _{\rho _{1}} \bigl(u(\nu ) \bigr) \,d\nu \biggr) \,dK_{i}(t) \\& \hphantom{b_{1}={}}{}- \frac{1}{\Delta } \Biggl( \sum _{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} t^{\beta _{1}-\delta _{i}-1} \,dK_{i}(t) \Biggr) \\& \hphantom{b_{1}={}}{}\times \biggl( \frac{1}{\varGamma (\beta _{1}-\gamma _{0})} \int _{0}^{1}(1- \nu )^{\beta _{1}-\gamma _{0}-1}\varphi _{\rho _{1}} \bigl(u(\nu ) \bigr) \,d\nu \biggr) \\& \hphantom{b_{1}={}}{}+ \frac{1}{\Delta } \Biggl( \sum _{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} t^{\beta _{1}-\delta _{i}-1} \,dK_{i}(t) \Biggr) \\& \hphantom{b_{1}={}}{}\times \Biggl( \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{0}^{t}(t-\nu )^{\beta _{2}-\gamma _{i}-1} \varphi _{\rho _{2}} \bigl(v(\nu ) \bigr) \,d\nu \biggr)\,dH_{i}(t) \Biggr). \end{aligned}$$
(14)

Now replacing the constants \(a_{1}\) and \(b_{1}\) given by (14) in (12) we find the solution \((x,y)\in C[0,1]\times C[0,1]\) of problem (6), (7) presented in (11). Conversely, we can easily prove that the functions x, y given by (11) satisfy the problem (6), (7). □

Lemma 2.2

If\(\Delta \neq0\), then the solution\((x,y)\)of problem (6), (7) given by (11) can be written as

$$ \textstyle\begin{cases} x(t)=- \int _{0}^{1}{\mathcal{G}}_{1}(t,\nu )\varphi _{\rho _{1}}(u( \nu )) \,d\nu - \int _{0}^{1} {\mathcal{G}}_{2}(t,\nu ) \varphi _{\rho _{2}}(v(\nu )) \,d\nu , \quad \forall t\in [0,1], \\ y(t)=- \int _{0}^{1}{\mathcal{G}}_{3}(t,\nu )\varphi _{\rho _{1}}(u( \nu )) \,d\nu - \int _{0}^{1} {\mathcal{G}}_{4}(t,\nu ) \varphi _{\rho _{2}}(v(\nu )) \,d\nu ,\quad \forall t\in [0,1], \end{cases} $$
(15)

where

$$ \begin{aligned} &{\mathcal{G}}_{1}(t,\nu )=g_{1}(t,\nu )+ \frac{t^{\beta _{1}-1}}{\Delta } \Biggl( \sum _{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} s^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \Biggr) \\ &\hphantom{{\mathcal{G}}_{1}(t,\nu )={}}{}\times \Biggl( \sum_{i=1}^{q} \int _{0}^{1} g_{1i}(\tau ,\nu ) \,dK_{i}(\tau ) \Biggr), \\ &{\mathcal{G}}_{2}(t,\nu )= \frac{t^{\beta _{1}-1}\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{2}-\delta _{0})} \sum _{i=1}^{p} \int _{0}^{1} g_{2i}(\tau , \nu ) \,dH_{i}(\tau ), \\ &{\mathcal{G}}_{3}(t,\nu )= \frac{t^{\beta _{2}-1}\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})} \sum _{i=1}^{q} \int _{0}^{1} g_{1i}(\tau , \nu ) \,dK_{i}(\tau ), \\ &{\mathcal{G}}_{4}(t,\nu )=g_{2}(t,\nu )+ \frac{t^{\beta _{2}-1}}{\Delta } \Biggl( \sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} s^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \Biggr) \\ &\hphantom{{\mathcal{G}}_{4}(t,\nu )={}}{}\times \Biggl( \sum_{i=1}^{p} \int _{0}^{1} g_{2i}(\tau ,\nu ) \,dH_{i}(\tau ) \Biggr), \end{aligned} $$
(16)

for all\((t,\nu )\in [0,1]\times [0,1]\)and

$$\begin{aligned}& g_{1}(t,s)= \frac{1}{\varGamma (\beta _{1})} \textstyle\begin{cases} t^{\beta _{1}-1}(1-s)^{\beta _{1}-\gamma _{0}-1}-(t-s)^{\beta _{1}-1},& 0\le s\le t\le 1, \\ t^{\beta _{1}-1}(1-s)^{\beta _{1}-\gamma _{0}-1}, & 0\le t\le s \le 1,\end{cases}\displaystyle \\& g_{1i}(\tau ,s)= \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \textstyle\begin{cases} {\tau }^{\beta _{1}-\delta _{i}-1}(1-s)^{\beta _{1}-\gamma _{0}-1}-( \tau -s)^{\beta _{1}-\delta _{i}-1}, & 0\le s\le \tau \le 1, \\ {\tau }^{\beta _{1}-\delta _{i}-1}(1-s)^{\beta _{1}-\gamma _{0}-1},& 0\le \tau \le s\le 1,\end{cases}\displaystyle \\& g_{2}(t,s)= \frac{1}{\varGamma (\beta _{2})} \textstyle\begin{cases} t^{\beta _{2}-1}(1-s)^{\beta _{2}-\delta _{0}-1}-(t-s)^{\beta _{2}-1}, & 0\le s\le t\le 1, \\ t^{\beta _{2}-1}(1-s)^{\beta _{2}-\delta _{0}-1},& 0\le t\le s \le 1,\end{cases}\displaystyle \\& g_{2j}(\tau ,s)= \frac{1}{\varGamma (\beta _{2}-\gamma _{j})} \textstyle\begin{cases} {\tau }^{\beta _{2}-\gamma _{j}-1}(1-s)^{\beta _{2}-\delta _{0}-1}-( \tau -s)^{\beta _{2}-\gamma _{j}-1}, & 0\le s\le \tau \le 1, \\ {\tau }^{\beta _{2}-\gamma _{j}-1}(1-s)^{\beta _{2}-\delta _{0}-1},& 0\le \tau \le s\le 1,\end{cases}\displaystyle \end{aligned}$$
(17)

for all\(i=1,\ldots ,q\)and\(j=1,\ldots ,p\).

Proof

For \(x(t)\) from (11) we have

$$\begin{aligned} x(t) =& \frac{1}{\varGamma (\beta _{1})} \int _{0}^{t} (t-s)^{\beta _{1}-1}\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \frac{t^{\beta _{1}-1}\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{1}-\gamma _{0})\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-s)^{\beta _{1}-\gamma _{0}-1} \varphi _{ \rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}+ \frac{t^{\beta _{1}}-1}{\Delta } \Biggl( \sum_{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} s^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \Biggr) \\ &{}\times \Biggl( \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} \biggl( \int _{0}^{s}(s-\tau )^{\beta _{1}-\delta _{i}-1} \varphi _{\rho _{1}} \bigl(u(\tau ) \bigr) \,d\tau \biggr) \,dK_{i}(s) \Biggr) \\ &{}+ \frac{t^{\beta _{1}-1}\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{2}-\delta _{0})} \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{0}^{s}(s-\tau )^{\beta _{2}-\gamma _{i}-1} \varphi _{\rho _{2}} \bigl(v(\tau ) \bigr) \,d\tau \biggr) \,dH_{i}(s) \\ &{}- \frac{t^{\beta _{1}-1}}{\Delta } \Biggl( \sum_{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} s^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \Biggr) \\ &{}\times \biggl( \frac{1}{\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-s)^{ \beta _{2}-\delta _{0}-1}\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \biggr) \\ =&- \frac{1}{\varGamma (\beta _{1})} \int _{0}^{t} \bigl[t^{\beta _{1}-1}(1-s)^{\beta _{1}-\gamma _{0}-1}-(t-s)^{\beta _{1}-1} \bigr]\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \frac{1}{\varGamma (\beta _{1})} \int _{t}^{1} t^{\beta _{1}-1}(1-s)^{\beta _{1}-\gamma _{0}-1} \varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}+ \frac{1}{\varGamma (\beta _{1})} \int _{0}^{1} t^{\beta _{1}-1}(1-s)^{\beta _{1}-\gamma _{0}-1} \varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \frac{t^{\beta _{1}-1}\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{1}-\gamma _{0})\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-s)^{\beta _{1}-\gamma _{0}-1} \varphi _{ \rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}+ \frac{t^{\beta _{1}-1}}{\Delta } \Biggl( \sum_{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} s^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \Biggr) \\ &{}\times \Biggl( \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} \biggl( \int _{0}^{s}(s-\tau )^{\beta _{1}-\delta _{i}-1} \varphi _{\rho _{1}} \bigl(u(\tau ) \bigr) \,d\tau \biggr) \,dK_{i}(s) \Biggr) \\ &{}+ \frac{t^{\beta _{1}-1}\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{2}-\delta _{0})} \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{\tau }^{1} (s-\tau )^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \biggr)\varphi _{\rho _{2}} \bigl(v(\tau ) \bigr) \,d\tau \\ &{}- \frac{t^{\beta _{1}-1}}{\Delta \varGamma (\beta _{2}-\delta _{0})} \sum_{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{0}^{1} \tau ^{ \beta _{2}-\gamma _{i}-1} \,dH_{i}(\tau ) \biggr) \\ &{}\times(1-s)^{\beta _{2}- \delta _{0}-1}\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ =&- \int _{0}^{1} g_{1}(t,s)\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds+ \frac{1}{\varGamma (\beta _{1})} \int _{0}^{1} t^{\beta _{1}-1}(1-s)^{\beta _{1}-\gamma _{0}-1} \varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \frac{t^{\beta _{1}-1}\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{1}-\gamma _{0})\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-s)^{\beta _{1}-\gamma _{0}-1} \varphi _{ \rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}+ \frac{t^{\beta _{1}-1}}{\Delta } \Biggl( \sum_{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} s^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \Biggr) \\ &{}\times \Biggl( \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} \biggl( \int _{0}^{s}(s-\tau )^{\beta _{1}-\delta _{i}-1} \varphi _{\rho _{1}} \bigl(u(\tau ) \bigr) \,d\tau \biggr) \,dK_{i}(s) \Biggr) \\ &{}- \frac{t^{\beta _{1}-1}\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{2}-\delta _{0})} \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \biggl[- \int _{0}^{1} \biggl( \int _{s}^{1}(\tau -s)^{\beta _{2}-\gamma _{i}-1} \,dH_{i}( \tau ) \biggr)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{}+ \int _{0}^{1} \biggl( \int _{0}^{1} \tau ^{\beta _{2}-\gamma _{i}-1}(1-s)^{\beta _{2}-\delta _{0}-1} \,dH_{i}( \tau ) \biggr)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \biggr], \quad \forall t \in [0,1]. \end{aligned}$$

Then we deduce

$$\begin{aligned} x(t) =&- \int _{0}^{1} g_{1}(t,s)\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \frac{1}{\Delta } \Biggl( \sum_{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} s^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \Biggr) \Biggl( \sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} s^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \Biggr) \\ &{}\times \frac{1}{\varGamma (\beta _{1})} \int _{0}^{1} t^{\beta _{1}-1}(1-s)^{\beta _{1}-\gamma _{0}-1} \varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}+ \frac{t^{\beta _{1}-1}}{\Delta } \Biggl( \sum_{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} s^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \Biggr) \\ &{}\times \Biggl( \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} \biggl( \int _{0}^{s}(s-\tau )^{\beta _{1}-\delta _{i}-1} \varphi _{\rho _{1}} \bigl(u(\tau ) \bigr) \,d\tau \biggr) \,dK_{i}(s) \Biggr) \\ &{}- \frac{t^{\beta _{1}-1}\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{2}-\delta _{0})} \sum_{i=1}^{p} \int _{0}^{1} \biggl( \int _{0}^{1} g_{2i}(\tau ,s) \,dH_{i}(\tau ) \biggr) \varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ =&- \int _{0}^{1} g_{1}(t,s)\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds- \frac{t^{\beta _{1}-1}}{\Delta } \Biggl( \sum _{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} s^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \Biggr) \\ &{}\times \Biggl[ \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \biggl( \int _{0}^{1} \tau ^{\beta _{1}-\delta _{i}-1} \,dK_{i}(\tau ) \biggr) \biggl( \int _{0}^{1} (1-s)^{\beta _{1}-\gamma _{0}-1}\varphi _{ \rho _{1}} \bigl(u(s) \bigr) \,ds \biggr) \\ &{}- \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} \biggl( \int _{0}^{s}(s-\tau )^{\beta _{1}-\delta _{i}-1} \varphi _{\rho _{1}} \bigl(u(\tau ) \bigr) \,d\tau \biggr) \,dK_{i}(s) \Biggr] \\ &{}- \int _{0}^{1} \Biggl( \frac{t^{\beta _{1}-1}\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{2}-\delta _{0})} \sum_{i=1}^{p} \int _{0}^{1} g_{2i}(\tau ,s) \,dH_{i}(\tau ) \Biggr)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ =&- \int _{0}^{1} g_{1}(t,s)\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds- \frac{t^{\beta _{1}-1}}{\Delta } \Biggl( \sum _{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} s^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \Biggr) \\ &{}\times \Biggl\{ \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \biggl[ \int _{0}^{1} \biggl( \int _{0}^{1} \tau ^{\beta _{1}-\delta _{i}-1}(1-s)^{ \beta _{1}-\gamma _{0}-1} \,dK_{i}(\tau ) \biggr)\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \int _{0}^{1} \biggl( \int _{s}^{1}(\tau -s)^{\beta _{1}-\delta _{i}-1} \,dK_{i}(\tau ) \biggr)\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \biggr] \Biggr\} \\ &{}- \int _{0}^{1} \Biggl( \frac{t^{\beta _{1}-1}\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{2}-\delta _{0})} \sum_{i=1}^{p} \int _{0}^{1} g_{2i}(\tau ,s) \,dH_{i}(\tau ) \Biggr)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ =&- \int _{0}^{1} g_{1}(t,s)\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds- \frac{t^{\beta _{1}-1}}{\Delta } \Biggl( \sum _{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} s^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \Biggr) \\ &{}\times \Biggl[ \sum_{i=1}^{q} \int _{0}^{1} \biggl( \int _{0}^{1} g_{1i}(\tau ,s) \,dK_{i}(\tau ) \biggr)\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \Biggr] \\ &{}- \int _{0}^{1} \Biggl( \frac{t^{\beta _{1}-1}\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{2}-\delta _{0})} \sum_{i=1}^{p} \int _{0}^{1} g_{2i}(\tau ,s) \,dH_{i}(\tau ) \Biggr)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ =&- \int _{0}^{1}{\mathcal{G}}_{1}(t,s) \varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds- \int _{0}^{1}{\mathcal{G}}_{2}(t,s) \varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds, \quad \forall t\in [0,1], \end{aligned}$$

where \(g_{1}\), \(g_{1i}\), \(i=1, \ldots ,q\), \(g_{2i}\), \(i=1,\ldots ,p\), are given by (17), and \({\mathcal{G}}_{1}\) and \({\mathcal{G}}_{2}\) are given by (16).

For \(y(t)\) we find

$$\begin{aligned} y(t) =& \frac{t^{\beta _{2}-1}\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})} \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} \biggl( \int _{0}^{s} (s-\tau )^{\beta _{1}-\delta _{i}-1} \varphi _{\rho _{1}} \bigl(u(\tau ) \bigr) \,d\tau \biggr) \,dK_{i}(s) \\ &{}- \frac{t^{\beta _{2}-1}}{\Delta } \Biggl( \sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} s^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \Biggr) \\ &{}\times \biggl( \frac{1}{\varGamma (\beta _{1}-\gamma _{0})} \int _{0}^{1} (1-s)^{ \beta _{1}-\gamma _{0}-1}\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \biggr) \\ &{}+ \frac{1}{\varGamma (\beta _{2})} \int _{0}^{t} (t-s)^{\beta _{2}-1}\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{} - \frac{t^{\beta _{2}-1}\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-s)^{\beta _{2}-\delta _{0}-1}\varphi _{ \rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{}+ \frac{t^{\beta _{2}-1}}{\Delta } \Biggl( \sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} s^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \Biggr) \\ &{}\times \Biggl( \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{0}^{s} (s-\tau )^{\beta _{2}-\gamma _{i}-1} \varphi _{\rho _{2}} \bigl(v(\tau ) \bigr) \,d\tau \biggr) \,dH_{i}(s) \Biggr) \\ =& \frac{t^{\beta _{2}-1}\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})} \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} \biggl( \int _{\tau }^{1}(s-\tau )^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \biggr)\varphi _{\rho _{1}} \bigl(u(\tau ) \bigr) \,d\tau \\ &{}- \frac{t^{\beta _{2}-1}}{\Delta } \sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})\varGamma (\beta _{1}-\gamma _{0})} \\ &{}\times \int _{0}^{1} \biggl( \int _{0}^{1} \tau ^{ \beta _{1}-\delta _{i}-1}(1-s)^{\beta _{1}-\gamma _{0}-1} \,dK_{i}( \tau ) \biggr)\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \frac{1}{\varGamma (\beta _{2})} \int _{0}^{t} \bigl[t^{ \beta _{2}-1}(1-s)^{\beta _{2}-\delta _{0}-1}-(t-s)^{\beta _{2}-1} \bigr] \varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{}- \frac{1}{\varGamma (\beta _{2})} \int _{t}^{1} t^{\beta _{2}-1}(1-s)^{\beta _{2}-\delta _{0}-1} \varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{}+ \frac{1}{\varGamma (\beta _{2})} \int _{0}^{1} t^{\beta _{2}-1}(1-s)^{\beta _{2}-\delta _{0}-1} \varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{}- \frac{t^{\beta _{2}-1}\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-s)^{\beta _{2}-\delta _{0}-1}\varphi _{ \rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{}+ \frac{t^{\beta _{2}-1}}{\Delta } \Biggl( \sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} s^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \Biggr) \\ &{}\times \Biggl( \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{0}^{s} (s-\tau )^{\beta _{2}-\gamma _{i}-1} \varphi _{\rho _{2}} \bigl(v(\tau ) \bigr) \,d\tau \biggr) \,dH_{i}(s) \Biggr) \\ = &\frac{t^{\beta _{2}-1}\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})} \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} \biggl( \int _{s}^{1} (\tau -s)^{\beta _{1}-\delta _{i}-1} \,dK_{i}( \tau ) \biggr)\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \frac{t^{\beta _{2}-1}\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})} \sum_{i=1}^{q} \frac{1}{\varGamma (\beta _{1}-\delta _{i})} \\ &{}\times \int _{0}^{1} \biggl( \int _{0}^{1} \tau ^{\beta _{1}-\delta _{i}-1}(1-s)^{ \beta _{1}-\gamma _{0}-1} \,dK_{i}( \tau ) \biggr)\varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \int _{0}^{1} g_{2}(t,s)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds+ \frac{1}{\varGamma (\beta _{2})} \int _{0}^{1} t^{\beta _{2}-1}(1-s)^{\beta _{2}-\delta _{0}-1} \varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{}- \frac{t^{\beta _{2}-1}\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})\varGamma (\beta _{2}-\delta _{0})} \int _{0}^{1}(1-s)^{\beta _{2}-\delta _{0}-1}\varphi _{ \rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{}+ \frac{t^{\beta _{2}-1}}{\Delta } \Biggl( \sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} s^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \Biggr) \\ &{}\times \Biggl( \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{0}^{s} (s-\tau )^{\beta _{2}-\gamma _{i}-1} \varphi _{\rho _{2}} \bigl(v(\tau ) \bigr) \,d\tau \biggr) \,dH_{i}(s) \Biggr),\quad \forall t\in [0,1]. \end{aligned}$$

Hence we obtain

$$\begin{aligned} y(t) =&- \frac{t^{\beta _{2}-1}\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})} \sum_{i=1}^{q} \int _{0}^{1} \biggl( \int _{0}^{1} g_{1i}(\tau ,s) \,dK_{i}(\tau ) \biggr) \varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \int _{0}^{1} g_{2}(t,s)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{} - \frac{1}{\Delta } \Biggl( \sum_{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1}s^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \Biggr) \Biggl( \sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} s^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \Biggr) \\ &{}\times \frac{1}{\varGamma (\beta _{2})} \int _{0}^{1} t^{\beta _{2}-1}(1-s)^{\beta _{2}-\delta _{0}-1} \varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{}+ \frac{t^{\beta _{2}-1}}{\Delta } \Biggl( \sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} s^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \Biggr) \\ &{}\times \Biggl( \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{0}^{s} (s-\tau )^{\beta _{2}-\gamma _{i}-1} \varphi _{\rho _{2}} \bigl(v(\tau ) \bigr) \,d\tau \biggr) \,dH_{i}(s) \Biggr) \\ =&- \frac{t^{\beta _{2}-1}\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})} \sum_{i=1}^{q} \int _{0}^{1} \biggl( \int _{0}^{1} g_{1i}(\tau ,s) \,dK_{i}(\tau ) \biggr) \varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \int _{0}^{1} g_{2}(t,s)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds - \frac{t^{\beta _{2}-1}}{\Delta } \Biggl( \sum _{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} s^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \Biggr) \\ &{}\times \Biggl[ \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{0}^{1} \tau ^{\beta _{2}-\gamma _{i}-1} (1-s)^{ \beta _{2}-\delta _{0}-1}\,dH_{i}(\tau ) \biggr)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{}- \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} \biggl( \int _{\tau }^{1}(s-\tau )^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \biggr)\varphi _{\rho _{2}} \bigl(v(\tau ) \bigr) \,d\tau \Biggr] \\ =&- \frac{t^{\beta _{2}-1}\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})} \sum_{i=1}^{q} \int _{0}^{1} \biggl( \int _{0}^{1} g_{1i}(\tau ,s) \,dK_{i}(\tau ) \biggr) \varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \int _{0}^{1} g_{2}(t,s)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds - \frac{t^{\beta _{2}-1}}{\Delta } \Biggl( \sum _{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} s^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \Biggr) \\ &{}\times \Biggl\{ \sum_{i=1}^{p} \frac{1}{\varGamma (\beta _{2}-\gamma _{i})} \biggl[ \int _{0}^{1} \biggl( \int _{0}^{1} \tau ^{\beta _{2}-\gamma _{i}-1} (1-s)^{\beta _{2}-\delta _{0}-1}\,dH_{i}(\tau ) \biggr) \varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \\ &{}- \int _{0}^{1} \biggl( \int _{s}^{1} (\tau -s)^{\beta _{2}-\gamma _{i}-1} \,dH_{i}( \tau ) \biggr)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \biggr] \Biggr\} \\ =&- \frac{t^{\beta _{2}-1}\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})} \sum_{i=1}^{q} \int _{0}^{1} \biggl( \int _{0}^{1} g_{1i}(\tau ,s) \,dK_{i}(\tau ) \biggr) \varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds \\ &{}- \int _{0}^{1} g_{2}(t,s)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds- \frac{t^{\beta _{2}-1}}{\Delta } \Biggl( \sum _{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} s^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \Biggr) \\ &{}\times \Biggl[ \sum_{i=1}^{p} \int _{0}^{1} \biggl( \int _{0}^{1} g_{2i}(\tau ,s) \,dH_{i}(\tau ) \biggr)\varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds \Biggr] \\ =&- \int _{0}^{1} {\mathcal{G}}_{3}(t,s) \varphi _{\rho _{1}} \bigl(u(s) \bigr) \,ds- \int _{0}^{1} {\mathcal{G}}_{4}(t,s) \varphi _{\rho _{2}} \bigl(v(s) \bigr) \,ds,\quad \forall t\in [0,1], \end{aligned}$$

where \(g_{1i}\), \(i=1,\ldots ,q\), \(g_{2}\), \(g_{2i}\), \(i=1,\ldots ,p\), are given by (17), and \({\mathcal{G}}_{3}\) and \({\mathcal{G}}_{4}\) are given by (16). □

Therefore, by (8), (9) and (15) we deduce the following lemma.

Lemma 2.3

If\(\Delta \neq0\), then the unique solution\((x,y)\in C[0,1]\times C[0,1]\)of problem (3), (2) is given by

$$ \textstyle\begin{cases} x(t)= \int _{0}^{1}{\mathcal{G}}_{1}(t,\nu )\varphi _{\rho _{1}}(I_{0+}^{ \alpha _{1}}\psi (\nu )) \,d\nu + \int _{0}^{1}{\mathcal{G}}_{2}(t, \nu )\varphi _{\rho _{2}}(I_{0+}^{\alpha _{2}}\chi (\nu )) \,d\nu , \quad \forall t\in [0,1], \\ y(t)= \int _{0}^{1} {\mathcal{G}}_{3}(t,\nu )\varphi _{\rho _{1}}(I_{0+}^{ \alpha _{1}}\psi (\nu )) \,d\nu + \int _{0}^{1} {\mathcal{G}}_{4}(t, \nu )\varphi _{\rho _{2}}(I_{0+}^{\alpha _{2}}\chi (\nu )) \,d\nu , \quad \forall t\in [0,1]. \end{cases} $$
(18)

In the next lemma, we present some properties of the functions \(g_{1}\), \(g_{2}\), \(g_{1i}\), \(i=1,\ldots ,q\), and \(g_{2i}\), \(i=1,\ldots ,p\).

Lemma 2.4

([19])

The functions\(g_{1}\), \(g_{2}\), \(g_{1i}\), \(i=1,\ldots ,q\), \(g_{2i}\), \(i=1,\ldots ,p\), given by (17) have the properties:

  1. (a)

    The functions\(g_{1}, g_{2}, g_{1i}, i=1,\ldots ,q\), \(g_{2i}, i=1,\ldots ,p\), are continuous on\([0,1]\times [0,1]\); \(g_{1}(t,\nu )\ge 0\), \(g_{2}(t,\nu )\ge 0\), \(g_{1i}(t,\nu )\ge 0\), \(i=1, \ldots ,q\), \(g_{2i}(t,\nu )\ge 0\), \(i=1,\ldots ,p\), for all\((t,\nu )\in [0,1]\times [0,1]\); \(g_{1}(t,\nu )>0\), \(g_{2}(t,\nu )>0\), \(g_{1i}(t,\nu )>0\), \(i=1,\ldots ,q\), \(g_{2i}(t,\nu )>0\), \(i=1,\ldots ,p\), for all\((t,\nu )\in (0,1)\times (0,1)\);

  2. (b)

    \(g_{1}(t,\nu )\le h_{1}(\nu )\)for all\((t,\nu )\in [0,1]\times [0,1]\), where\(h_{1}(\nu )=\frac{1}{\varGamma (\beta _{1})}(1-\nu )^{ \beta _{1}-\gamma _{0}-1}(1-(1-\nu )^{\gamma _{0}})\), \(\forall \nu \in [0,1]\);

  3. (c)

    \(g_{1}(t,\nu )\ge t^{\beta _{1}-1}h_{1}(\nu )\)for all\((t,\nu )\in [0,1]\times [0,1]\);

  4. (d)

    \(g_{2}(t,\nu )\le h_{2}(\nu )\)for all\((t,\nu )\in [0,1]\times [0,1]\), where\(h_{2}(\nu )=\frac{1}{\varGamma (\beta _{2})}(1-\nu )^{ \beta _{2}-\delta _{0}-1}(1-(1-\nu )^{\delta _{0}})\), \(\forall \nu \in [0,1]\);

  5. (e)

    \(g_{2}(t,\nu )\ge t^{\beta _{2}-1}h_{2}(\nu )\)for all\((t,\nu )\in [0,1]\times [0,1]\);

  6. (f)

    \(g_{1i}(t,\nu )\le \frac{1}{\varGamma (\beta _{1}-\delta _{i})}t^{\beta _{1}- \delta _{i}-1}(1-\nu )^{\beta _{1}-\gamma _{0}-1}\)for all\((t,\nu )\in [0,1]\times [0,1]\), \(i=1,\ldots ,q\);

  7. (g)

    \(g_{1i}(t,\nu )\ge t^{\beta _{1}-\delta _{i}-1}h_{1i}(\nu )\)for all\((t,\nu )\in [0,1]\times [0,1]\), where\(h_{1i}(\nu )=\frac{1}{\varGamma (\beta _{1}-\delta _{i})}(1-\nu )^{ \beta _{1}-\gamma _{0}-1}(1-(1-\nu )^{\gamma _{0}-\delta _{i}})\), \(\forall \nu \in [0,1]\), \(i=1,\ldots ,q\);

  8. (h)

    \(g_{2i}(t,\nu )\le \frac{1}{\varGamma (\beta _{2}-\gamma _{i})}t^{\beta _{2}- \gamma _{i}-1}(1-\nu )^{\beta _{2}-\delta _{0}-1}\)for all\((t,\nu )\in [0,1]\times [0,1]\), \(i=1,\ldots ,p\);

  9. (i)

    \(g_{2i}(t,\nu )\ge t^{\beta _{2}-\gamma _{i}-1}h_{2i}(\nu )\)for all\((t,\nu )\in [0,1]\times [0,1]\), where\(h_{2i}(\nu )=\frac{1}{\varGamma (\beta _{2}-\gamma _{i})}(1-\nu )^{ \beta _{2}-\delta _{0}-1}(1-(1-\nu )^{\delta _{0}-\gamma _{i}})\), \(\forall \nu \in [0,1]\), \(i=1,\ldots ,p\).

By using Lemma 2.4, we obtain the following properties for the functions \({\mathcal{G}}_{i}\), \(i=1,\ldots ,4\).

Lemma 2.5

Assume that\(\Delta >0\), \(H_{i}\), \(i=1,\ldots ,p\), \(K_{i}\), \(i=1,\ldots ,q\), are nondecreasing functions. Then the functions\({\mathcal{G}}_{i}\), \(i=1,\ldots ,4\), given by (16) have the properties:

  1. (a)

    \({\mathcal{G}}_{i}:[0,1]\times [0,1]\to [0,\infty )\), \(i=1,\ldots ,4\), are continuous functions;

  2. (b)

    \({\mathcal{G}}_{1}(t,\nu )\le {\mathcal{J}}_{1}(\nu )\)for all\((t,\nu )\in [0,1]\times [0,1]\), where

    $$\begin{aligned} {\mathcal{J}}_{1}(\nu ) =&h_{1}(\nu )+ \frac{1}{\Delta } \Biggl(\sum_{i=1}^{p} \frac{\varGamma (\beta _{2})}{\varGamma (\beta _{2}-\gamma _{i})} \int _{0}^{1} s^{\beta _{2}-\gamma _{i}-1} \,dH_{i}(s) \Biggr) \\ &{}\times\Biggl(\sum_{i=1}^{q} \int _{0}^{1} g_{1i}(\tau ,\nu ) \,dK_{i}(\tau ) \Biggr), \quad \forall \nu \in [0,1]; \end{aligned}$$
  3. (c)

    \({\mathcal{G}}_{1}(t,\nu )\ge t^{\beta _{1}-1}{\mathcal{J}}_{1}(\nu )\)for all\((t,\nu )\in [0,1]\times [0,1]\);

  4. (d)

    \({\mathcal{G}}_{2}(t,\nu )\le {\mathcal{J}}_{2}(\nu )\), for all\((t,\nu )\in [0,1]\times [0,1]\), where

    $$ {\mathcal{J}}_{2}(\nu )=\frac{\varGamma (\beta _{2})}{\Delta \varGamma (\beta _{2}-\delta _{0})} \sum _{i=1}^{p} \int _{0}^{1} g_{2i}(\tau , \nu ) \,dH_{i}(\tau ), \quad \forall \nu \in [0,1]; $$
  5. (e)

    \({\mathcal{G}}_{2}(t,\nu )=t^{\beta _{1}-1}{\mathcal{J}}_{2}(\nu )\)for all\((t,\nu )\in [0,1]\times [0,1]\);

  6. (f)

    \({\mathcal{G}}_{3}(t,\nu )\le {\mathcal{J}}_{3}(\nu )\)for all\((t,\nu )\in [0,1]\times [0,1]\), where

    $$ {\mathcal{J}}_{3}(\nu )=\frac{\varGamma (\beta _{1})}{\Delta \varGamma (\beta _{1}-\gamma _{0})} \sum _{i=1}^{q} \int _{0}^{1} g_{1i}(\tau , \nu ) \,dK_{i}(\tau ), \quad \forall \nu \in [0,1]; $$
  7. (g)

    \({\mathcal{G}}_{3}(t,\nu )=t^{\beta _{2}-1}{\mathcal{J}}_{3}(\nu )\)for all\((t,\nu )\in [0,1]\times [0,1]\);

  8. (h)

    \({\mathcal{G}}_{4}(t,\nu )\le {\mathcal{J}}_{4}(\nu )\)for all\((t,\nu )\in [0,1]\times [0,1]\), where

    $$\begin{aligned} {\mathcal{J}}_{4}(\nu ) =&h_{2}(\nu )+ \frac{1}{\Delta } \Biggl(\sum_{i=1}^{q} \frac{\varGamma (\beta _{1})}{\varGamma (\beta _{1}-\delta _{i})} \int _{0}^{1} s^{\beta _{1}-\delta _{i}-1} \,dK_{i}(s) \Biggr) \\ &{}\times\Biggl(\sum_{i=1}^{p} \int _{0}^{1} g_{2i}(\tau ,\nu ) \,dH_{i}(\tau ) \Biggr),\quad \forall \nu \in [0,1]; \end{aligned}$$
  9. (i)

    \({\mathcal{G}}_{4}(t,\nu )\ge t^{\beta _{2}-1}{\mathcal{J}}_{4}(\nu )\), for all\((t,\nu )\in [0,1]\times [0,1]\).

Existence of positive solutions

In this section we present sufficient conditions for the functions f and g, and intervals for the parameters λ and μ such that problem (1), (2) has at least one positive solution.

We give now the basic assumptions that we will use in the main results.

\((I1)\):

\(\alpha _{1}, \alpha _{2}\in (0,1]\), \(\beta _{1}\in (n-1,n]\), \(\beta _{2}\in (m-1,m]\), \(n, m\in \mathbb{N}\), \(n, m\ge 3\), \(p, q\in \mathbb{N}\), \(\gamma _{i}\in \mathbb{R}\) for all \(i=0,1,\ldots ,p\), \(0\le \gamma _{1}<\gamma _{2}<\cdots <\gamma _{p}\le \delta _{0}< \beta _{2}-1\), \(\delta _{0}\ge 1\), \(\delta _{i}\in \mathbb{R}\) for all \(i=0,1,\ldots ,q\), \(0\le \delta _{1}<\delta _{2}<\cdots <\delta _{q}\le \gamma _{0}< \beta _{1}-1\), \(\gamma _{0}\ge 1\), \(H_{i}:[0,1]\to \mathbb{R}\), \(i=1,\ldots ,p\), and \(K_{j}:[0,1]\to \mathbb{R}\), \(j=1,\ldots ,q\), are nondecreasing functions, \(\exists i_{0}\in \{1,\ldots ,p\}\), such that \(H_{i_{0}}(1)>H_{i_{0}}(0)\), \(\exists j_{0}\in \{1,\ldots ,q\}\), such that \(K_{j_{0}}(1)>K_{j_{0}}(0)\), \(\lambda >0\), \(\mu >0\), \(\Delta >0\) (Δ is given by (10)), \(\varrho _{i}>1\), \(\varphi _{\varrho _{i}}(s)=|s|^{\varrho _{i}-2}s\), \(\varphi _{\varrho _{i}}^{-1}=\varphi _{\rho _{i}}\), \(\rho _{i}=\frac{\varrho _{i}}{\varrho _{i}-1}\), \(i=1,2\).

\((I2)\):

The functions \(f, g:[0,1]\times \mathbb{R}_{+}\times \mathbb{R}_{+}\to \mathbb{R}_{+}\) are continuous.

For \([\theta _{1},\theta _{2}]\subset [0,1]\) with \(0<\theta _{1}<\theta _{2}\le 1\), we introduce the following extreme limits:

$$\begin{aligned}& {\mathfrak{f}}_{0}^{s}= \limsup_{ \substack{x+y\to 0^{+}\\x,y\ge 0}} \max_{t\in [0,1]} \frac{f(t,x,y)}{\varphi _{\varrho _{1}}(x+y)}, \qquad { \mathfrak{g}}_{0}^{s}= \limsup _{ \substack{x+y\to 0^{+}\\x,y\ge 0}} \max_{t\in [0,1]} \frac{g(t,x,y)}{\varphi _{\varrho _{2}}(x+y)}, \\& {\mathfrak{f}}_{0}^{i}= \liminf_{ \substack{x+y\to 0^{+}\\x,y\ge 0}} \min_{t\in [\theta _{1}, \theta _{2}]} \frac{f(t,x,y)}{\varphi _{\varrho _{1}}(x+y)},\qquad {\mathfrak{g}}_{0}^{i}= \liminf_{\substack{x+y\to 0^{+}\\x,y\ge 0}} \min_{t\in [\theta _{1},\theta _{2}]} \frac{g(t,x,y)}{\varphi _{\varrho _{2}}(x+y)}, \\& {\mathfrak{f}}_{\infty }^{s}= \limsup_{ \substack{x+y\to \infty\\x,y\ge 0}} \max_{t\in [0,1]} \frac{f(t,x,y)}{\varphi _{\varrho _{1}}(x+y)},\qquad { \mathfrak{g}}_{\infty }^{s}= \limsup_{ \substack{x+y\to \infty \\x,y\ge 0}} \max_{t\in [0,1]} \frac{g(t,x,y)}{\varphi _{\varrho _{2}}(x+y)}, \\& {\mathfrak{f}}_{\infty }^{i}= \liminf_{ \substack{x+y\to \infty \\x,y\ge 0}} \min_{t\in [ \theta _{1},\theta _{2}]} \frac{f(t,x,y)}{\varphi _{\varrho _{1}}(x+y)},\qquad {\mathfrak{g}}_{ \infty }^{i}= \liminf_{ \substack{x+y\to \infty \\x,y\ge 0}} \min_{t\in [ \theta _{1},\theta _{2}]} \frac{g(t,x,y)}{\varphi _{\varrho _{2}}(x+y)}. \end{aligned}$$

By using Lemma 2.3 (Eqs. (18)), \((x,y)\) is a solution of problem (1), (2) if and only if \((x,y)\) is a solution of the following nonlinear system of integral equations:

$$ \textstyle\begin{cases} x(t)=\lambda ^{\rho _{1}-1} \int _{0}^{1}{\mathcal{G}}_{1}(t, \nu )\varphi _{\rho _{1}}(I_{0+}^{\alpha _{1}}f(\nu ,x(\nu ),y(\nu ))) \,d\nu \\ \hphantom{x(t)={}}{} +\mu ^{\rho _{2}-1} \int _{0}^{1} {\mathcal{G}}_{2}(t, \nu )\varphi _{\rho _{2}}(I_{0+}^{\alpha _{2}}g(\nu ,x(\nu ),y(\nu ))) \,d\nu ,\quad t\in [0,1], \\ y(t)=\lambda ^{\rho _{1}-1} \int _{0}^{1}{\mathcal{G}}_{3}(t, \nu )\varphi _{\rho _{1}}(I_{0+}^{\alpha _{1}}f(\nu ,x(\nu ),y(\nu ))) \,d\nu \\ \hphantom{y(t)={}}{} +\mu ^{\rho _{2}-1} \int _{0}^{1} {\mathcal{G}}_{4}(t, \nu )\varphi _{\rho _{2}}(I_{0+}^{\alpha _{2}}g(\nu ,x(\nu ),y(\nu ))) \,d\nu , \quad t\in [0,1]. \end{cases} $$

We consider the Banach space \({\mathcal{X}}=C[0,1]\) with the supremum norm

$$ \|x\|=\sup_{t\in [0,1]}\bigl|x(t)\bigr|, $$

and the Banach space \({\mathcal{Y}}={\mathcal{X}}\times {\mathcal{X}}\) with the norm \(\|(x,y)\|_{\mathcal{Y}}=\|x\|+\|y\|\). We define the cones

$$\begin{aligned}& {\mathcal{P}}_{1}= \bigl\{ x\in { \mathcal{X}}, x(t)\ge t^{\beta _{1}-1} \Vert x \Vert , \forall t\in [0,1] \bigr\} \subset {\mathcal{X}}, \\& {\mathcal{P}}_{2}= \bigl\{ y\in {\mathcal{X}}, y(t)\ge t^{\beta _{2}-1} \Vert y \Vert , \forall t\in [0,1] \bigr\} \subset { \mathcal{X}}, \end{aligned}$$

and \({\mathcal{P}}={\mathcal{P}}_{1}\times {\mathcal{P}}_{2}\subset {\mathcal{Y}}\).

We also define the operators \({\mathcal{A}}_{1}, {\mathcal{A}}_{2}:{\mathcal{Y}}\to {\mathcal{X}}\) and \({\mathcal{A}}:{\mathcal{Y}}\to {\mathcal{Y}}\) by

$$\begin{aligned}& \begin{aligned} {\mathcal{A}}_{1}(x,y) (t)={}&\lambda ^{\rho _{1}-1} \int _{0}^{1}{ \mathcal{G}}_{1}(t, \nu )\varphi _{\rho _{1}} \bigl(I_{0+}^{\alpha _{1}}f \bigl( \nu ,x( \nu ),y(\nu ) \bigr) \bigr) \,d\nu \\ &{}+\mu ^{\rho _{2}-1} \int _{0}^{1} {\mathcal{G}}_{2}(t, \nu )\varphi _{\rho _{2}} \bigl(I_{0+}^{\alpha _{2}}g \bigl( \nu ,x(\nu ),y(\nu ) \bigr) \bigr) \,d\nu , \quad t\in [0,1], \end{aligned} \\& \begin{aligned} {\mathcal{A}}_{2}(x,y) (t)={}&\lambda ^{\rho _{1}-1} \int _{0}^{1}{ \mathcal{G}}_{3}(t, \nu )\varphi _{\rho _{1}} \bigl(I_{0+}^{\alpha _{1}}f \bigl( \nu ,x( \nu ),y(\nu ) \bigr) \bigr) \,d\nu \\ &{}+\mu ^{\rho _{2}-1} \int _{0}^{1} {\mathcal{G}}_{4}(t, \nu )\varphi _{\rho _{2}} \bigl(I_{0+}^{\alpha _{2}}g \bigl( \nu ,x(\nu ),y(\nu ) \bigr) \bigr) \,d\nu ,\quad t\in [0,1], \end{aligned} \end{aligned}$$

and \({\mathcal{A}}(x,y)=({\mathcal{A}}_{1}(x,y),{\mathcal{A}}_{2}(x,y))\), \((x,y)\in {\mathcal{Y}}\). The pair \((x,y)\) is a solution of problem (1), (2) if and only if \((x,y)\) is a fixed point of the operator \({\mathcal{A}}\).

Lemma 3.1

If\((I1)\)and\((I2)\)hold, then\({\mathcal{A}}:{\mathcal{P}}\to {\mathcal{P}}\)is a completely continuous operator.

Proof

Let \((x,y)\in {\mathcal{P}}\) be an arbitrary element. Because \({\mathcal{A}}_{1}(x,y)\) and \({\mathcal{A}}_{2}(x,y)\) satisfy the problem (3), (2) for \(\psi (t)=\lambda f(t,x(t),y(t))\) and \(\chi (t)=\mu g(t,x(t),y(t))\), \(t\in [0,1]\), then by Lemma 2.5 we obtain

$$\begin{aligned}& \begin{aligned} {\mathcal{A}}_{1}(x,y) (t)\le{}& \lambda ^{\rho _{1}-1} \int _{0}^{1}{ \mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \bigl(I_{0+}^{\alpha _{1}}f \bigl(\nu ,x( \nu ),y(\nu ) \bigr) \bigr) \,d\nu \\ &{}+\mu ^{\rho _{2}-1} \int _{0}^{1} {\mathcal{J}}_{2}( \nu )\varphi _{\rho _{2}} \bigl(I_{0+}^{\alpha _{2}}g \bigl( \nu ,x(\nu ),y(\nu ) \bigr) \bigr) \,d\nu , \quad \forall t\in [0,1], \end{aligned} \\& \begin{aligned} {\mathcal{A}}_{2}(x,y) (t)\le{}& \lambda ^{\rho _{1}-1} \int _{0}^{1}{ \mathcal{J}}_{3}( \nu )\varphi _{\rho _{1}} \bigl(I_{0+}^{\alpha _{1}}f \bigl( \nu ,x( \nu ),y(\nu ) \bigr) \bigr) \,d\nu \\ &{}+\mu ^{\rho _{2}-1} \int _{0}^{1} {\mathcal{J}}_{4}( \nu )\varphi _{\rho _{2}} \bigl(I_{0+}^{\alpha _{2}}g \bigl( \nu ,x(\nu ),y(\nu ) \bigr) \bigr) \,d\nu , \quad \forall t\in [0,1], \end{aligned} \end{aligned}$$

and then

$$\begin{aligned}& \begin{aligned} \bigl\Vert {\mathcal{A}}_{1}(x,y) \bigr\Vert \le{}& \lambda ^{\rho _{1}-1} \int _{0}^{1}{ \mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \bigl(I_{0+}^{\alpha _{1}}f \bigl(\nu ,x( \nu ),y(\nu ) \bigr) \bigr) \,d\nu \\ &{}+\mu ^{\rho _{2}-1} \int _{0}^{1} {\mathcal{J}}_{2}( \nu )\varphi _{\rho _{2}} \bigl(I_{0+}^{\alpha _{2}}g \bigl( \nu ,x(\nu ),y(\nu ) \bigr) \bigr) \,d\nu , \end{aligned} \\& \begin{aligned} \bigl\Vert {\mathcal{A}}_{2}(x,y) \bigr\Vert \le {}&\lambda ^{\rho _{1}-1} \int _{0}^{1}{ \mathcal{J}}_{3}( \nu )\varphi _{\rho _{1}} \bigl(I_{0+}^{\alpha _{1}}f \bigl( \nu ,x( \nu ),y(\nu ) \bigr) \bigr) \,d\nu \\ &{}+\mu ^{\rho _{2}-1} \int _{0}^{1} {\mathcal{J}}_{4}( \nu )\varphi _{\rho _{2}} \bigl(I_{0+}^{\alpha _{2}}g \bigl( \nu ,x(\nu ),y(\nu ) \bigr) \bigr) \,d\nu . \end{aligned} \end{aligned}$$

Hence by using again Lemma 2.5, we deduce

$$\begin{aligned}& \begin{aligned} {\mathcal{A}}_{1}(x,y) (t)\ge{}& \lambda ^{\rho _{1}-1} \int _{0}^{1}t^{ \beta _{1}-1}{ \mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \bigl(I_{0+}^{\alpha _{1}}f \bigl( \nu ,x(\nu ),y(\nu ) \bigr) \bigr) \,d\nu \\ &{}+\mu ^{\rho _{2}-1} \int _{0}^{1} t^{\beta _{1}-1}{ \mathcal{J}}_{2}(\nu )\varphi _{\rho _{2}} \bigl(I_{0+}^{\alpha _{2}}g \bigl(\nu ,x( \nu ),y(\nu ) \bigr) \bigr) \,d\nu \\ \ge{}& t^{\beta _{1}-1} \bigl\Vert { \mathcal{A}}_{1}(x,y) \bigr\Vert , \quad \forall t\in [0,1], \end{aligned} \\& \begin{aligned} {\mathcal{A}}_{2}(x,y) (t)\ge{}& \lambda ^{\varrho _{1}-1} \int _{0}^{1}t^{ \beta _{2}-1}{ \mathcal{J}}_{3}(\nu )\varphi _{\rho _{1}} \bigl(I_{0+}^{\alpha _{1}}f \bigl( \nu ,x(\nu ),y(\nu ) \bigr) \bigr) \,d\nu \\ &{}+\mu ^{\rho _{2}-1} \int _{0}^{1} t^{\beta _{2}-1}{ \mathcal{J}}_{4}(\nu )\varphi _{\rho _{2}} \bigl(I_{0+}^{\alpha _{2}}g \bigl(\nu ,x( \nu ),y(\nu ) \bigr) \bigr) \,d\nu \\ \ge{}& t^{\beta _{2}-1} \bigl\Vert {\mathcal{A}}_{2}(x,y) \bigr\Vert ,\quad \forall t\in [0,1]. \end{aligned} \end{aligned}$$

Therefore \({\mathcal{A}}(x,y)=({\mathcal{A}}_{1}(x,y),{\mathcal{A}}_{2}(x,y))\in {\mathcal{P}}\), and then \({\mathcal{A}}({\mathcal{P}})\subset {\mathcal{P}}\). By using the continuity of the functions \(f, g, {\mathcal{G}}_{i}, i=1,\ldots,4\), and the Ascoli–Arzela theorem, we can prove that \({\mathcal{A}}_{1}\) and \({\mathcal{A}}_{2}\) are completely continuous operators (compact operators, that is, they map bounded sets into relatively compact sets, and continuous), and then \({\mathcal{A}}\) is a completely continuous operator. □

For \([\theta _{1},\theta _{2}]\subset [0,1]\) with \(0<\theta _{1}<\theta _{2}\le 1\), we denote

$$ \begin{aligned} &L_{1}= \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}} \int _{0}^{1}\nu ^{\alpha _{1}(\rho _{1}-1)}{ \mathcal{J}}_{1}( \nu ) \,d\nu , \\ &L_{2}= \frac{1}{(\varGamma (\alpha _{2}+1))^{\rho _{2}-1}} \int _{0}^{1}\nu ^{\alpha _{2}(\rho _{2}-1)}{ \mathcal{J}}_{2}( \nu ) \,d\nu , \\ &L_{3}= \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}} \int _{0}^{1}\nu ^{\alpha _{1}(\rho _{1}-1)}{ \mathcal{J}}_{3}( \nu ) \,d\nu , \\ &L_{4}= \frac{1}{(\varGamma (\alpha _{2}+1))^{\rho _{2}-1}} \int _{0}^{1}\nu ^{\alpha _{2}(\rho _{2}-1)}{ \mathcal{J}}_{4}( \nu ) \,d\nu , \\ &\widetilde{L}_{1}= \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}} \int _{ \theta _{1}}^{\theta _{2}}(\nu -\theta _{1})^{\alpha _{1}(\rho _{1}-1)}{ \mathcal{J}}_{1}(\nu ) \,d\nu , \\ &\widetilde{L}_{2}= \frac{1}{(\varGamma (\alpha _{2}+1))^{\rho _{2}-1}} \int _{ \theta _{1}}^{\theta _{2}}(\nu -\theta _{1})^{\alpha _{2}(\rho _{2}-1)}{ \mathcal{J}}_{2}(\nu ) \,d\nu , \\ &\widetilde{L}_{3}= \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}} \int _{ \theta _{1}}^{\theta _{2}}(\nu -\theta _{1})^{\alpha _{1}(\rho _{1}-1)}{ \mathcal{J}}_{3}(\nu ) \,d\nu , \\ &\widetilde{L}_{4}= \frac{1}{(\varGamma (\alpha _{2}+1))^{\rho _{2}-1}} \int _{ \theta _{1}}^{\theta _{2}}(\nu -\theta _{1})^{\alpha _{2}(\rho _{2}-1)}{ \mathcal{J}}_{4}(\nu ) \,d\nu , \end{aligned} $$
(19)

where \({\mathcal{J}}_{i}\), \(i=1,\ldots,4\), are defined in Lemma 2.5.

For \({\mathfrak{f}}_{0}^{s}, {\mathfrak{g}}_{0}^{s}, {\mathfrak{f}}_{ \infty }^{i}, {\mathfrak{g}}_{\infty }^{i}\in (0,\infty )\) and numbers \(c_{1}, c_{2}\in [0,1]\), \(c_{3}, c_{4}\in (0,1)\), \(a\in [0,1]\) and \(b\in (0,1)\), we introduce the numbers

$$\begin{aligned}& A=\max \biggl\{ \frac{1}{{\mathfrak{f}}_{\infty }^{i}} \varphi _{\varrho _{1}} \biggl( \frac{ac_{1}}{\zeta \zeta _{1}\widetilde{L}_{1}} \biggr), \frac{1}{{\mathfrak{f}}_{\infty }^{i}} \varphi _{ \varrho _{1}} \biggl( \frac{(1-a)c_{2}}{\zeta \zeta _{2}\widetilde{L}_{3}} \biggr) \biggr\} , \\& B=\min \biggl\{ \frac{1}{{\mathfrak{f}}_{0}^{s}}\varphi _{ \varrho _{1}} \biggl( \frac{b c_{3}}{L_{1}} \biggr), \frac{1}{{\mathfrak{f}}_{0}^{s}}\varphi _{\varrho _{1}} \biggl( \frac{(1-b)c_{4}}{L_{3}} \biggr) \biggr\} , \\& C=\max \biggl\{ \frac{1}{{\mathfrak{g}}_{\infty }^{i}} \varphi _{\varrho _{2}} \biggl( \frac{a(1-c_{1})}{\zeta \zeta _{1}\widetilde{L}_{2}} \biggr), \frac{1}{{\mathfrak{g}}_{\infty }^{i}}\varphi _{ \varrho _{2}} \biggl( \frac{(1-a)(1-c_{2})}{\zeta \zeta _{2}\widetilde{L}_{4}} \biggr) \biggr\} , \\& D=\min \biggl\{ \frac{1}{{\mathfrak{g}}_{0}^{s}}\varphi _{ \varrho _{2}} \biggl( \frac{b (1-c_{3})}{L_{2}} \biggr), \frac{1}{{\mathfrak{g}}_{0}^{s}}\varphi _{\varrho _{2}} \biggl( \frac{(1-b)(1-c_{4})}{L_{4}} \biggr) \biggr\} , \\& E=\min \biggl\{ \frac{1}{{\mathfrak{f}}_{0}^{s}}\varphi _{ \varrho _{1}} \biggl( \frac{b}{L_{1}} \biggr), \frac{1}{{\mathfrak{f}}_{0}^{s}} \varphi _{\varrho _{1}} \biggl( \frac{1-b}{L_{3}} \biggr) \biggr\} , \\& F=\min \biggl\{ \frac{1}{{\mathfrak{g}}_{0}^{s}}\varphi _{ \varrho _{2}} \biggl( \frac{b}{L_{2}} \biggr), \frac{1}{{\mathfrak{g}}_{0}^{s}} \varphi _{\varrho _{2}} \biggl( \frac{1-b}{L_{4}} \biggr) \biggr\} , \end{aligned}$$

where \(\zeta _{1}=\theta _{1}^{\beta _{1}-1}\), \(\zeta _{2}=\theta _{1}^{\beta _{2}-1}\), \(\zeta =\min \{\zeta _{1},\zeta _{2}\}\).

Theorem 3.1

Assume that\((I1)\)and\((I2)\)hold, \([\theta _{1},\theta _{2}]\subset [0,1]\)with\(0<\theta _{1}<\theta _{2}\le 1\), \(c_{1}, c_{2}\in [0,1]\), \(c_{3}, c_{4}\in (0,1)\), \(a\in [0,1]\)and\(b\in (0,1)\).

  1. (1)

    If\({\mathfrak{f}}_{0}^{s}, {\mathfrak{g}}_{0}^{s}, {\mathfrak{f}}_{ \infty }^{i}, {\mathfrak{g}}_{\infty }^{i}\in (0,\infty )\), \(A< B\)and\(C< D\), then, for each\(\lambda \in (A,B)\)and\(\mu \in (C,D)\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  2. (2)

    If\({\mathfrak{f}}_{0}^{s}=0\), \({\mathfrak{g}}_{0}^{s}, {\mathfrak{f}}_{\infty }^{i}, { \mathfrak{g}}_{\infty }^{i}\in (0,\infty )\)and\(C< F\), then, for each\(\lambda \in (A,\infty )\)and\(\mu \in (C,F)\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  3. (3)

    If\({\mathfrak{g}}_{0}^{s}=0\), \({\mathfrak{f}}_{0}^{s}, {\mathfrak{f}}_{\infty }^{i}, { \mathfrak{g}}_{\infty }^{i}\in (0,\infty )\)and\(A< E\), then, for each\(\lambda \in (A,E)\)and\(\mu \in (C,\infty )\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  4. (4)

    If\({\mathfrak{f}}_{0}^{s}={\mathfrak{g}}_{0}^{s}=0\), \({\mathfrak{f}}_{\infty }^{i}, {\mathfrak{g}}_{\infty }^{i}\in (0, \infty )\), then, for each\(\lambda \in (A,\infty )\)and\(\mu \in (C,\infty )\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  5. (5)

    If\({\mathfrak{f}}_{0}^{s}, {\mathfrak{g}}_{0}^{s}\in (0,\infty )\)and at least one of\({\mathfrak{f}}_{\infty }^{i}\), \({\mathfrak{g}}_{\infty }^{i}\)is ∞, then, for each\(\lambda \in (0,B)\)and\(\mu \in (0,D)\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  6. (6)

    If\({\mathfrak{f}}_{0}^{s}=0\), \({\mathfrak{g}}_{0}^{s}\in (0,\infty )\)and at least one of\({\mathfrak{f}}_{\infty }^{i}\), \({\mathfrak{g}}_{\infty }^{i}\)is ∞, then, for each\(\lambda \in (0,\infty )\)and\(\mu \in (0,F)\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  7. (7)

    If\({\mathfrak{f}}_{0}^{s}\in (0,\infty )\), \({\mathfrak{g}}_{0}^{s}=0\)and at least one of\({\mathfrak{f}}_{\infty }^{i}\), \({\mathfrak{g}}_{\infty }^{i}\)is ∞, then, for each\(\lambda \in (0,E)\)and\(\mu \in (0,\infty )\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  8. (8)

    If\({\mathfrak{f}}_{0}^{s}={\mathfrak{g}}_{0}^{s}=0\)and at least one of\({\mathfrak{f}}_{\infty }^{i}\), \({\mathfrak{g}}_{\infty }^{i}\)is ∞, then, for each\(\lambda \in (0,\infty )\)and\(\mu \in (0,\infty )\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

Proof

We consider the cone \({\mathcal{P}}\subset {\mathcal{Y}}\) and the operators \({\mathcal{A}}_{1}\), \({\mathcal{A}}_{2}\) and \({\mathcal{A}}\). The proofs of the above cases are similar, and for this reason we will prove only two cases, namely (1) and (6).

Case (1). We have \({\mathfrak{f}}_{0}^{s}, {\mathfrak{g}}_{0}^{s}, {\mathfrak{f}}_{ \infty }^{i}, {\mathfrak{g}}_{\infty }^{i}\in (0,\infty )\), \(A< B\) and \(C< D\). Let \(\lambda \in (A,B)\) and \(\mu \in (C,D)\). We consider \(\epsilon >0\) such that \(\epsilon <{\mathfrak{f}}_{\infty }^{i}\), \(\epsilon <{\mathfrak{g}}_{\infty }^{i}\) and

$$\begin{aligned}& \max \biggl\{ \frac{1}{{\mathfrak{f}}_{\infty }^{i}-\epsilon }\varphi _{\varrho _{1}} \biggl( \frac{a c_{1}}{\zeta \zeta _{1}\widetilde{L}_{1}} \biggr), \frac{1}{{\mathfrak{f}}_{\infty }^{i}-\epsilon } \varphi _{\varrho _{1}} \biggl( \frac{(1-a)c_{2}}{\zeta \zeta _{2}\widetilde{L}_{3}} \biggr) \biggr\} \\& \quad \le \lambda\le \min \biggl\{ \frac{1}{{\mathfrak{f}}_{0}^{s}+\epsilon }\varphi _{\varrho _{1}} \biggl( \frac{bc_{3}}{L_{1}} \biggr), \frac{1}{{\mathfrak{f}}_{0}^{s}+\epsilon }\varphi _{\varrho _{1}} \biggl( \frac{(1-b)c_{4}}{L_{3}} \biggr) \biggr\} , \\& \max \biggl\{ \frac{1}{{\mathfrak{g}}_{\infty }^{i}-\epsilon }\varphi _{\varrho _{2}} \biggl( \frac{a(1-c_{1})}{\zeta \zeta _{1}\widetilde{L}_{2}} \biggr), \frac{1}{{\mathfrak{g}}_{\infty }^{i}-\epsilon } \varphi _{\varrho _{2}} \biggl( \frac{(1-a)(1-c_{2})}{\zeta \zeta _{2}\widetilde{L}_{4}} \biggr) \biggr\} \\& \quad \le \mu\le \min \biggl\{ \frac{1}{{\mathfrak{g}}_{0}^{s}+\epsilon }\varphi _{\varrho _{2}} \biggl( \frac{b(1-c_{3})}{L_{2}} \biggr), \frac{1}{{\mathfrak{g}}_{0}^{s}+\epsilon }\varphi _{\varrho _{2}} \biggl( \frac{(1-b)(1-c_{4})}{L_{4}} \biggr) \biggr\} . \end{aligned}$$

By using \((I2)\) and the definitions of \({\mathfrak{f}}_{0}^{s}\) and \({\mathfrak{g}}_{0}^{s}\), we find that there exists \(r_{1}>0\) such that

$$ f(t,x,y)\le \bigl({\mathfrak{f}}_{0}^{s}+\epsilon \bigr)\varphi _{\varrho _{1}}(x+y),\qquad g(t,x,y)\le \bigl({\mathfrak{g}}_{0}^{s}+ \epsilon \bigr)\varphi _{ \varrho _{2}}(x+y), $$

for all \(t\in [0,1]\) and \(x, y\ge 0\), \(x+y\le r_{1}\).

We introduce the set \(\varLambda _{1}=\{(x,y)\in {\mathcal{Y}}, \|(x,y)\|_{\mathcal{Y}}< r_{1}\}\). Let \((x,y)\in {\mathcal{P}}\cap \partial \varLambda _{1}\), that is \((x,y)\in {\mathcal{P}}\) with \(\|(x,y)\|_{\mathcal{Y}}=r_{1}\) or \(\|x\|+\|y\|=r_{1}\). Then \(x(t)+y(t)\le r_{1}\) for all \(t\in [0,1]\), and by Lemma 2.5, we deduce

$$\begin{aligned}& {\mathcal{A}}_{1}(x,y) (t) \\& \quad \le \lambda ^{\rho _{1}-1} \int _{0}^{1} {\mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{0}^{\nu }(\nu -s)^{ \alpha _{1}-1}f \bigl(s,x(s),y(s) \bigr) \,ds \biggr) \,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1} \int _{0}^{1} {\mathcal{J}}_{2}( \nu ) \varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{0}^{\nu }(\nu -s)^{ \alpha _{2}-1}g \bigl(s,x(s),y(s) \bigr) \,ds \biggr) \,d\nu \\& \quad \le \lambda ^{\rho _{1}-1} \int _{0}^{1} {\mathcal{J}}_{1}( \nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{0}^{\nu } (\nu -s)^{ \alpha _{1}-1} \bigl({ \mathfrak{f}}_{0}^{s}+\epsilon \bigr)\varphi _{\varrho _{1}} \bigl(x(s)+y(s) \bigr) \,ds \biggr)\,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1} \int _{0}^{1}{\mathcal{J}}_{2}(\nu ) \varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{0}^{\nu }(\nu -s)^{ \alpha _{2}-1} \bigl({ \mathfrak{g}}_{0}^{s}+\epsilon \bigr)\varphi _{\varrho _{2}} \bigl(x(s)+y(s) \bigr) \,ds \biggr)\,d\nu \\& \quad \le \lambda ^{\rho _{1}-1}\varphi _{\rho _{1}} \bigl({ \mathfrak{f}}_{0}^{s}+ \epsilon \bigr) \int _{0}^{1} {\mathcal{J}}_{1}(\nu )\varphi _{ \rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{0}^{\nu } (\nu -s)^{\alpha _{1}-1}\varphi _{ \varrho _{1}} \bigl( \Vert x \Vert + \Vert y \Vert \bigr) \,ds \biggr)\,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1}\varphi _{\rho _{2}} \bigl({\mathfrak{g}}_{0}^{s}+ \epsilon \bigr) \int _{0}^{1} {\mathcal{J}}_{2}( \nu )\varphi _{ \rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{0}^{\nu } (\nu -s)^{\alpha _{2}-1}\varphi _{ \varrho _{2}} \bigl( \Vert x \Vert + \Vert y \Vert \bigr) \,ds \biggr)\,d\nu \\& \quad = \lambda ^{\rho _{1}-1}\varphi _{\rho _{1}} \bigl({ \mathfrak{f}}_{0}^{s}+ \epsilon \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{0}^{1} {\mathcal{J}}_{1}( \nu ) \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}}{ \nu }^{\alpha _{1}(\rho _{1}-1)} \,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1}\varphi _{\rho _{2}} \bigl({\mathfrak{g}}_{0}^{s}+ \epsilon \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{0}^{1} {\mathcal{J}}_{2}( \nu ) \frac{1}{(\varGamma (\alpha _{2}+1))^{\rho _{2}-1}}{ \nu }^{\alpha _{2}(\rho _{2}-1)} \,d\nu \\& \quad = \bigl[\varphi _{\rho _{1}} \bigl(\lambda \bigl({\mathfrak{f}}_{0}^{s}+ \epsilon \bigr) \bigr)L_{1}+ \varphi _{\rho _{2}} \bigl(\mu \bigl({\mathfrak{g}}_{0}^{s}+\epsilon \bigr) \bigr)L_{2} \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \\& \quad \le \bigl[bc_{3}+b(1-c_{3}) \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}=b \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}},\quad \forall t\in [0,1]. \end{aligned}$$

Therefore \(\|{\mathcal{A}}_{1}(x,y)\|\le b\|(x,y)\|_{\mathcal{Y}}\).

In a similar manner we obtain

$$\begin{aligned} {\mathcal{A}}_{2}(x,y) (t) \le& \lambda ^{\rho _{1}-1}\varphi _{\rho _{1}} \bigl({ \mathfrak{f}}_{0}^{s}+ \epsilon \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{0}^{1} {\mathcal{J}}_{3}( \nu ) \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}}{\nu }^{\alpha _{1}( \rho _{1}-1)} \,d\nu \\ &{}+\mu ^{\rho _{2}-1}\varphi _{\rho _{2}} \bigl({\mathfrak{g}}_{0}^{s}+ \epsilon \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{0}^{1} {\mathcal{J}}_{4}( \nu ) \frac{1}{(\varGamma (\alpha _{2}+1))^{\rho _{2}-1}}{ \nu }^{\alpha _{2}(\rho _{2}-1)} \,d\nu \\ =& \bigl[\varphi _{\rho _{1}} \bigl(\lambda \bigl({\mathfrak{f}}_{0}^{s}+ \epsilon \bigr) \bigr)L_{3}+ \varphi _{\rho _{2}} \bigl(\mu \bigl({\mathfrak{g}}_{0}^{s}+\epsilon \bigr) \bigr)L_{4} \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \\ \le& \bigl[(1-b)c_{4}+(1-b) (1-c_{4}) \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}=(1-b) \bigl\Vert (x,y) \bigr\Vert _{ \mathcal{Y}},\quad \forall t\in [0,1]. \end{aligned}$$

Then \(\|{\mathcal{A}}_{2}(x,y)\|\le (1-b)\|(x,y)\|_{\mathcal{Y}}\).

Therefore, for \((x,y)\in {\mathcal{P}}\cap \partial \varLambda _{1}\), we conclude

$$ \bigl\Vert {\mathcal{A}}(x,y) \bigr\Vert _{\mathcal{Y}}= \bigl\Vert {\mathcal{A}}_{1}(x,y) \bigr\Vert + \bigl\Vert {\mathcal{A}}_{2}(x,y) \bigr\Vert \le b \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}+(1-b) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}= \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}. $$
(20)

By the definitions of \({\mathfrak{f}}_{\infty }^{i}\) and \({\mathfrak{g}}_{\infty }^{i}\), we find that there exists \(r_{2}'>0\) such that

$$ f(t,x,y)\ge \bigl({\mathfrak{f}}_{\infty }^{i}-\epsilon \bigr)\varphi _{ \varrho _{1}}(x+y),\qquad g(t,x,y)\ge \bigl({\mathfrak{g}}_{\infty }^{i}- \epsilon \bigr)\varphi _{\varrho _{2}}(x+y), $$

for all \(t\in [\theta _{1},\theta _{2}]\) and \(x, y\ge 0\), \(x+y\ge r_{2}'\).

We consider \(r_{2}=\max \{2r_{1},r_{2}'/\zeta \}\) and we introduce the set \(\varLambda _{2}=\{(x,y)\in {\mathcal{Y}}, \|(x,y)\|_{\mathcal{Y}}< r_{2}\}\). Then, for \((x,y)\in {\mathcal{P}}\cap \partial \varLambda _{2}\), we deduce

$$ \begin{aligned} x(t)+y(t)&\ge \min_{t\in [\theta _{1},\theta _{2}]}t^{ \beta _{1}-1} \Vert x \Vert + \min_{t\in [\theta _{1},\theta _{2}]}t^{ \beta _{2}-1} \Vert y \Vert =\theta _{1}^{\beta _{1}-1} \Vert x \Vert +\theta _{1}^{ \beta _{2}-1} \Vert y \Vert \\ &=\zeta _{1} \Vert x \Vert +\zeta _{2} \Vert y \Vert \ge \zeta \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}=\zeta r_{2} \ge r_{2}', \quad \forall t\in [\theta _{1},\theta _{2}]. \end{aligned} $$

Therefore, by Lemma 2.5, we obtain

$$\begin{aligned}& {\mathcal{A}}_{1}(x,y) (\theta _{1}) \\& \quad \ge \lambda ^{\rho _{1}-1} \int _{0}^{1}\theta _{1}^{\beta _{1}-1}{ \mathcal{J}}_{1}( \nu )\varphi _{\rho _{1}} \bigl(I_{0+}^{\alpha _{1}}f \bigl(\nu ,x(\nu ),y(\nu ) \bigr) \bigr) \,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1} \int _{0}^{1}\theta _{1}^{\beta _{1}-1}{ \mathcal{J}}_{2}(\nu )\varphi _{\rho _{2}} \bigl(I_{0+}^{\alpha _{2}}g \bigl(\nu ,x( \nu ),y(\nu ) \bigr) \bigr) \,d\nu \\& \quad \ge \lambda ^{\rho _{1}-1}\theta _{1}^{\beta _{1}-1} \int _{\theta _{1}}^{\theta _{2}}{\mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{\theta _{1}}^{\nu }(\nu -s)^{\alpha _{1}-1}f \bigl(s,x(s),y(s) \bigr) \,ds \biggr)\,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1}\theta _{1}^{\beta _{1}-1} \int _{ \theta _{1}}^{\theta _{2}}{\mathcal{J}}_{2}( \nu )\varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{\theta _{1}}^{\nu }(\nu -s)^{\alpha _{2}-1}g \bigl(s,x(s),y(s) \bigr) \,ds \biggr)\,d\nu \\& \quad \ge \lambda ^{\rho _{1}-1}\theta _{1}^{\beta _{1}-1} \int _{\theta _{1}}^{\theta _{2}}{\mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{\theta _{1}}^{\nu }(\nu -s)^{\alpha _{1}-1} \bigl({ \mathfrak{f}}_{ \infty }^{i}-\epsilon \bigr)\varphi _{\varrho _{1}} \bigl(x(s)+y(s) \bigr) \,ds \biggr)\,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1}\theta _{1}^{\beta _{1}-1} \int _{ \theta _{1}}^{\theta _{2}}{\mathcal{J}}_{2}( \nu )\varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{\theta _{1}}^{\nu }(\nu -s)^{\alpha _{2}-1} \bigl({ \mathfrak{g}}_{ \infty }^{i}-\epsilon \bigr)\varphi _{\varrho _{1}} \bigl(x(s)+y(s) \bigr) \,ds \biggr)\,d\nu \\& \quad \ge \lambda ^{\rho _{1}-1}\theta _{1}^{\beta _{1}-1} \int _{\theta _{1}}^{\theta _{2}}{\mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{\theta _{1}}^{\nu }(\nu -s)^{\alpha _{1}-1} \bigl({ \mathfrak{f}}_{ \infty }^{i}-\epsilon \bigr)\varphi _{\varrho _{1}} \bigl(\zeta \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \bigr) \,ds \biggr)\,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1}\theta _{1}^{\beta _{1}-1} \int _{ \theta _{1}}^{\theta _{2}}{\mathcal{J}}_{2}( \nu )\varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{\theta _{1}}^{\nu }(\nu -s)^{\alpha _{2}-1} \bigl({ \mathfrak{g}}_{ \infty }^{i}-\epsilon \bigr)\varphi _{\varrho _{1}} \bigl(\zeta \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \bigr) \,ds \biggr)\,d\nu \\& \quad =\zeta \zeta _{1}\varphi _{\rho _{1}} \bigl(\lambda \bigl({ \mathfrak{f}}_{ \infty }^{i}-\epsilon \bigr) \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{ \theta _{1}}^{\theta _{2}}{\mathcal{J}}_{1}(\nu ) \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}}(\nu -\theta _{1})^{ \alpha _{1}(\rho _{1}-1)}\,d\nu \\& \qquad {}+\zeta \zeta _{1}\varphi _{\rho _{2}} \bigl(\mu \bigl({ \mathfrak{g}}_{\infty }^{i}- \epsilon \bigr) \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{\theta _{1}}^{ \theta _{2}}{\mathcal{J}}_{2}( \nu ) \frac{1}{(\varGamma (\alpha _{2}+1))^{\rho _{2}-1}}(\nu -\theta _{1})^{ \alpha _{2}(\rho _{2}-1)}\,d\nu \\& \quad = \bigl[\zeta \zeta _{1}\varphi _{\rho _{1}} \bigl(\lambda \bigl({\mathfrak{f}}_{ \infty }^{i}-\epsilon \bigr) \bigr) \widetilde{L}_{1}+\zeta \zeta _{1}\varphi _{ \rho _{2}}\bigl(\mu \bigl({\mathfrak{g}}_{\infty }^{i}- \epsilon \bigr)\bigr)\widetilde{L}_{2} \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \\& \quad \ge \bigl[ac_{1}+a(1-c_{1}) \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}=a \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}. \end{aligned}$$

Then \(\|{\mathcal{A}}_{1}(x,y)\|\ge {\mathcal{A}}_{1}(x,y)(\theta _{1})\ge a\|(x,y) \|_{\mathcal{Y}}\).

In a similar manner, we deduce

$$\begin{aligned}& {\mathcal{A}}_{2}(x,y) (\theta _{1}) \\& \quad \ge \zeta \zeta _{2}\varphi _{\rho _{1}} \bigl( \lambda \bigl({\mathfrak{f}}_{\infty }^{i}-\epsilon \bigr) \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{\theta _{1}}^{\theta _{2}}{\mathcal{J}}_{3}( \nu ) \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}}(\nu - \theta _{1})^{\alpha _{1}(\rho _{1}-1)}\,d\nu \\& \qquad {}+\zeta \zeta _{2}\varphi _{\rho _{2}} \bigl(\mu \bigl({ \mathfrak{g}}_{\infty }^{i}- \epsilon \bigr) \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{\theta _{1}}^{ \theta _{2}}{\mathcal{J}}_{4}( \nu ) \frac{1}{(\varGamma (\alpha _{2}+1))^{\rho _{2}-1}}(\nu -\theta _{1})^{ \alpha _{2}(\rho _{2}-1)}\,d\nu \\& \quad = \bigl[\zeta \zeta _{2}\varphi _{\rho _{1}} \bigl(\lambda \bigl({\mathfrak{f}}_{ \infty }^{i}-\epsilon \bigr) \bigr) \widetilde{L}_{3}+\zeta \zeta _{2}\varphi _{ \rho _{2}}\bigl(\mu \bigl({\mathfrak{g}}_{\infty }^{i}- \epsilon \bigr)\bigr)\widetilde{L}_{4} \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \\& \quad \ge \bigl[(1-a)c_{2}+(1-a) (1-c_{2}) \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}=(1-a) \bigl\Vert (x,y) \bigr\Vert _{ \mathcal{Y}}. \end{aligned}$$

Hence \(\|{\mathcal{A}}_{2}(x,y)\|\ge {\mathcal{A}}_{2}(x,y)(\theta _{1})\ge (1-a)\|(x,y) \|_{\mathcal{Y}}\).

Therefore for \((x,y)\in {\mathcal{P}}\cap \partial \varLambda _{2}\) we find

$$ \bigl\Vert {\mathcal{A}}(x,y) \bigr\Vert _{\mathcal{Y}}= \bigl\Vert {\mathcal{A}}_{1}(x,y) \bigr\Vert + \bigl\Vert {\mathcal{A}}_{2}(x,y) \bigr\Vert \ge a \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}+(1-a) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}= \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}. $$
(21)

By Lemma 3.1, Eqs. (20), (21) and the Guo–Krasnosel’skii fixed point theorem, we conclude that \({\mathcal{A}}\) has a fixed point \((x,y)\in {\mathcal{P}}\cap (\overline{\varLambda }_{2}\setminus \varLambda _{1})\) such that \(r_{1}\le \|x\|+\|y\|\le r_{2}\), \(x(t)\ge t^{\beta _{1}-1}\|x\|\), \(y(t)\ge t^{\beta _{2}-1}\|y\|\) for all \(t\in [0,1]\). If \(\|x\|>0\) then \(x(t)>0\) for all \(t\in (0,1]\) and if \(\|y\|>0\) then \(y(t)>0\) for all \(t\in (0,1]\). Therefore \((x,y)\) is a positive solution for problem (1), (2).

Case (6). We have \({\mathfrak{f}}_{0}^{s}=0\), \({\mathfrak{g}}_{0}^{s}\in (0,\infty )\) and \({\mathfrak{f}}_{\infty }^{i}=\infty \). Let \(\lambda \in (0,\infty )\) and \(\mu \in (0,F)\). We choose \(\widetilde{c}_{3}\in (0,1-\varphi _{\rho _{2}}(\mu {\mathfrak{g}}_{0}^{s}) \frac{L_{2}}{b})\) and \(\widetilde{c}_{4}\in (0,1-\varphi _{\rho _{2}}(\mu {\mathfrak{g}}_{0}^{s}) \frac{L_{4}}{1-b})\). The choice of \(\widetilde{c}_{3}\) and \(\widetilde{c}_{4}\) is possible because \(\mu <\frac{1}{{\mathfrak{g}}_{0}^{s}}\varphi _{\varrho _{2}}( \frac{b}{L_{2}})\) and \(\mu <\frac{1}{{\mathfrak{g}}_{0}^{s}}\varphi _{\varrho _{2}}( \frac{1-b}{L_{4}})\). Let \(\epsilon >0\) be such that

$$ \begin{aligned} &\epsilon \varphi _{\varrho _{1}} \biggl( \frac{1}{\zeta \zeta _{1}\widetilde{L}_{1}} \biggr)\le \lambda \le \min \biggl\{ \frac{1}{\epsilon } \varphi _{\varrho _{1}} \biggl( \frac{b\widetilde{c}_{3}}{L_{1}} \biggr), \frac{1}{\epsilon }\varphi _{\varrho _{1}} \biggl( \frac{(1-b)\widetilde{c}_{4}}{L_{3}} \biggr) \biggr\} , \\ &\mu \le \min \biggl\{ \frac{1}{{\mathfrak{g}}_{0}^{s} +\epsilon }\varphi _{\varrho _{2}} \biggl( \frac{b(1-\widetilde{c}_{3})}{L_{2}} \biggr), \frac{1}{{\mathfrak{g}}_{0}^{s}+\epsilon } \biggl( \frac{(1-b)(1-\widetilde{c}_{4})}{L_{4}} \biggr) \biggr\} . \end{aligned} $$

By using \((I2)\) and the definitions of \({\mathfrak{f}}_{0}^{s}\) and \({\mathfrak{g}}_{0}^{s}\), we find that there exists \(r_{1}>0\) such that

$$ f(t,x,y)\le \epsilon \varphi _{\varrho _{1}}(x+y), \qquad g(t,x,y)\le \bigl({ \mathfrak{g}}_{0}^{s}+\epsilon \bigr)\varphi _{\varrho _{2}}(x+y) $$

for all \(t\in [0,1]\) and \(x, y\ge 0\), \(x+y\le r_{1}\).

We introduce the set \(\varLambda _{1}=\{(x,y)\in {\mathcal{Y}}, \|(x,y)\|_{\mathcal{Y}}< r_{1}\}\). As in the proof of Case (1), for any \((x,y)\in {\mathcal{P}}\cap \partial \varLambda _{1}\), we deduce

$$\begin{aligned}& \begin{aligned} {\mathcal{A}}_{1}(x,y) (t)&\le \bigl[ \varphi _{\rho _{1}}(\lambda \epsilon )L_{1}+ \varphi _{\rho _{2}} \bigl(\mu \bigl({\mathfrak{g}}_{0}^{s}+ \epsilon \bigr) \bigr)L_{2} \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \\ &\le \bigl[b\widetilde{c}_{3}+b(1-\widetilde{c}_{3}) \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}=b \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}, \quad \forall t\in [0,1], \end{aligned} \\& \begin{aligned} {\mathcal{A}}_{2}(x,y) (t)&\le \bigl[\varphi _{\rho _{1}}( \lambda \epsilon )L_{3}+ \varphi _{\rho _{2}} \bigl(\mu \bigl({ \mathfrak{g}}_{0}^{s}+\epsilon \bigr) \bigr)L_{4} \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \\ &\le \bigl[(1-b)\widetilde{c}_{4}+(1-b) (1-\widetilde{c}_{4}) \bigr] \bigl\Vert (x,y) \bigr\Vert _{ \mathcal{Y}}=(1-b) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}},\quad \forall t\in [0,1], \end{aligned} \end{aligned}$$

and so \(\|{\mathcal{A}}(x,y)\|_{\mathcal{Y}}\le \|(x,y)\|_{\mathcal{Y}}\).

For the second part, by the definition of \({\mathfrak{f}}_{\infty }^{i}\), we find that there exists \(r_{2}'>0\) such that

$$ f(t,x,y)\ge \frac{1}{\epsilon }\varphi _{\varrho _{1}}(x+y),\quad \forall t\in [ \theta _{1},\theta _{2}], x, y\ge 0, x+y\ge r_{2}'. $$

We consider \(r_{2}=\max \{2 r_{1},r_{2}'/\zeta \}\) and we introduce the set \(\varLambda _{2}=\{(x,y)\in {\mathcal{Y}}, \|(x,y)\|_{\mathcal{Y}}< r_{2}\}\). Then, for any \((x,y)\in {\mathcal{P}}\cap \partial \varLambda _{2}\), we obtain as in Case (1) that \(x(t)+y(t)\ge \zeta \|(x,y)\|_{\mathcal{Y}}=\zeta r_{2}\ge r_{2}'\) for all \(t\in [\theta _{1},\theta _{2}]\).

Hence by Lemma 2.5 we deduce

$$\begin{aligned}& {\mathcal{A}}_{1}(x,y) (\theta _{1}) \\& \quad \ge \lambda ^{\rho _{1}-1} \int _{0}^{1}{\theta }_{1}^{\beta _{1}-1}{ \mathcal{J}}_{1}( \nu )\varphi _{\rho _{1}} \bigl(I_{0+}^{\alpha _{1}}f \bigl(\nu ,x(\nu ),y(\nu ) \bigr) \bigr) \,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1} \int _{0}^{1} \theta _{1}^{\beta _{1}-1}{ \mathcal{J}}_{2}(\nu )\varphi _{\rho _{2}} \bigl(I_{0+}^{\alpha _{2}}g \bigl(\nu ,x( \nu ),y(\nu ) \bigr) \bigr) \,d\nu \\& \quad \ge \lambda ^{\rho _{1}-1} \int _{0}^{1} \theta _{1}^{ \beta _{1}-1}{ \mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \bigl(I_{0+}^{\alpha _{1}}f \bigl( \nu ,x(\nu ),y(\nu ) \bigr) \bigr) \,d\nu \\& \quad \ge \lambda ^{\rho _{1}-1}\theta _{1}^{\beta _{1}-1} \int _{\theta _{1}}^{\theta _{2}}{\mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{\theta _{1}}^{\nu } (\nu -s)^{\alpha _{1}-1}f \bigl(s,x(s),y(s) \bigr) \,ds \biggr)\,d\nu \\& \quad \ge \lambda ^{\rho _{1}-1}\theta _{1}^{\beta _{1}-1} \int _{\theta _{1}}^{\theta _{2}}{\mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{\theta _{1}}^{\nu } (\nu -s)^{\alpha _{1}-1} \frac{1}{\epsilon }\varphi _{\varrho _{1}} \bigl(x(s)+y(s) \bigr)\,ds \biggr) \,d\nu \\& \quad \ge \lambda ^{\rho _{1}-1}\theta _{1}^{\beta _{1}-1} \int _{\theta _{1}}^{\theta _{2}}{\mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{\theta _{1}}^{\nu } (\nu -s)^{\alpha _{1}-1} \frac{1}{\epsilon }\varphi _{\varrho _{1}} \bigl(\zeta \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \bigr) \,ds \biggr)\,d\nu \\& \quad =\zeta \zeta _{1}\varphi _{\rho _{1}} \biggl( \frac{\lambda }{\epsilon } \biggr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{\theta _{1}}^{\theta _{2}}{\mathcal{J}}_{1}(\nu ) \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}}(\nu -\theta _{1})^{ \alpha _{1}(\rho _{1}-1)}\,d\nu \\& \quad =\zeta \zeta _{1}\varphi _{\rho _{1}} \biggl( \frac{\lambda }{\epsilon } \biggr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \widetilde{L}_{1} \ge \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}. \end{aligned}$$

Then we conclude that \(\|{\mathcal{A}}_{1}(x,y)\|\ge {\mathcal{A}}_{1}(x,y)(\theta _{1})\ge \|(x,y)\|_{ \mathcal{Y}}\) and \(\|{\mathcal{A}}(x,y)\|_{\mathcal{Y}}\ge \|{\mathcal{A}}_{1}(x,y)\|\ge \|(x,y)\|_{ \mathcal{Y}}\). Therefore we obtain the conclusion of the theorem. □

Next for \({\mathfrak{f}}_{0}^{i}, {\mathfrak{g}}_{0}^{i}, {\mathfrak{f}}_{ \infty }^{s}, {\mathfrak{g}}_{\infty }^{s}\in (0,\infty )\) and numbers \(c_{1}, c_{2}\in [0,1]\), \(c_{3}, c_{4}\in (0,1)\), \(a\in [0,1]\) and \(b\in (0,1)\), we introduce the numbers

$$\begin{aligned}& \widetilde{A}=\max \biggl\{ \frac{1}{{\mathfrak{f}}_{0}^{i}} \varphi _{\varrho _{1}} \biggl( \frac{ac_{1}}{\zeta \zeta _{1}\widetilde{L}_{1}} \biggr), \frac{1}{{\mathfrak{f}}_{0}^{i}}\varphi _{\varrho _{1}} \biggl( \frac{(1-a)c_{2}}{\zeta \zeta _{2}\widetilde{L}_{3}} \biggr) \biggr\} , \\& \widetilde{B}=\min \biggl\{ \frac{1}{{\mathfrak{f}}_{\infty }^{s}}\varphi _{\varrho _{1}} \biggl( \frac{b c_{3}}{L_{1}} \biggr), \frac{1}{{\mathfrak{f}}_{\infty }^{s}}\varphi _{\varrho _{1}} \biggl( \frac{(1-b)c_{4}}{L_{3}} \biggr) \biggr\} , \\& \widetilde{C}=\max \biggl\{ \frac{1}{{\mathfrak{g}}_{0}^{i}}\varphi _{\varrho _{2}} \biggl( \frac{a(1-c_{1})}{\zeta \zeta _{1}\widetilde{L}_{2}} \biggr), \frac{1}{{\mathfrak{g}}_{0}^{i}}\varphi _{ \varrho _{2}} \biggl( \frac{(1-a)(1-c_{2})}{\zeta \zeta _{2}\widetilde{L}_{4}} \biggr) \biggr\} , \\& \widetilde{D}=\min \biggl\{ \frac{1}{{\mathfrak{g}}_{\infty }^{s}}\varphi _{\varrho _{2}} \biggl( \frac{b (1-c_{3})}{L_{2}} \biggr), \frac{1}{{\mathfrak{g}}_{\infty }^{s}}\varphi _{\varrho _{2}} \biggl( \frac{(1-b)(1-c_{4})}{L_{4}} \biggr) \biggr\} , \\& \widetilde{E}=\min \biggl\{ \frac{1}{{\mathfrak{f}}_{\infty }^{s}}\varphi _{\varrho _{1}} \biggl( \frac{b}{L_{1}} \biggr), \frac{1}{{\mathfrak{f}}_{\infty }^{s}} \varphi _{\varrho _{1}} \biggl( \frac{1-b}{L_{3}} \biggr) \biggr\} , \\& \widetilde{F}=\min \biggl\{ \frac{1}{{\mathfrak{g}}_{\infty }^{s}}\varphi _{\varrho _{2}} \biggl( \frac{b}{L_{2}} \biggr), \frac{1}{{\mathfrak{g}}_{\infty }^{s}} \varphi _{\varrho _{2}} \biggl( \frac{1-b}{L_{4}} \biggr) \biggr\} . \end{aligned}$$

Theorem 3.2

Assume that\((I1)\)and\((I2)\)hold, \([\theta _{1},\theta _{2}]\subset [0,1]\)with\(0<\theta _{1}<\theta _{2}\le 1\), \(c_{1}, c_{2}\in [0,1]\), \(c_{3}, c_{4}\in (0,1)\), \(a\in [0,1]\)and\(b\in (0,1)\).

  1. (1)

    If\({\mathfrak{f}}_{0}^{i}, {\mathfrak{g}}_{0}^{i}, {\mathfrak{f}}_{ \infty }^{s}, {\mathfrak{g}}_{\infty }^{s}\in (0,\infty )\), \(\widetilde{A}<\widetilde{B}\)and\(\widetilde{C}<\widetilde{D}\), then, for each\(\lambda \in (\widetilde{A},\widetilde{B})\)and\(\mu \in (\widetilde{C},\widetilde{D})\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  2. (2)

    If\({\mathfrak{f}}_{0}^{i}, {\mathfrak{g}}_{0}^{i}, {\mathfrak{f}}_{ \infty }^{s}\in (0,\infty )\), \({\mathfrak{g}}_{\infty }^{s}=0\)and\(\widetilde{A}<\widetilde{E}\), then, for each\(\lambda \in (\widetilde{A},\widetilde{E})\)and\(\mu \in (\widetilde{C},\infty )\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  3. (3)

    If\({\mathfrak{f}}_{0}^{i}, {\mathfrak{g}}_{0}^{i}, {\mathfrak{g}}_{ \infty }^{s}\in (0,\infty )\), \({\mathfrak{f}}_{\infty }^{s}=0\)and\(\widetilde{C}<\widetilde{F}\), then, for each\(\lambda \in (\widetilde{A},\infty )\)and\(\mu \in (\widetilde{C},\widetilde{F})\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  4. (4)

    If\({\mathfrak{f}}_{0}^{i}, {\mathfrak{g}}_{0}^{i}\in (0,\infty )\), \({\mathfrak{f}}_{\infty }^{s}={\mathfrak{g}}_{\infty }^{s}=0\), then, for each\(\lambda \in (\widetilde{A},\infty )\)and\(\mu \in (\widetilde{C},\infty )\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  5. (5)

    If\({\mathfrak{f}}_{\infty }^{s}, {\mathfrak{g}}_{\infty }^{s}\in (0, \infty )\)and at least one of\({\mathfrak{f}}_{0}^{i}\), \({\mathfrak{g}}_{0}^{i}\)is ∞, then, for each\(\lambda \in (0,\widetilde{B})\)and\(\mu \in (0,\widetilde{D})\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  6. (6)

    If\({\mathfrak{f}}_{\infty }^{s}\in (0,\infty )\), \({\mathfrak{g}}_{\infty }^{s}=0\)and at least one of\({\mathfrak{f}}_{0}^{i}\), \({\mathfrak{g}}_{0}^{i}\)is ∞, then, for each\(\lambda \in (0,\widetilde{E})\)and\(\mu \in (0,\infty )\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  7. (7)

    If\({\mathfrak{f}}_{\infty }^{s}=0\), \({\mathfrak{g}}_{\infty }^{s}\in (0,\infty )\)and at least one of\({\mathfrak{f}}_{0}^{i}\), \({\mathfrak{g}}_{0}^{i}\)is ∞, then, for each\(\lambda \in (0,\infty )\)and\(\mu \in (0,\widetilde{F})\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

  8. (8)

    If\({\mathfrak{f}}_{\infty }^{s}={\mathfrak{g}}_{\infty }^{s}=0\)and at least one of\({\mathfrak{f}}_{0}^{i}\), \({\mathfrak{g}}_{0}^{i}\)is ∞, then, for each\(\lambda \in (0,\infty )\)and\(\mu \in (0,\infty )\), the problem (1), (2) has at least one positive solution\((x(t),y(t))\), \(t\in [0,1]\).

Proof

We consider again the cone \({\mathcal{P}}\subset {\mathcal{Y}}\) and the operators \({\mathcal{A}}_{1}\), \({\mathcal{A}}_{2}\) and \({\mathcal{A}}\). The proofs of the above cases are similar, and for this reason we will prove only two cases, namely (1) and (6).

Case (1). We have \({\mathfrak{f}}_{0}^{i}, {\mathfrak{g}}_{0}^{i}, {\mathfrak{f}}_{ \infty }^{s}, {\mathfrak{g}}_{\infty }^{s}\in (0,\infty )\), \(\widetilde{A}<\widetilde{B}\) and \(\widetilde{C}<\widetilde{D}\). Let \(\lambda \in (\widetilde{A},\widetilde{B})\) and \(\mu \in (\widetilde{C},\widetilde{D})\). We consider \(\epsilon >0\) such that \(\epsilon <{\mathfrak{f}}_{0}^{i}\), \(\epsilon <{\mathfrak{g}}_{0}^{i}\) and

$$\begin{aligned}& \max \biggl\{ \frac{1}{{\mathfrak{f}}_{0}^{i}-\epsilon } \varphi _{\varrho _{1}} \biggl( \frac{ac_{1}}{\zeta \zeta _{1}\widetilde{L}_{1}} \biggr), \frac{1}{{\mathfrak{f}}_{0}^{i}-\epsilon } \varphi _{ \varrho _{1}} \biggl( \frac{(1-a)c_{2}}{\zeta \zeta _{2}\widetilde{L}_{3}} \biggr) \biggr\} \\& \quad \le \lambda\le \min \biggl\{ \frac{1}{{\mathfrak{f}}_{\infty }^{s}+\epsilon }\varphi _{\varrho _{1}} \biggl( \frac{bc_{3}}{L_{1}} \biggr), \frac{1}{{\mathfrak{f}}_{\infty }^{s}+\epsilon }\varphi _{\varrho _{1}} \biggl( \frac{(1-b)c_{4}}{L_{3}} \biggr) \biggr\} , \\& \max \biggl\{ \frac{1}{{\mathfrak{g}}_{0}^{i}-\epsilon } \varphi _{\varrho _{2}} \biggl( \frac{a(1-c_{1})}{\zeta \zeta _{1}\widetilde{L}_{2}} \biggr), \frac{1}{{\mathfrak{g}}_{0}^{i}-\epsilon }\varphi _{ \varrho _{2}} \biggl( \frac{(1-a)(1-c_{2})}{\zeta \zeta _{2}\widetilde{L}_{4}} \biggr) \biggr\} \\& \quad \le \mu\le \min \biggl\{ \frac{1}{{\mathfrak{g}}_{\infty }^{s}+\epsilon }\varphi _{\varrho _{2}} \biggl( \frac{b(1-c_{3})}{L_{2}} \biggr), \frac{1}{{\mathfrak{g}}_{\infty }^{s}+\epsilon } \biggl( \frac{(1-b)(1-c_{4})}{L_{4}} \biggr) \biggr\} . \end{aligned}$$

By using \((I2)\) and the definitions of \({\mathfrak{f}}_{0}^{i}\) and \({\mathfrak{g}}_{0}^{i}\) we deduce that there exists \(r_{3}>0\) such that

$$ f(t,x,y)\ge \bigl({\mathfrak{f}}_{0}^{i}-\epsilon \bigr)\varphi _{\varrho _{1}}(x+y),\qquad g(t,x,y)\ge \bigl({\mathfrak{g}}_{0}^{i}- \epsilon \bigr)\varphi _{ \varrho _{2}}(x+y), $$

for all \(t\in [\theta _{1},\theta _{2}]\), \(x, y\ge 0\), \(x+y\le r_{3}\).

We introduce the set \(\varLambda _{3}=\{(x,y)\in {\mathcal{Y}}, \|(x,y)\|_{\mathcal{Y}}< r_{3}\}\). Let \((x,y)\in {\mathcal{P}}\cap \partial \varLambda _{3}\), that is \((x,y)\in {\mathcal{P}}\) with \(\|(x,y)\|_{\mathcal{Y}}=r_{3}\) or \(\|x\|+\|y\|=r_{3}\). Then \(x(t)+y(t)\le r_{3}\) for all \(t\in [0,1]\), and \(x(t)+y(t)\ge \zeta \|(x,y)\|_{\mathcal{Y}}\) for all \(t\in [\theta _{1},\theta _{2}]\) (see the proof of Case (1) in Theorem 3.1). So, by Lemma 2.5, we obtain

$$\begin{aligned}& {\mathcal{A}}_{1}(x,y) (\theta _{1}) \\& \quad \ge \lambda ^{\rho _{1}-1}\theta _{1}^{ \beta _{1}-1} \int _{\theta _{1}}^{\theta _{2}}{\mathcal{J}}_{1}( \nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{\theta _{1}}^{\nu } (\nu -s)^{\alpha _{1}-1}f \bigl(s,x(s),y(s) \bigr) \,ds \biggr)\,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1}\theta _{1}^{\beta _{1}-1} \int _{ \theta _{1}}^{\theta _{2}}{\mathcal{J}}_{2}( \nu )\varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{\theta _{1}}^{\nu }(\nu -s)^{\alpha _{2}-1}g \bigl(s,x(s),y(s) \bigr) \,ds \biggr)\,d\nu \\& \quad \ge \lambda ^{\rho _{1}-1}\theta _{1}^{\beta _{1}-1} \int _{\theta _{1}}^{\theta _{2}}{\mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{\theta _{1}}^{\nu }(\nu -s)^{\alpha _{1}-1} \bigl({ \mathfrak{f}}_{0}^{i}- \epsilon \bigr)\varphi _{\varrho _{1}} \bigl(x(s)+y(s) \bigr) \,ds \biggr)\,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1}\theta _{1}^{\beta _{1}-1} \int _{ \theta _{1}}^{\theta _{2}}{\mathcal{J}}_{2}( \nu )\varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{\theta _{1}}^{\nu } (\nu -s)^{\alpha _{2}-1} \bigl({ \mathfrak{g}}_{0}^{i}- \epsilon \bigr)\varphi _{\varrho _{2}} \bigl(x(s)+y(s) \bigr) \,ds \biggr)\,d\nu \\& \quad \ge \lambda ^{\rho _{1}-1}\theta _{1}^{\beta _{1}-1} \int _{\theta _{1}}^{\theta _{2}}{\mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{\theta _{1}}^{\nu } (\nu -s)^{\alpha _{1}-1} \bigl({ \mathfrak{f}}_{0}^{i}- \epsilon \bigr)\varphi _{\varrho _{1}} \bigl(\zeta \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \bigr) \,ds \biggr)\,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1}\theta _{1}^{\beta _{1}-1} \int _{ \theta _{1}}^{\theta _{2}}{\mathcal{J}}_{2}( \nu )\varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{\theta _{1}}^{\nu } (\nu -s)^{\alpha _{2}-1} \bigl({ \mathfrak{g}}_{0}^{i}- \epsilon \bigr)\varphi _{\varrho _{2}} \bigl(\zeta \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \bigr) \,ds \biggr)\,d\nu \\& \quad =\zeta \zeta _{1}\varphi _{\rho _{1}} \bigl(\lambda \bigl({ \mathfrak{f}}_{0}^{i}- \epsilon \bigr) \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{\theta _{1}}^{ \theta _{2}}{\mathcal{J}}_{1}(\nu ) \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}}(\nu -\theta _{1})^{ \alpha _{1}(\rho _{1}-1)} \,d\nu \\& \qquad {}+\zeta \zeta _{1}\varphi _{\rho _{2}} \bigl(\mu \bigl({ \mathfrak{g}}_{0}^{i}- \epsilon \bigr) \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{\theta _{1}}^{ \theta _{2}}J_{2}(\nu ) \frac{1}{(\varGamma (\alpha _{2}+1))^{\rho _{2}-1}}(\nu -\theta _{1})^{ \alpha _{2}(\rho _{2}-1)} \,d\nu \\& \quad = \bigl[\zeta \zeta _{1}\varphi _{\rho _{1}} \bigl(\lambda \bigl({\mathfrak{f}}_{0}^{i}- \epsilon \bigr) \bigr) \widetilde{L}_{1}+\zeta \zeta _{1}\varphi _{\rho _{2}}\bigl(\mu \bigl({ \mathfrak{g}}_{0}^{i}- \epsilon \bigr)\bigr)\widetilde{L}_{2} \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \\& \quad \ge \bigl[ac_{1}+a(1-c_{1}) \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}=a \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}. \end{aligned}$$

Then \(\|{\mathcal{A}}_{1}(x,y)\|\ge {\mathcal{A}}_{1}(x,y)(\theta _{1})\ge a\|(x,y) \|_{\mathcal{Y}}\).

In a similar manner we find

$$\begin{aligned}& {\mathcal{A}}_{2}(x,y) (\theta _{1}) \\& \quad \ge \zeta \zeta _{2}\varphi _{\rho _{1}} \bigl( \lambda \bigl({\mathfrak{f}}_{0}^{i}-\epsilon \bigr) \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{\theta _{1}}^{\theta _{2}}{\mathcal{J}}_{3}( \nu ) \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}}(\nu - \theta _{1})^{\alpha _{1}(\rho _{1}-1)} \,d\nu \\& \qquad {}+\zeta \zeta _{2}\varphi _{\rho _{2}} \bigl(\mu \bigl({ \mathfrak{g}}_{0}^{i}- \epsilon \bigr) \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{\theta _{1}}^{ \theta _{2}}J_{4}(\nu ) \frac{1}{(\varGamma (\alpha _{2}+1))^{\rho _{2}-1}}(\nu -\theta _{1})^{ \alpha _{2}(\rho _{2}-1)} \,d\nu \\& \quad = \bigl[\zeta \zeta _{2}\varphi _{\rho _{1}} \bigl(\lambda \bigl({\mathfrak{f}}_{0}^{i}- \epsilon \bigr) \bigr) \widetilde{L}_{3}+\zeta \zeta _{2}\varphi _{\rho _{2}}\bigl(\mu \bigl({ \mathfrak{g}}_{0}^{i}- \epsilon \bigr)\bigr)\widetilde{L}_{4} \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \\& \quad \ge \bigl[(1-a)c_{2}+(1-a) (1-c_{2}) \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}=(1-a) \bigl\Vert (x,y) \bigr\Vert _{ \mathcal{Y}}. \end{aligned}$$

So, \(\|{\mathcal{A}}_{2}(x,y)\|\ge {\mathcal{A}}_{2}(x,y)(\theta _{1})\ge (1-a)\|(x,y) \|_{\mathcal{Y}}\).

Then, for \((x,y)\in {\mathcal{P}}\cap \partial \varLambda _{3}\), we conclude

$$ \bigl\Vert {\mathcal{A}}(x,y) \bigr\Vert _{\mathcal{Y}}= \bigl\Vert {\mathcal{A}}_{1}(x,y) \bigr\Vert + \bigl\Vert {\mathcal{A}}_{2}(x,y) \bigr\Vert \ge a \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}+(1-a) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}= \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}. $$
(22)

We introduce now the functions \({\mathfrak{f}}^{*}, {\mathfrak{g}}^{*}:[0,1]\times \mathbb{R}_{+} \to \mathbb{R}_{+}\) by \({\mathfrak{f}}^{*}(t,u)=\max_{0\le x+y\le u} f(t,x,y)\), \(g^{*}(t,u)=\max_{0\le x+y\le u}g(t,x,y)\) for all \(t\in [0,1]\) and \(u\in \mathbb{R}_{+}\). Then

$$ f(t,x,y)\le {\mathfrak{f}}^{*}(t,u),\qquad g(t,x,y)\le { \mathfrak{g}}^{*}(t,u),\quad \forall t\in [0,1], x,y\ge 0, x+y\le u. $$

The functions \({\mathfrak{f}}^{*}(t,\cdot )\), \({\mathfrak{g}}^{*}(t,\cdot )\) are nondecreasing for every \(t\in [0,1]\) and they satisfy the conditions

$$ \limsup_{u\to \infty }\max_{t\in [0,1]} \frac{{\mathfrak{f}}^{*}(t,u)}{\varphi _{\varrho _{1}}(u)}={ \mathfrak{f}}_{\infty }^{s},\qquad \limsup _{u\to \infty }\max_{t\in [0,1]} \frac{{\mathfrak{g}}^{*}(t,u)}{\varphi _{\varrho _{2}}(u)}={ \mathfrak{g}}_{\infty }^{s}. $$

Then for \(\epsilon >0\) there exists \(r_{4}'>0\) such that for all \(u\ge r_{4}'\) and \(t\in [0,1]\) we obtain

$$\begin{aligned}& \frac{{\mathfrak{f}}^{*}(t,u)}{\varphi _{\varrho _{1}}(u)}\le \limsup_{u\to \infty } \max_{t\in [0,1]} \frac{f^{*}(t,u)}{\varphi _{\varrho _{1}}(u)}+\epsilon ={ \mathfrak{f}}_{ \infty }^{s}+\epsilon , \\& \frac{{\mathfrak{g}}^{*}(t,u)}{\varphi _{\varrho _{2}}(u)}\le \limsup_{u\to \infty }\max _{t\in [0,1]} \frac{{\mathfrak{g}}^{*}(t,u)}{\varphi _{\varrho _{2}}(u)}+\epsilon ={ \mathfrak{g}}_{\infty }^{s}+\epsilon , \end{aligned}$$

and hence \({\mathfrak{f}}^{*}(t,u)\le ({\mathfrak{f}}_{\infty }^{s}+\epsilon ) \varphi _{\varrho _{1}}(u)\) and \({\mathfrak{g}}^{*}(t,u)\le ({\mathfrak{g}}_{\infty }^{s}+\epsilon ) \varphi _{\varrho _{2}}(u)\).

We consider \(r_{4}=\max \{2r_{3},r_{4}'\}\) and we introduce the set \(\varLambda _{4}=\{(x,y)\in {\mathcal{Y}}, \|(x,y)\|_{\mathcal{Y}}< r_{4}\}\). Let \((x,y)\in {\mathcal{P}}\cap \partial \varLambda _{4}\). By the definitions of \({\mathfrak{f}}^{*}\) and \({\mathfrak{g}}^{*}\) we deduce

$$ \begin{aligned} &f \bigl(t,x(t),y(t) \bigr)\le {\mathfrak{f}}^{*} \bigl(t, \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \bigr), \\ & g \bigl(t,x(t),y(t) \bigr) \le {\mathfrak{g}}^{*} \bigl(t, \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \bigr), \quad \forall t\in [0,1]. \end{aligned} $$
(23)

Then for all \(t\in [0,1]\) we find

$$\begin{aligned}& {\mathcal{A}}_{1}(x,y) (t) \\& \quad \le \lambda ^{\rho _{1}-1} \int _{0}^{1}{ \mathcal{J}}_{1}(\nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{0}^{\nu }(\nu -s)^{ \alpha _{1}-1}f \bigl(s,x(s),y(s) \bigr) \,ds \biggr)\,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1} \int _{0}^{1} {\mathcal{J}}_{2}( \nu ) \varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{0}^{\nu } (\nu -s)^{ \alpha _{2}-1}g \bigl(s,x(s),y(s) \bigr) \,ds \biggr)\,d\nu \\& \quad \le \lambda ^{\rho _{1}-1} \int _{0}^{1} {\mathcal{J}}_{1}( \nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{0}^{\nu } (\nu -s)^{ \alpha _{1}-1}{ \mathfrak{f}}^{*} \bigl(s, \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \bigr) \,ds \biggr)\,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1} \int _{0}^{1}{\mathcal{J}}_{2}(\nu ) \varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{0}^{\nu }(\nu -s)^{ \alpha _{2}-1}{ \mathfrak{g}}^{*} \bigl(s, \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \bigr) \,ds \biggr)\,d\nu \\& \quad \le \lambda ^{\rho _{1}-1} \int _{0}^{1}{\mathcal{J}}_{1}( \nu )\varphi _{\rho _{1}} \biggl( \frac{1}{\varGamma (\alpha _{1})} \int _{0}^{\nu } (\nu -s)^{ \alpha _{1}-1} \bigl({ \mathfrak{f}}_{\infty }^{s}+\epsilon \bigr)\varphi _{ \varrho _{1}} \bigl( \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \bigr) \,ds \biggr)\,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1} \int _{0}^{1}{\mathcal{J}}_{2}(\nu ) \varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{0}^{\nu } (\nu -s)^{ \alpha _{2}-1} \bigl({ \mathfrak{g}}_{\infty }^{s}+\epsilon \bigr)\varphi _{ \varrho _{2}} \bigl( \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \bigr) \,ds \biggr)\,d\nu \\& \quad =\varphi _{\rho _{1}} \bigl(\lambda \bigl({\mathfrak{f}}_{\infty }^{s}+ \epsilon \bigr) \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{0}^{1} \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}}{\nu }^{\alpha _{1}( \rho _{1}-1)}{\mathcal{J}}_{1}(\nu ) \,d\nu \\& \qquad {}+\varphi _{\rho _{2}} \bigl(\mu \bigl({\mathfrak{g}}_{\infty }^{s}+ \epsilon \bigr) \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{0}^{1} \frac{1}{(\varGamma (\alpha _{2}+1))^{\rho _{2}-1}}{\nu }^{\alpha _{2}( \rho _{2}-1)}{\mathcal{J}}_{2}(\nu ) \,d\nu \\& \quad = \bigl[\varphi _{\rho _{1}} \bigl(\lambda \bigl({\mathfrak{f}}_{\infty }^{s}+ \epsilon \bigr) \bigr)L_{1}+\varphi _{\rho _{2}} \bigl(\mu \bigl({\mathfrak{g}}_{\infty }^{s}+ \epsilon \bigr) \bigr)L_{2} \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \\& \quad \le \bigl[bc_{3}+b(1-c_{3}) \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}=b \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}, \end{aligned}$$

and so \(\|{\mathcal{A}}_{1}(x,y)\|\le b\|(x,y)\|_{\mathcal{Y}}\).

In a similar manner, we obtain

$$ \begin{aligned} {\mathcal{A}}_{2}(x,y) (t)\le{}& \varphi _{\rho _{1}} \bigl(\lambda \bigl({\mathfrak{f}}_{ \infty }^{s}+ \epsilon \bigr) \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{0}^{1} \frac{1}{(\varGamma (\alpha _{1}+1))^{\rho _{1}-1}}{\nu }^{ \alpha _{1}(\rho _{1}-1)}{\mathcal{J}}_{3}(\nu ) \,d\nu \\ &{}+\varphi _{\rho _{2}} \bigl(\mu \bigl({\mathfrak{g}}_{\infty }^{s}+ \epsilon \bigr) \bigr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \int _{0}^{1} \frac{1}{(\varGamma (\alpha _{2}+1))^{\rho _{2}-1}}{\nu }^{\alpha _{2}( \rho _{2}-1)}{\mathcal{J}}_{4}(\nu ) \,d\nu \\ ={}& \bigl[\varphi _{\rho _{1}} \bigl(\lambda \bigl({\mathfrak{f}}_{\infty }^{s}+ \epsilon \bigr) \bigr)L_{3}+\varphi _{\rho _{2}} \bigl(\mu \bigl({\mathfrak{g}}_{\infty }^{s}+ \epsilon \bigr) \bigr)L_{4} \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \\ \le{}& \bigl[(1-b)c_{4}+(1-b) (1-c_{4}) \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}=(1-b) \bigl\Vert (x,y) \bigr\Vert _{ \mathcal{Y}}, \end{aligned} $$

and then \(\|{\mathcal{A}}_{2}(x,y)\|\le (1-b)\|(x,y)\|_{\mathcal{Y}}\).

Therefore, for \((x,y)\in {\mathcal{P}}\cap \partial \varLambda _{4}\), we deduce

$$ \bigl\Vert {\mathcal{A}}(x,y) \bigr\Vert _{\mathcal{Y}}= \bigl\Vert {\mathcal{A}}_{1}(x,y) \bigr\Vert + \bigl\Vert {\mathcal{A}}_{2}(x,y) \bigr\Vert \le b \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}+(1-b) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}= \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}. $$
(24)

By using Lemma 3.1, Eqs. (23), (24) and the Guo–Krasnosel’skii fixed point theorem, we conclude that \({\mathcal{A}}\) has a fixed point \((x,y)\in {\mathcal{P}}\cap (\overline{\varLambda }_{4}\setminus \varLambda _{3})\), which is a positive solution of problem (1), (2).

Case (6). We have \({\mathfrak{f}}_{\infty }^{s}\in (0,\infty )\), \({\mathfrak{g}}_{\infty }^{s}=0\) and \({\mathfrak{g}}_{0}^{i}=\infty \). Let \(\lambda \in (0,\widetilde{E})\) and \(\mu \in (0,\infty )\). We choose \(\widetilde{c}_{3}\in (\varphi _{\rho _{1}}(\lambda {\mathfrak{f}}_{ \infty }^{s})\frac{L_{1}}{b},1)\) and \(\widetilde{c}_{4}\in (\varphi _{\rho _{1}}(\lambda {\mathfrak{f}}_{ \infty }^{s})\frac{L_{3}}{1-b},1)\). The choice of \(\widetilde{c}_{3}\) and \(\widetilde{c}_{4}\) is possible because \(\lambda <\frac{1}{{\mathfrak{f}}_{\infty }^{s}}\varphi _{\varrho _{1}}( \frac{b}{L_{1}})\) and \(\lambda <\frac{1}{{\mathfrak{f}}_{\infty }^{s}}\varphi _{\varrho _{1}}( \frac{1-b}{L_{3}})\). Let \(\epsilon >0\) be such that

$$\begin{aligned}& \lambda \le \min \biggl\{ \frac{1}{{\mathfrak{f}}_{\infty }^{s}+\epsilon } \varphi _{\varrho _{1}} \biggl( \frac{b\widetilde{c}_{3}}{L_{1}} \biggr), \frac{1}{{\mathfrak{f}}_{\infty }^{s}+\epsilon } \varphi _{\varrho _{1}} \biggl( \frac{(1-b)\widetilde{c}_{4}}{L_{3}} \biggr) \biggr\} , \\& \varepsilon \varphi _{\varrho _{2}} \biggl( \frac{1}{\zeta \zeta _{1}\widetilde{L}_{2}} \biggr)\le \mu \le \min \biggl\{ \frac{1}{\epsilon }\varphi _{\varrho _{2}} \biggl( \frac{b(1-\widetilde{c}_{3})}{L_{2}} \biggr), \frac{1}{\epsilon } \varphi _{\varrho _{2}} \biggl( \frac{(1-b)(1-\widetilde{c}_{4})}{L_{4}} \biggr) \biggr\} . \end{aligned}$$

By \((I2)\) and the definition of \({\mathfrak{g}}_{0}^{i}\) we find that there exists \(r_{3}>0\) such that

$$ g(t,x,y)\ge \frac{1}{\epsilon }\varphi _{\varrho _{2}}(x+y), \quad \forall t\in [ \theta _{1},\theta _{2}], x,y\ge 0, x+y\le r_{3}. $$

We define the set \(\varLambda _{3}=\{(x,y)\in {\mathcal{Y}}, \|(x,y)\|_{\mathcal{Y}}< r_{3}\}\). Let \((x,y)\in {\mathcal{P}}\cap \partial \varLambda _{3}\), that is \(\|x\|+\|y\|=r_{3}\). Because \(x(t)+y(t)\le \|x\|+\|y\|=r_{3}\) for all \(t\in [0,1]\), then by Lemma 2.5, we find

$$\begin{aligned}& {\mathcal{A}}_{1}(x,y) (\theta _{1}) \\& \quad \ge \lambda ^{\rho _{1}-1} \int _{0}^{1} \theta _{1}^{\beta _{1}-1}{ \mathcal{J}}_{1}( \nu )\varphi _{\rho _{1}} \bigl(I_{0+}^{\alpha _{1}}f \bigl(\nu ,x(\nu ),y(\nu ) \bigr) \bigr) \,d\nu \\& \qquad {}+\mu ^{\rho _{2}-1} \int _{0}^{1} \theta _{1}^{\beta _{1}-1}{ \mathcal{J}}_{2}(\nu )\varphi _{\rho _{2}} \bigl(I_{0+}^{\alpha _{2}}g \bigl(\nu ,x( \nu ),y(\nu ) \bigr) \bigr) \,d\nu \\& \quad \ge \mu ^{\rho _{2}-1}\theta _{1}^{\beta _{1}-1} \int _{ \theta _{1}}^{\theta _{2}}{\mathcal{J}}_{2}(s) \varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{ \theta _{1}}^{\nu } (\nu -s)^{\alpha _{2}-1}g \bigl(s,x(s),y(s) \bigr) \,ds \biggr)\,d\nu \\& \quad \ge \mu ^{\rho _{2}-1}\theta _{1}^{\beta _{1}-1} \int _{ \theta _{1}}^{\theta _{2}}{\mathcal{J}}_{2}( \nu )\varphi _{\rho _{2}} \biggl( \frac{1}{\varGamma (\alpha _{2})} \int _{\theta _{1}}^{\nu } (\nu -s)^{\alpha _{2}-1} \frac{1}{\epsilon }\varphi _{\varrho _{2}} \bigl(x(s)+y(s) \bigr) \,ds \biggr)\,d\nu \\& \quad \ge \mu ^{\rho _{2}-1}\theta _{1}^{\beta _{1}-1} \int _{ \theta _{1}}^{\theta _{2}}{\mathcal{J}}_{2}( \nu )\varphi _{\rho _{2}} \biggl(\frac{1}{\varGamma (\alpha _{2})} \int _{\theta _{1}}^{ \nu }(\nu -s)^{\alpha _{2}-1} \frac{1}{\epsilon }\varphi _{\varrho _{2}} \bigl( \zeta \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \bigr) \,ds \biggr)\,d\nu \\& \quad = \zeta \zeta _{1}\varphi _{\rho _{2}} \biggl( \frac{\mu }{\epsilon } \biggr) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \widetilde{L}_{2}\ge \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}. \end{aligned}$$

Hence we deduce \(\|{\mathcal{A}}_{1}(x,y)\|\ge {\mathcal{A}}_{1}(x,y)(\theta _{1})\ge \|(x,y)\|_{ \mathcal{Y}}\) and \(\|{\mathcal{A}}(x,y)\|_{\mathcal{Y}}\ge \|{\mathcal{A}}_{1}(x,y)\|\ge \|(x,y)\|_{ \mathcal{Y}}\).

For the second part of the proof, we use the functions \({\mathfrak{f}}^{*}\) and \({\mathfrak{g}}^{*}\) from Case (1), which satisfy here the conditions

$$ \limsup_{u\to \infty }\max_{t\in [0,1]} \frac{{\mathfrak{f}}^{*}(t,u)}{\varphi _{\varrho _{1}}(u)}={ \mathfrak{f}}_{\infty }^{s}, \qquad \lim _{u\to \infty } \max_{t\in [0,1]} \frac{{\mathfrak{g}}^{*}(t,u)}{\varphi _{\varrho _{2}}(u)}=0. $$

Then for \(\epsilon >0\) there exists \(r_{4}'>0\) such that for all \(u\ge r_{4}'\) and \(t\in [0,1]\) we obtain

$$ \begin{aligned} &\frac{{\mathfrak{f}}^{*}(t,u)}{\varphi _{\varrho _{1}}(u)}\le \limsup_{u\to \infty } \max_{t\in [0,1]} \frac{{\mathfrak{f}}^{*}(t,u)}{\varphi _{\varrho _{1}}(u)}+\epsilon ={ \mathfrak{f}}_{\infty }^{s}+\epsilon , \\ &\frac{{\mathfrak{g}}^{*}(t,u)}{\varphi _{\varrho _{2}}(u)}\le \lim_{u\to \infty }\max _{t\in [0,1]} \frac{{\mathfrak{g}}^{*}(t,u)}{\varphi _{\varrho _{2}}(u)}+\epsilon = \epsilon , \end{aligned} $$

and hence \({\mathfrak{f}}^{*}(t,u)\le ({\mathfrak{f}}_{\infty }^{s}+\epsilon ) \varphi _{\varrho _{1}}(u)\) and \(g^{*}(t,u)\le \epsilon \varphi _{\varrho _{2}}(u)\).

We define \(r_{4}=\max \{2r_{3},r_{4}'\}\) and we introduce the set \(\varLambda _{4}=\{(x,y)\in {\mathcal{Y}}, \|(x,y)\|_{\mathcal{Y}}< r_{4}\}\). Let \((x,y)\in {\mathcal{P}}\cap \partial \varLambda _{4}\). By the definitions of \({\mathfrak{f}}^{*}\) and \({\mathfrak{g}}^{*}\) we deduce Eq. (23). Besides, in a similar manner to that used in the proof of Case (1), we find

$$\begin{aligned}& \begin{aligned} {\mathcal{A}}_{1}(x,y) (t)&\le \varphi _{\rho _{1}} \bigl(\lambda \bigl({\mathfrak{f}}_{ \infty }^{s}+ \epsilon \bigr) \bigr)L_{1} \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}+\varphi _{\rho _{2}}( \mu \epsilon )L_{2} \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \\ &\le \bigl[b \widetilde{c}_{3}+b(1-\widetilde{c}_{3}) \bigr] \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}=b \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}, \quad \forall t\in [0,1], \end{aligned} \\& \begin{aligned} {\mathcal{A}}_{2}(x,y) (t)&\le \varphi _{\rho _{1}} \bigl( \lambda \bigl({\mathfrak{f}}_{ \infty }^{s}+\epsilon \bigr) \bigr)L_{3} \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}+ \varphi _{\rho _{2}}( \mu \epsilon )L_{4} \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}} \\ &\le \bigl[(1-b) \widetilde{c}_{4}+(1-b) (1-\widetilde{c}_{4}) \bigr] \bigl\Vert (x,y) \bigr\Vert _{ \mathcal{Y}}=(1-b) \bigl\Vert (x,y) \bigr\Vert _{\mathcal{Y}}, \quad \forall t\in [0,1], \end{aligned} \end{aligned}$$

and then \(\|{\mathcal{A}}(x,y)\|_{\mathcal{Y}}\le b\|(x,y)\|_{\mathcal{Y}}+(1-b)\|(x,y)\|_{ \mathcal{Y}}=\|(x,y)\|_{\mathcal{Y}}\).

Therefore we obtain the conclusion of the theorem. □

Nonexistence of the positive solutions

In this section we give intervals for λ and μ for which there exist no positive solutions of problem (1), (2). By using similar arguments to those used in the proofs of Theorems 4.1–4.4 from [30], we obtain the following theorems for problem (1), (2).

Theorem 4.1

Assume that\((I1)\)and\((I2)\)hold. If there exist positive numbers\(T_{1}\), \(T_{2}\)such that

$$ f(t,x,y)\le T_{1}\varphi _{\varrho _{1}}(x+y),\qquad g(t,x,y)\le T_{2} \varphi _{\varrho _{2}}(x+y),\quad \forall t\in [0,1], x, y\ge 0, $$
(25)

then there exist positive constants\(\lambda _{0}\)and\(\mu _{0}\)such that for every\(\lambda \in (0,\lambda _{0})\)and\(\mu \in (0,\mu _{0})\)the boundary value problem (1), (2) has no positive solution.

In the proof of Theorem 4.1 we define \(\lambda _{0}=\min \{ (T_{1}\varphi _{\varrho _{1}}(4L_{1}))^{-1},(T_{1} \varphi _{\varrho _{1}}(4L_{3}))^{-1} \} \) and \(\mu _{0}=\min \{ (T_{2}\varphi _{\varrho _{2}}(4L_{2}))^{-1},(T_{2} \varphi _{\varrho _{2}}(4L_{4}))^{-1} \} \), where \(L_{i}\), \(i=1,\ldots ,4\), are given by (19).

Remark 4.1

If \({\mathfrak{f}}_{0}^{s}, {\mathfrak{g}}_{0}^{s}, {\mathfrak{f}}_{ \infty }^{s}, {\mathfrak{g}}_{\infty }^{s}<\infty \), then there exist positive constants \(T_{1}\), \(T_{2}\) such that relation (25) holds, and so we obtain the conclusion of Theorem 4.1.

Theorem 4.2

Assume that\((I1)\)and\((I2)\)hold. If there exist positive numbers\(\theta _{1}\), \(\theta _{2}\)with\(0<\theta _{1}<\theta _{2}\le 1\)and\(q_{1}>0\)such that

$$ f(t,x,y)\ge q_{1}\varphi _{\varrho _{1}}(x+y),\quad \forall t\in [ \theta _{1},\theta _{2}], x, y\ge 0, $$
(26)

then there exists a positive constant\(\lambda _{0}'\)such that for every\(\lambda >\lambda _{0}'\)and\(\mu >0\), the boundary value problem (1), (2) has no positive solution.

In the proof of Theorem 4.2 we define \(\lambda _{0}'=\min \{ (q_{1}\varphi _{\varrho _{1}}(\zeta \zeta _{1}\widetilde{L}_{1}))^{-1},(q_{1}\varphi _{\varrho _{1}}( \zeta \zeta _{2}\widetilde{L}_{3}))^{-1} \} \), where \(\widetilde{L}_{1}\) and \(\widetilde{L}_{3}\) are given by (19).

Remark 4.2

If for \(\theta _{1}\), \(\theta _{2}\) with \(0<\theta _{1}<\theta _{2}\le 1\), we have \({\mathfrak{f}}_{0}^{i}, {\mathfrak{f}}_{\infty }^{i}>0\), and \(f(t,x,y)>0\) for all \(t\in [\theta _{1},\theta _{2}]\) and \(x,y\ge 0\) with \(x+y>0\), then Eq. (26) holds, and therefore we obtain the conclusion of Theorem 4.2.

Theorem 4.3

Assume that\((I1)\)and\((I2)\)hold. If there exist positive numbers\(\theta _{1}\), \(\theta _{2}\)with\(0<\theta _{1}<\theta _{2}\le 1\)and\(q_{2}>0\)such that

$$ g(t,x,y)\ge q_{2}\varphi _{\varrho _{2}}(x+y),\quad \forall t\in [ \theta _{1},\theta _{2}], x, y\ge 0, $$
(27)

then there exists a positive constant\(\mu _{0}'\)such that for every\(\mu >\mu _{0}'\)and\(\lambda >0\)the boundary value problem (1), (2) has no positive solution.

In the proof of Theorem 4.3 we define \(\mu _{0}'=\min \{ (q_{2}\varphi _{\varrho _{2}}(\zeta \zeta _{1} \widetilde{L}_{2}))^{-1},(q_{2}\varphi _{\varrho _{2}}(\zeta \zeta _{2} \widetilde{L}_{4}))^{-1} \} \), where \(\widetilde{L}_{2}\) and \(\widetilde{L}_{4}\) are given by (19).

Remark 4.3

If for \(\theta _{1}\), \(\theta _{2}\) with \(0<\theta _{1}<\theta _{2}\le 1\), we have \({\mathfrak{g}}_{0}^{i}, {\mathfrak{g}}_{\infty }^{i}>0\), and \(g(t,x,y)>0\) for all \(t\in [\theta _{1},\theta _{2}]\) and \(x,y\ge 0\) with \(x+y>0\), then Eq. (27) holds, and hence we deduce the conclusion of Theorem 4.3.

Theorem 4.4

Assume that\((I1)\)and\((I2)\)hold. If there exist positive numbers\(\theta _{1}\), \(\theta _{2}\)with\(0<\theta _{1}<\theta _{2}\le 1\)and\(q_{1}, q_{2}>0\)such that

$$ f(t,x,y)\ge q_{1}\varphi _{\varrho _{1}}(x+y),\qquad g(t,x,y)\ge q_{2} \varphi _{\varrho _{2}}(x+y),\quad \forall t\in [ \theta _{1}, \theta _{2}], x, y\ge 0, $$
(28)

then there exist positive constants\(\widetilde{\lambda }_{0}\)and\(\widetilde{\mu }_{0}\)such that for every\(\lambda >\widetilde{\lambda }_{0}\)and\(\mu >\widetilde{\mu }_{0}\)the boundary value problem (1), (2) has no positive solution.

In the proof of Theorem 4.4 we define \(\widetilde{\lambda }_{0}=(q_{1}\varphi _{\varrho _{1}}(2\zeta \zeta _{1} \widetilde{L}_{1}))^{-1}\) and \(\widetilde{\mu }_{0}=(q_{2}\varphi _{\varrho _{2}}(2\zeta \zeta _{2} \widetilde{L}_{4}))^{-1}\).

Remark 4.4

If for \(\theta _{1}\), \(\theta _{2}\) with \(0<\theta _{1}<\theta _{2}\le 1\), we have \({\mathfrak{f}}_{0}^{i}, {\mathfrak{f}}_{\infty }^{i}, { \mathfrak{g}}_{0}^{i}, {\mathfrak{g}}_{\infty }^{i}>0\), and \(f(t,x,y)>0\) and \(g(t,x,y)>0\) for all \(t\in [\theta _{1},\theta _{2}]\) and \(x,y\ge 0\) with \(x+y>0\), then Eq. (28) holds, and so we obtain the conclusion of Theorem 4.4.

An example

Let \(\alpha _{1}=1/2\), \(\alpha _{2}=1/3\), \(\beta _{1}=10/3\), \(n=4\), \(\beta _{2}=17/4\), \(m=5\), \(p=2\), \(q=1\), \(\delta _{0}=11/5\), \(\gamma _{1}=5/6\), \(\gamma _{2}=3/2\), \(\gamma _{0}=9/8\), \(\delta _{1}=3/7\), \(H_{1}(t)=4t\) for all \(t\in [0,1]\), \(H_{2}(t)=\{1, t\in [0,1/4); 8, t\in [1/4,1]\}\), \(K_{1}(t)=\{1, t\in [0,1/2); 3, t\in [1/2,1]\}\), \(\varrho _{1}=5\), \(\rho _{1}=5/4\), \(\varrho _{2}=3\), \(\rho _{2}=3/2\), \(\varphi _{\varrho _{1}}(s)=|s|^{3}s\), \(\varphi _{\rho _{1}}(s)=|s|^{-3/4}s\), \(\varphi _{\varrho _{2}}(s)=|s|s\), \(\varphi _{\rho _{2}}(s)=|s|^{-1/2}s\).

We consider the system of fractional differential equations

$$ \textstyle\begin{cases} D_{0+}^{1/2} (\varphi _{5} (D_{0+}^{10/3}x(t) ) )+\lambda (3-t)^{a_{0}} (e^{(x(t)+y(t))^{4}}-1 )=0,\quad t\in (0,1), \\ D_{0+}^{1/3} (\varphi _{3} (D_{0+}^{17/4}y(t) ) )+\mu (t+2)^{b_{0}}(x^{3}(t)+y^{3}(t))=0,\quad t\in (0,1), \end{cases} $$
(29)

with the coupled nonlocal boundary conditions

$$ \textstyle\begin{cases} x(0)=x'(0)=x''(0)=0, \qquad D_{0+}^{10/3}x(0)=0, \\ D_{0+}^{9/8}x(1)=4 \int _{0}^{1} D_{0+}^{5/6}y(t) \,dt+7D_{0+}^{3/2}y (\frac{1}{4} ), \\ y(0)=y'(0)=y''(0)=y'''(0)=0, \qquad D_{0+}^{17/4}y(0)=0, \\ D_{0+}^{11/5}y(1)=2D_{0+}^{3/7}x (\frac{1}{2} ), \end{cases} $$
(30)

where \(a_{0}, b_{0}>0\).

Here we have \(f(t,x,y)=(3-t)^{a_{0}}(e^{(x+y)^{4}}-1)\), \(g(t,x,y)=(t+2)^{b_{0}}(x^{3}+y^{3})\) for all \(t\in [0,1]\) and \(x, y\ge 0\). We obtain \(\Delta \approx 15.18283383>0\), and then the assumptions \((I1)\) and \((I2)\) are satisfied. Besides, we find

$$\begin{aligned}& g_{1}(t,s)= \frac{1}{\varGamma (10/3)} \textstyle\begin{cases} t^{7/3}(1-s)^{29/24}-(t-s)^{7/3},& 0\le s\le t\le 1, \\ t^{7/3}(1-s)^{29/24}, &0\le t\le s\le 1, \end{cases}\displaystyle \\& g_{11}(t,s)= \frac{1}{\varGamma (61/21)} \textstyle\begin{cases} t^{40/21}(1-s)^{29/24}-(t-s)^{40/21}, & 0\le s\le t\le 1, \\ t^{40/21}(1-s)^{29/24}, &0\le t\le s\le 1, \end{cases}\displaystyle \\& g_{2}(t,s)= \frac{1}{\varGamma (17/4)} \textstyle\begin{cases} t^{13/4}(1-s)^{21/20}-(t-s)^{13/4},& 0\le s\le t\le 1, \\ t^{13/4}(1-s)^{21/20},& 0\le t\le s\le 1,\end{cases}\displaystyle \\& g_{21}(t,s)= \frac{1}{\varGamma (41/12)} \textstyle\begin{cases} t^{29/12}(1-s)^{21/20}-(t-s)^{29/12}, & 0\le s\le t\le 1, \\ t^{29/12}(1-s)^{21/20}, &0\le t\le s\le 1,\end{cases}\displaystyle \\& g_{22}(t,s)= \frac{1}{\varGamma (11/4)} \textstyle\begin{cases} t^{7/4}(1-s)^{21/20}-(t-s)^{7/4}, & 0\le s\le t\le 1, \\ t^{7/4}(1-s)^{21/20},& 0\le t\le s\le 1,\end{cases}\displaystyle \\& {\mathcal{G}}_{1}(t,s)=g_{1}(t,s)+ \frac{2t^{7/3}}{\Delta } \biggl[\frac{48\varGamma (17/4)}{41\varGamma (41/12)}+7 \biggl(\frac{1}{4} \biggr)^{7/4} \frac{\varGamma (17/4)}{\varGamma (11/4)} \biggr] g_{11} \biggl(\frac{1}{2},s \biggr), \\& {\mathcal{G}}_{2}(t,s)= \frac{t^{7/3}\varGamma (17/4)}{\Delta \varGamma (41/20)} \biggl[4 \int _{0}^{1} g_{21}(\tau ,s) \,d \tau +7 g_{22} \biggl(\frac{1}{4},s \biggr) \biggr], \\& {\mathcal{G}}_{3}(t,s)= \frac{2t^{13/4}\varGamma (10/3)}{\Delta \varGamma (53/24)}g_{11} \biggl( \frac{1}{2},s \biggr), \\& {\mathcal{G}}_{4}(t,s)=g_{2}(t,s)+ \frac{t^{13/4}\varGamma (10/3)}{\Delta \varGamma (61/21)2^{19/21}} \biggl[4 \int _{0}^{1} g_{21}(\tau ,s) \,d \tau +7 g_{22} \biggl(\frac{1}{4},s \biggr) \biggr], \end{aligned}$$

for all \((t,s)\in [0,1]\times [0,1]\). For the functions \(h_{1}\), \(h_{2}\) and \({\mathcal{J}}_{i}\), \(i=1,\ldots ,4\), we obtain

$$\begin{aligned}& h_{1}(s)=\frac{1}{\varGamma (10/3)} \bigl[(1-s)^{29/24}-(1-s)^{7/3} \bigr], \\& h_{2}(s)= \frac{1}{\varGamma (17/4)} \bigl[(1-s)^{21/20}-(1-s)^{13/4} \bigr], \\& \quad \text{for all } s\in [0,1], \\& {\mathcal{J}}_{1}(s)= \textstyle\begin{cases} \frac{1}{\varGamma (10/3)} [(1-s)^{29/24}-(1-s)^{7/3} ]+ \frac{1}{\Delta } [\frac{48\varGamma (17/4)}{41\varGamma (41/12)}+7 (\frac{1}{4} )^{7/4}\frac{\varGamma (17/4)}{\varGamma (11/4)} ] \\ \quad {} \times \frac{2}{\varGamma (61/21)} [ (\frac{1}{2} )^{40/21}(1-s)^{29/24}- (\frac{1}{2}-s )^{40/21} ],& 0\le s < \frac{1}{2}, \\ \frac{1}{\varGamma (10/3)} [(1-s)^{29/24}-(1-s)^{7/3} ]+ \frac{1}{\Delta } [\frac{48\varGamma (17/4)}{41\varGamma (41/12)}+7 (\frac{1}{4} )^{7/4}\frac{\varGamma (17/4)}{\varGamma (11/4)} ] \\ \quad {} \times \frac{2}{\varGamma (61/21)} (\frac{1}{2} )^{40/21}(1-s)^{29/24},& \frac{1}{2}\le s \le 1, \end{cases}\displaystyle \\& {\mathcal{J}}_{2}(s)= \textstyle\begin{cases} \frac{\varGamma (17/4)}{\Delta \varGamma (41/20)} \{ \frac{48}{41 \varGamma (41/12)} [(1-s)^{21/20}-(1-s)^{41/12} ] \\ \quad {} +\frac{7}{\varGamma (11/4)} [ (\frac{1}{4} )^{7/4}(1-s)^{21/20}- (\frac{1}{4}-s )^{7/4} ] \} , &0\le s< \frac{1}{4}, \\ \frac{\varGamma (17/4)}{\Delta \varGamma (41/20)} \{ \frac{48}{41 \varGamma (41/12)} [(1-s)^{21/20}-(1-s)^{41/12} ] \\ \quad {} +\frac{7}{\varGamma (11/4)} (\frac{1}{4} )^{7/4}(1-s)^{21/20} \} , &\frac{1}{4}\le s \le 1, \end{cases}\displaystyle \\& {\mathcal{J}}_{3}(s)= \textstyle\begin{cases} \frac{2\varGamma (10/3)}{\Delta \varGamma (53/24)\varGamma (61/21)} [ (\frac{1}{2} )^{40/21}(1-s)^{29/24}- (\frac{1}{2}-s )^{40/21} ],& 0\le s< \frac{1}{2}, \\ \frac{2\varGamma (10/3)}{\Delta \varGamma (53/24)\varGamma (61/21)} ( \frac{1}{2} )^{40/21}(1-s)^{29/24}, &\frac{1}{2}\le s\le 1, \end{cases}\displaystyle \\& {\mathcal{J}}_{4}(s)= \textstyle\begin{cases} \frac{1}{\varGamma (17/4)} [(1-s)^{21/20}-(1-s)^{13/4} ] \\ \quad {}+ \frac{\varGamma (10/3)}{\Delta \varGamma (61/21)2^{19/21}} \{ \frac{48}{41 \varGamma (41/12)} [(1-s)^{21/20} \\ \quad {} -(1-s)^{41/12} ]+\frac{7}{\varGamma (11/4)} [ (\frac{1}{4} )^{7/4}(1-s)^{21/20}- ( \frac{1}{4}-s )^{7/4} ] \} ,& 0\le s< \frac{1}{4}, \\ \frac{1}{\varGamma (17/4)} [(1-s)^{21/20}-(1-s)^{13/4} ] \\ \quad {}+ \frac{\varGamma (10/3)}{\Delta \varGamma (61/21)2^{19/21}} \{ \frac{48}{41 \varGamma (41/12)} [(1-s)^{21/20} \\ \quad {} -(1-s)^{41/12} ]+\frac{7}{\varGamma (11/4)} (\frac{1}{4} )^{7/4}(1-s)^{21/20} \} ,& \frac{1}{4}\le s\le 1. \end{cases}\displaystyle \end{aligned}$$

We choose \(\theta _{1}=1/4\) and \(\theta _{2}=3/4\), and hence we obtain \(\zeta _{1}=(1/4)^{7/3}\), \(\zeta _{2}=(1/4)^{13/4}\) and \(\zeta =\zeta _{2}\). Besides, we deduce \({\mathfrak{f}}_{0}^{s}=3^{a_{0}}\), \({\mathfrak{f}}_{\infty }^{i}=\infty \), \({\mathfrak{g}}_{0}^{s}=0\), \({\mathfrak{g}}_{\infty }^{i}=\infty \), \(L_{1}\approx 0.08207184\), \(L_{3}\approx 0.01229905\), \(\widetilde{L}_{1}\approx 0.05185073\), \(\widetilde{L}_{3}\approx 0.00775575\).

By Theorem 3.1(7), if we consider \(b=1/2\), then, for any \(\lambda \in (0,E)\) and \(\mu \in (0,\infty )\) with \(E=3^{-a_{0}}(2L_{1})^{-4}\), the problem (29), (30) has a positive solution \((x(t),y(t))\), \(t\in [0,1]\). For example, if \(a_{0}=1\) we find \(E\approx 459.179\).

We can also use Theorem 4.2, because \(f(t,x,y)\ge q_{1}(x+y)^{4}\) for all \(t\in [1/4,3/4]\) and \(x, y\ge 0\), with \(q_{1}=(9/4)^{a_{0}}\). If \(a_{0}=1\), we obtain \(\lambda _{0}'=(q_{1}(\zeta \zeta _{1}\widetilde{L}_{1})^{4})^{-1} \approx 1.71714\times 10^{18}\), and then we deduce then, for every \(\lambda >\lambda _{0}'\) and \(\mu >0\), that the boundary value problem (29), (30) has no positive solution.

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Alexandru Tudorache, Faculty of Automatic Control and Computer Engineering, Gh. Asachi Technical University, Str. Prof.dr.doc. Dimitrie Mangeron, nr.27, Iasi 700050, Romania. Rodica Luca, Department of Mathematics, Gh. Asachi Technical University, Blvd. Carol I, nr.11, Iasi 700506, Romania.

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Tudorache, A., Luca, R. Positive solutions for a system of Riemann–Liouville fractional boundary value problems with p-Laplacian operators. Adv Differ Equ 2020, 292 (2020). https://doi.org/10.1186/s13662-020-02750-6

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MSC

  • 34A08
  • 34B15
  • 34B10
  • 34B18
  • 45G15

Keywords

  • Systems of Riemann–Liouville fractional differential equations
  • Nonlocal coupled boundary conditions
  • Nonnegative nonlinearities
  • Existence of solutions
  • Nonexistence of solutions