Existence of positive solutions for p-Laplacian systems involving left and right fractional derivatives

Abstract

Purpose

The paper deals with the existence of positive solutions for a coupled system of nonlinear fractional differential equations with p-Laplacian operator and involving both right Riemann–Liouville and left Caputo-type fractional derivatives. The existence results are obtained by the help of Guo–Krasnosel'skii fixed-point theorem on a cone in the sublinear case. In addition, an example is included to illustrate the main results.

Design/methodology/approach

Fixed-point theorems.

Findings

No finding.

Originality/value

The obtained results are original.

Citation

Ramdane, S. and Guezane-Lakoud, A. (2021), "Existence of positive solutions for p-Laplacian systems involving left and right fractional derivatives", Arab Journal of Mathematical Sciences, Vol. 27 No. 2, pp. 235-248. https://doi.org/10.1108/AJMS-10-2020-0086

1. Introduction

In this paper, we consider the following coupled system of nonlinear fractional differential equations with p-Laplacian operator:

$$\{\begin{array}{l}{}^R{{\displaystyle D}}_{{{\displaystyle 1}}^{-}}^{\alpha }{\phi }_p\left({}^C{{\displaystyle D}}_{0^{+}}^{{\beta }_1}u(t)\right)+a_1(t)f_1\left(u(t),v(t)\right)=0,t\in \left[0,1\right]\text{,}\hfill \\ {}^R{{\displaystyle D}}_{1^{-}}^{\alpha }{\phi }_p\left({}^C{{\displaystyle D}}_{0^{+}}^{{\beta }_2}v(t)\right)+a_2(t)f_2\left(u(t),v(t)\right)=0,t\in \left[0,1\right]\text{,}\hfill \\ {\phi }_p\left({}^C{{\displaystyle D}}_{0^{+}}^{{\beta }_1}u(1)\right)=0,{{\displaystyle u}}^{\prime }(0)=0,{\eta }_{1}u(1)-u(0)={\displaystyle \underset{0}{\overset{1}{\int }}{\mathit{g}}_1\left(s,u(s),v(s)\right)ds\text{,}}\hfill \\ {\phi }_p\left({}^C{{\displaystyle D}}_{0^{+}}^{{\beta }_2}v(1)\right)=0,{{\displaystyle v}}^{\prime }(0)=0,{\eta }_{2}v(1)-v(0)={\displaystyle \underset{0}{\overset{1}{\int }}{g}_2\left(s,u(s),v(s)\right)ds\text{.}}\hfill \end{array}$$

where $0<\alpha <1,1<{\beta }_i<2,{\eta }_i>1,\left(i=1,2\right)$ and ${\phi }_p(s)={{\displaystyle |s|}}^{p-2^{}}s,p>1,{\phi }_q={\left({\phi }_p\right)}^{-1},\frac{1}{p}+\frac{1}{q}=1,{}^{ℝ}{{\displaystyle D}}_{1^{-}}^{\alpha }$ the right Riemann–Liouville fractional derivative, ${}^C{{\displaystyle D}}_{0^{+}}^{{\beta }_i}$ denotes the left Caputo fractional derivative of order ${\beta }_i\text{,}$ the functions $a_i\in C\left(\left[0,1\right],{ℝ}^{+}\right),f_i\in C\left({ℝ}^{+}\times {ℝ}^{+},{ℝ}^{+}\right),g_i\in C\left(\left[0,1\right]\times {ℝ}^{+}\times {ℝ}^{+},{ℝ}^{+}\right)$ for $i=1,2.$

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of a complex medium, polymer rheology, etc. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. For the basic theory and recent development of subject, see [1, 2, 3]. Recently, a linear boundary value problem involving both the right Caputo and the left Riemann–Liouville fractional derivatives have been studied by many authors [4, 5] Many people pay attention to the existence and multiplicity of solutions or positive solutions for boundary value problems of nonlinear fractional differential equations by means of some fixed-point theorems [6–13].

In [14], by applying Guo–Krasnosel'skiî's fixed-point theorem, Guezane-Lakoud and Ashyralyev discussed the existence of positive solutions for the following fractional BVP

$$\{\begin{array}{l}{D}_{0^{+}}^{q}u(t)=f\left(t,u(t)\right),t\in \left[0,1\right],1<q<2\hfill \\ {{\displaystyle u}}^{\prime }(0)=0,u(0)-\alpha u(1)={\displaystyle \underset{0}{\overset{1}{\int }}g\left(s,u(s)\right)ds\text{.}}\hfill \end{array}$$

where $f:[\mathrm{0,1}]\times \mathrm{ℝ}\to \mathrm{ℝ}$ is a given function, $\alpha \in {\mathrm{ℝ}}^{+},D_{0^{+}}^q$ denotes the Caputo's fractional derivative of order q.

On the other hand, the study of coupled systems involving fractional differential equations is also important as such systems occur in various problems, see [13, 15, 16] and the references therein.

In the interesting paper [17], Liu studied by the help of Picard iterative method and Schaefer's fixed-point theorem, the existence of solutions for four classes of boundary value problems for impulsive fractional differential equations.

In [12], relying on the Guo–Krasnosel'skiî's fixed-point theorem, Li and Wei discussed existence of positive solutions for the following coupled system of mixed higher-order nonlinear singular fractional differential equations with integral boundary conditions

$$\{\begin{array}{l}{D}_{0^{+}}^{{\alpha }_1}u(t)+a_1(t)f_1\left(t,u(t),v(t)\right)=0,t\in \left[0,1\right]\hfill \\ D_{0^{+}}^{{\alpha }_2}v(t)+a_2(t)f_2\left(t,u(t)\right)=0,t\in \left[0,1\right]\hfill \\ u^{(j)}(0)=v^{(k)}(0)=0,0\le j\le n_1-\mathrm{2,0}\le k\le n_2-2\hfill \\ u(1)={\displaystyle \underset{0}{\overset{1}{\int }}{h}_1(s)u(s)ds},v(1)={\displaystyle \underset{0}{\overset{1}{\int }}{h}_2(s)v(s)ds}\hfill \end{array}$$

where $n_i-1<{\alpha }_i<n_i,n_i\ge 3,D_{0^{+}}^{{\alpha }_i}$ are the standard Riemann–Liouville fractional derivative, $a_i(t)\in C\left[0,1\right]$ may be singular at $t=0,$ and/or $t=1,h_i\in L_1\left[0,1\right]$ are nonnegative $\left(i=1,2\right)$.

On the other hand, differential equations with p-Laplacian operator have been widely studied owing to its importance in theory and application of mathematics and physics, such in non-Newtonian mechanics, nonlinear elasticity and glaciology, population biology, nonlinear flow laws. There are a very large number of papers devoted to the existence of solutions of the p-Laplacian operator, see for example [18–25].

In [26] G. Q. Chai, studied the existence of positive solutions for the boundary-value problem of nonlinear fractional differential equations with p-Laplacian operator

$$\{\begin{array}{l}{D}_{0^{+}}^{\beta }{\phi }_p\left(D_{0^{+}}^{\alpha }u(t)\right)+f\left(t,u(t)\right)=0, 0<t<1,\hfill \\ D_{0^{+}}^{\alpha }u(0)=0,D_{0^{+}}^{\alpha }u(1)+\sigma D_{0^{+}}^{\gamma }u(1)=0,u(0)=0.\hfill \end{array}$$

where $1<\alpha <2,0<\beta <1,{\phi }_p(s)={{\displaystyle |s|}}^{p-2}s,p>1,{\phi }_q={\left({\phi }_p\right)}^{-1},\frac{1}{p}+\frac{1}{q}=1,D_{0^{+}}^{\alpha },D_{0^{+}}^{\beta }$ are the standard Riemann–Liouville fractional derivatives, $0<\gamma \le 1\text{,}$ The function $f:\left[0,1\right]\times {ℝ}^{+}\to {ℝ}^{+}$ is continuous.

The rest of the paper is organized as follows. In Section 2, we present preliminaries and lemmas. Section 3, we investigate the existence of a solution for the corresponding fractional linear boundary value problem. Finally, Section 4 is devoted to the existence of positive solutions under some sufficient conditions on the nonlinear terms, then we give an example to illustrate our results.

2. Preliminaries

In this section, we recall the basic definitions and lemmas from fractional calculus theory, see [2, 3], for more details.

Let $\alpha >0,\left[a,b\right]$ be a finite interval of $\mathrm{ℝ}$ and g a real function on $\left(a,b\right)\text{.}$ The left and right Riemann–Liouville fractional integral of the function g are defined, respectively, by

$$I_{a+}^{\alpha }f(t)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{a}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}g(s)ds},I_{b-}^{\alpha }f(t)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{t}{\overset{b}{\int }}{\left(s-t\right)}^{\alpha -1}g(s)ds\text{,}}$$

provided that the right-hand side exists.

The right Riemann–Liouville fractional derivative and the left Caputo fractional derivative of order $\alpha >0$ of g are, respectively

$${}^R{{\displaystyle D}}_{b^{-}}^{\alpha }f(t)={\left(\frac{-d}{dt}\right)}^n{I}_{b-}^{n-\alpha }g(t),{}^C{D}_{a^{+}}^{\alpha }f(t)=I_{a^{+}}^{n-\alpha }{g}^{(n)}(t)\text{,}$$

where $n<\alpha <n+1,n=[\alpha ]+1\text{,}$ provided that the right-hand side exists.

For the properties of Riemann–Liouville fractional derivative and Caputo fractional derivative, we obtain the following statement. Let $u\in L^1\left(0,1\right)$ then

(2.1)$$I_{1^{-}}^{\alpha R}{D}_{1^{-}}^{\alpha }u(t)=u(t)+{\displaystyle \sum _{i=1}^n{a}_i{\left(1-t\right)}^{\alpha -i}}$$

(2.2)$$I_{0^{+}}^{\alpha C}{D}_{0^{+}}^{\alpha }u(t)=u(t)+{\displaystyle \sum _{k=0}^{n-1}{b}_k{t}^k}$$

where $a_i,b_k\in ℝ,i=0,\dots n\text{,}$ and $k=0,\dots n-1.$

We also need the following lemma and theorem to obtain our results.

Lemma 2.1.

[26] Let $c>0,\gamma >0.$ for any $x,y\in \left[0,c\right]$ we have

• (1)if $\gamma >1\text{,}$ then $\left|x^{\gamma }-y^{\gamma }\right|\le \gamma c^{\gamma -1}\left|x-y\right|\text{,}$

• (2)if $0<\gamma \le 1\text{,}$ then $\left|x^{\gamma }-y^{\gamma }\right|\le {\left|x-y\right|}^{\gamma }\text{.}$

Theorem 2.1.

[27] (Guo–Krasnoselskiî's) Let E be a Banach space, and let $K\subset E$, be a cone. Assume ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ are open subsets of E with $0\in {\mathrm{\Omega }}_1,{\overline{\mathrm{\Omega }}}_1\subset {\mathrm{\Omega }}_2$ and let $T:K\cap \left({\overline{\mathrm{\Omega }}}_2\backslash {\mathrm{\Omega }}_1\right)\to K$, be a completely continuous operator such that

• (1)$\Vert Tu\Vert \le \Vert u\Vert ,u\in K\cap \partial {\mathrm{\Omega }}_1$, and $||Tu||\ge \Vert u\Vert ,u\in K\cap \partial {\mathrm{\Omega }}_2\text{,}$ or

• (2)$\Vert Tu\Vert \ge \Vert u\Vert ,u\in K\cap \partial {\mathrm{\Omega }}_1$ and $||Tu||\le \Vert u\Vert ,u\in K\cap \partial {\mathrm{\Omega }}_2\text{.}$

Then T has a fixed point in $K\cap \left({\overline{\mathrm{\Omega }}}_2\backslash {\mathrm{\Omega }}_1\right)$

3. Linear boundary value problem

Lemma 3.1.

Assume that $y\in C\left(0,1\right)\cap L_1\left(0,1\right)$ and $1<{\beta }_i<2,i=1,2$, the unique solution of the boundary value problem

(3.1)$${}^C{{\displaystyle D}}_{0^{+}}^{{\beta }_i}u(t)+y(t)=0,t\in \left[0,1\right],$$

(3.2)$${{\displaystyle u}}^{\prime }(0)=0,{\eta }_{i}u(1)-u(0)={\displaystyle \underset{0}{\overset{1}{\int }}{g}_i(s)ds}$$

is given by

(3.3)$$u(t)={\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(t,s\right)y(s)ds}+\frac{1}{{\eta }_i-1}{\displaystyle \underset{0}{\overset{1}{\int }}{g}_i(s)ds}$$

where

(3.4)$$G_i\left(t,s\right)=\frac{1}{\mathrm{\Gamma }\left({\beta }_i\right)}\{\begin{array}{ll}\frac{{\eta }_i}{{\eta }_i-1}{\left(1-s\right)}^{{\beta }_i-1}-{\left(t-s\right)}^{{\beta }_i-1},\hfill & 0\le s\le t\le 1.\hfill \\ \frac{{\eta }_i}{{\eta }_i-1}{\left(1-s\right)}^{{\beta }_i-1},\hfill & 0\le t\le s\le 1.\hfill \end{array}$$

Proof. We apply (2.2) to equation (3.1) to get

(3.5)$$u(t)=-I_{0^{+}}^{{\beta }_i}y(t)+c_1+c_{2}t,t\in [\mathrm{0,1}]$$

thanks to boundary condition (3.2) we obtain $c_2=0,$ and

$$c_1=\frac{1}{{\eta }_i-1}\left[\frac{{\eta }_i}{\mathrm{\Gamma }\left({\beta }_i\right)}{\displaystyle \underset{0}{\overset{1}{\int }}{\left(1-s\right)}^{{\beta }_i-1}y(s)ds}+{\displaystyle \underset{0}{\overset{1}{\int }}{g}_i(s)ds}\right]\text{.}$$

So, the unique solution of the problem (3.1) is

$$\begin{array}{l}u(t)=\frac{1}{\mathrm{\Gamma }\left({\beta }_i\right)}\left[-{\displaystyle \underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{{\beta }_i-1}y(s)ds}+\frac{{\eta }_i}{{\eta }_i-1}{\displaystyle \underset{0}{\overset{1}{\int }}{\left(1-s\right)}^{{\beta }_i-1}y(s)ds}\right]+\frac{1}{{\eta }_i-1}{\displaystyle \underset{0}{\overset{1}{\int }}{g}_i(s)ds}\\ ={\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(t,s\right)y(s)ds}+\frac{1}{{\eta }_i-1}{\displaystyle \underset{0}{\overset{1}{\int }}{g}_i(s)ds\text{.}}\end{array}$$

The proof is completed.▪

Lemma 3.2.

If $y\in C\left(0,1\right)\cap L_1\left(0,1\right)$, then the boundary value problem

(3.6)$${}^R{{\displaystyle D}}_{1^{-}}^{\alpha }{\phi }_p\left({}^C{{\displaystyle D}}_{0^{+}}^{{\beta }_i}u(t)\right)+y(t)=0,0\le t\le 1$$

(3.7)$${\phi }_p\left({}^C{{\displaystyle D}}_{0^{+}}^{{\beta }_i}u(1)\right)=0$$

(3.8)$${{\displaystyle u}}^{\prime }(0)=0,{\eta }_{i}u(1)-u(0)={\displaystyle \underset{0}{\overset{1}{\int }}{g}_i(s)ds}$$

has an unique solution

$$u(t)={\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(t,s\right){\phi }_q\left({\displaystyle \underset{s}{\overset{1}{\int }}\frac{{(\tau -s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}y(\tau )d\tau }\right)ds}+\frac{1}{{\eta }_i-1}\underset{0}{\overset{1}{\int }}{\displaystyle g_i(s)ds}$$

where $G_i\left(t,s\right)$ is defined as (3.4).

Proof. From Eqs (3.6) and (2.1), we have

(3.9)$${\phi }_p\left({}^C{{\displaystyle D}}_{0^{+}}^{{\beta }_i}u(t)\right)=-I_{1^{-}}^{\alpha }y(t)+C_1{\left(1-t\right)}^{\alpha -1},C_1\in ℝ\text{.}$$

By the boundary conditions (3.7) we get $C_1=0,$ consequently,

$${\phi }_p\left({}^C{{\displaystyle D}}_{0^{+}}^{{\beta }_i}u(t)\right)=-I_{1^{-}}^{\alpha }y(t)$$

and then

(3.10)$${}^C{{\displaystyle D}}_{0^{+}}^{{\beta }_i}u(t)+{\phi }_q\left(\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{t}{\overset{1}{\int }}{(s-t)}^{\alpha -1}y(s)ds}\right)=0,t\in \left[0,1\right]\text{.}$$

Thus, the fractional boundary value problem (3.1)–(3.2) is equivalent to the following problem

$$\begin{array}{l}{}^C{{\displaystyle D}}_{0^{+}}^{{\beta }_i}u(t)+{\phi }_q\left(\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{s}{\overset{1}{\int }}{(s-t)}^{\alpha -1}y(s)ds}\right)=0,t\in \left[0,1\right]\hfill \\ {{\displaystyle u}}^{\prime }(0)=0;{\eta }_{i}u(1)-u(0)={\displaystyle \underset{0}{\overset{1}{\int }}{g}_i(s)ds\text{.}}\hfill \end{array}$$

Lemma 3.1 implies that the problem (3.6), (3.7) and (3.8) has an unique solution

$$u(t)={\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(t,s\right){\phi }_q\left(\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{s}{\overset{1}{\int }}{(\tau -s)}^{\alpha -1}y(\tau )d\tau }\right)ds}+\frac{1}{{\eta }_i-1}{\displaystyle \underset{\text{0}}{\overset{1}{\int }}{g}_i(s)ds\text{,}}$$

the proof is achieved.▪

Lemma 3.3.

The functions $G_i\left(t,s\right),i=1,2$ are continuous on $\left[0,1\right]\times \left[0,1\right]$ and satisfy the following properties:

1. $G_i\left(t,s\right)>0$ for $t,s\in [0,1),i=1,2$

2. $\frac{1}{{\eta }_i}{G}_i\left(s,s\right)\le G_i\left(t,s\right)\le G_i\left(s,s\right),i=1,2$ for $\left(t,s\right)\in [0,1)\times [0,1)$.

Proof. (1) Observing the expression of $G_i\left(t,s\right)$, it is easy to see that $G_i\left(t,s\right)>0\text{,}$ for $t,s\in [0,1),i=1,2$

• (2)First, $G_i\left(t,s\right)\le G_i\left(s,s\right)$ for $t,s\in [\mathrm{0,1})$

Second, setting

$$\begin{array}{l}{g}_{i,1}\left(t,s\right)=\frac{{\eta }_i}{\left({\eta }_i-1\right)\mathrm{\Gamma }\left({\beta }_i\right)}{\left(1-s\right)}^{{\beta }_i-1}-\frac{{\left(t-s\right)}^{{\beta }_i-1}}{\mathrm{\Gamma }\left({\beta }_i\right)},s\le t\hfill \\ g_{i,2}(s)=\frac{{\eta }_i}{\left({\eta }_i-1\right)\mathrm{\Gamma }\left({\beta }_i\right)}{\left(1-s\right)}^{{\beta }_i-1},t\le s\hfill \end{array}$$

for given $s\in [0,1),g_{i,1}\left(t,s\right)$ is decreasing as a function of t, then,

$$\begin{array}{l}{g}_{i,1}\left(t,s\right)\ge g_{i,1}\left(1,s\right)\\ =\frac{1}{\left({\eta }_i-1\right)\mathrm{\Gamma }\left({\beta }_i\right)}{\left(1-s\right)}^{{\beta }_i-1}\\ \ge \frac{1}{{\eta }_i}{G}_i\left(s,s\right)\text{,}\end{array}$$

and $g_{i,2}(s)\ge \frac{1}{{\eta }_i}{G}_i\left(s,s\right)\text{.}$▪

4. Existence of positive solutions

We need to introduce some notations for the forthcoming discussion. Let $X=C\left[0,1\right]\times C\left[0,1\right]$ be the Banach space endowed with the norm

$$\Vert \left(x_1,x_2\right)\Vert =\max \left({{\displaystyle \Vert x_i\Vert }}_{\infty },i=1,2\right)$$

where ${{\displaystyle \Vert x_i\Vert }}_{\infty }=\underset{t\in \left[0,1\right]}{\max }\left|x_i(t)\right|$

Define the cone $P\subset X$ by

(4.1)$$P=\left\{\left(x_1,x_2\right)\in X:x_i(t)\ge 0,t\in \left[0,1\right],\underset{t\in [\mathrm{0,1}]}{\min }{x}_i(t)\ge \frac{1}{{\eta }_i}{{\displaystyle \Vert x_i\Vert }}_{\infty },i=1,2\right\}$$

Let us introduce the following notations

$$\begin{array}{l}{A}_{\delta ,i}=\underset{\left(|u|+|v|\right)\to \delta }{\lim }\frac{f_i\left(u,v\right)}{{\left(|u|+|v|\right)}^{p-1}},\left(\delta =0^{+} or +\infty \right),\\ E_i={\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(s,s\right)ds,}\\ F_i=\frac{a_i^{q-1}}{{\left(\mathrm{\Gamma }(\alpha )\right)}^{q-1}}{\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(s,s\right){\left(\underset{s}{\overset{1}{{\displaystyle \int }}}{(\tau -s)}^{\alpha -1}d\tau \right)}^{q-1}ds},\text{where}{a}_i=\underset{t\in \left[0.1\right]}{\max }{a}_i(t)\end{array}$$

By simple calculation, we get

$$\begin{array}{l}{E}_i=\frac{{\eta }_i}{\left({\eta }_i-1\right)\mathrm{\Gamma }\left({\beta }_i+1\right)}\\ F_i=\frac{{\eta }_i{a}_i^{q-1}}{\left({\eta }_i-1\right){(\mathrm{\Gamma }(\alpha +1))}^{q-1}\mathrm{\Gamma }\left({\beta }_i\right)\left[\alpha \left(q-1\right)\left({\beta }_i-1\right)+1\right]},i=1,2\end{array}$$

We make the following assumption:

(H): There exist two nonnegative functions $c_1,c_2\in L^1\left[0,1\right]$ and two constants $b_1,b_2>0$ such that

$$g_1\left(t,u,v\right)\le b_1{c}_1(t)\left(u+v\right),g_2\left(t,u,v\right)\le b_2{c}_2(t)\left(u+v\right)\text{,}$$

for $\left(u,v\right)\in {ℝ}^{+}\times {ℝ}^{+}\text{,}$ with ${{\displaystyle \Vert c_i\Vert }}_{L^1}\le \frac{{\eta }_i-1}{2b_i},i=1,2.$

Lemma 4.1.

The system $(S)$ has a positive solution $\left(u,v\right)$ if and only if $\left(u,v\right)$ is a positive solution for the following system of integral equations:

(4.2)$$\{\begin{array}{l}u(t)={\displaystyle \underset{0}{\overset{1}{\int }}{G}_1\left(t,s\right){\left(\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{s}{\overset{1}{\int }}{(\tau -s)}^{\alpha -1}{a}_1(\tau )f_1\left(u(\tau ),v(\tau )\right)d\tau }\right)}^{q-1}ds}\hfill \\ +\frac{1}{{\eta }_1-1}{\displaystyle \underset{0}{\overset{1}{\int }}{g}_1\left(s,u(s),v(s)\right)ds}\hfill \\ v(t)={\displaystyle \underset{0}{\overset{1}{\int }}{G}_2\left(t,s\right){\left(\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{s}{\overset{1}{\int }}{(\tau -s)}^{\alpha -1}{a}_2(\tau )f_2\left(u(\tau ),v(\tau )\right)d\tau }\right)}^{q-1}ds}\hfill \\ +\frac{1}{{\eta }_2-1}{\displaystyle \underset{0}{\overset{1}{\int }}{g}_2\left(s,u(s),v(s)\right)ds\text{.}}\hfill \end{array}$$

Proof. Easily obtained by Lemma 3.2, then we omit it.▪

Define the operator

(4.3)$$\begin{array}{l}T:P\to C\left[0,1\right]\times C\left[0,1\right]\\ T\left(u,v\right)=(T_1(u,v),T_2\left(u,v\right))\text{,}\end{array}$$

where $T_i:P\to C\left[0,1\right]$ and

(4.4)$$\begin{array}{l}{T}_i\left(u,v\right)={\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(t,s\right){\left(\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{s}{\overset{1}{\int }}{(\tau -s)}^{\alpha -1}{a}_i(\tau )f_i\left(u(\tau ),v(\tau )\right)d\tau }\right)}^{q-1}ds}\\ +\frac{1}{{\eta }_i-1}{\displaystyle \underset{0}{\overset{1}{\int }}{g}_i\left(s,u(s),v(s)\right)ds\text{.}}\end{array}$$

Then, by Lemma 4.1, the existence of solutions for problem $(S)$ is translated into the existence of fixed points for $T\left(u,v\right)=\left(u,v\right)$, thus the fixed point of the operator T coincides with the solution of problem $(S)$.

Lemma 4.2.

Let $T:P\to X$ be the operator defined by (4.3). Then T is completely continuous and $TP\subset P$.

Proof. First, we shall show that $TP\subset P.$ We have for each $t\in \left[0,1\right]\text{,}$

$$\begin{array}{l}\left|T_i\left(u(t),v(t)\right)\right|\le {\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(t,s\right){\left(\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{s}{\overset{1}{\int }}{(\tau -s)}^{\alpha -1}{a}_i(\tau )f_i\left(u(\tau ),v(\tau )\right)d\tau }\right)}^{q-1}ds}\\ +\frac{1}{{\eta }_i-1}{\displaystyle \underset{0}{\overset{1}{\int }}{g}_i\left(s,u(s),v(s)\right)ds}\end{array}$$

Taking the supremum over $\left[0,1\right]$, we get

$$\begin{array}{l}{{\displaystyle \Vert T_i\left(u,v\right)\Vert }}_{\infty }\le {\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(t,s\right){\left(\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{s}{\overset{1}{\int }}{(\tau -s)}^{\alpha -1}{a}_i(\tau )f_i\left(u(\tau ),v(\tau )\right)d\tau }\right)}^{q-1}ds}\\ +\frac{1}{{\eta }_i-1}{\displaystyle \underset{0}{\overset{1}{\int }}{g}_i\left(s,u(s),v(s)\right)ds\text{.}}\end{array}$$

On the other side, we have

$$\begin{array}{l}{T}_i\left(u(t),v(t)\right)\ge \frac{1}{{\eta }_i}{\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(t,s\right){\left(\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{s}{\overset{1}{\int }}{(\tau -s)}^{\alpha -1}{a}_i(\tau )f_i\left(u(\tau ),v(\tau )\right)d\tau }\right)}^{q-1}ds}\\ +\frac{1}{{\eta }_i-1}{\displaystyle \underset{0}{\overset{1}{\int }}{g}_i\left(s,u(s),v(s)\right)ds}\end{array}$$

Since ${\eta }_i>1$, then,

$$T_i\left(u(t),v(t)\right)\ge \frac{1}{{\eta }_i}{{\displaystyle \Vert T_i\left(u,v\right)\Vert }}_{\infty }\text{.}$$

That is $TP\subset P.$

Second, we shall proof that T is completely continuous that will be done in two steps.

• Step 1: By the continuity of the functions $f_i$ and $g_i$ it yields for $n\ge N,$

$$\begin{array}{l}\left|f_i\left(u_n(\tau ),v_n(\tau )\right)-f_i\left(u(\tau ),v(\tau )\right)\right|<\epsilon \text{,}\\ \left|g_i\left(s,u_n(s),v_n(\tau )\right)-g_i\left(s,u(s),v(\tau )\right)\right|<\epsilon \text{.}\end{array}$$

• (1)If $1<q\le 2\text{,}$ then from Lemma 2.1

$$\begin{array}{l}\left|{\left({\displaystyle \underset{s}{\overset{1}{\int }}{(\tau -s)}^{\alpha -1}{a}_i(\tau )f_i\left(u(\tau ),v(\tau )\right)d\tau }\right)}^{q-1}-{\left({\displaystyle \underset{s}{\overset{1}{\int }}{(\tau -s)}^{\alpha -1}{a}_i(\tau )f_i\left(u(\tau ),v(\tau )\right)d\tau }\right)}^{q-1}\right|\hfill \\ \le {\left({\displaystyle \underset{s}{\overset{1}{\int }}{(\tau -s)}^{\alpha -1}{a}_i(\tau )\left|f_i\left(u_n(\tau ),v(\tau )\right)-f_i\left(u(\tau ),v(\tau )\right)\right|d\tau }\right)}^{q-1}\hfill \\ <{\left[\frac{\epsilon }{\alpha }{a}_i\right]}^{q-1}\text{.}\hfill \end{array}$$

Then,

$$\left|T_i\left(u_n,v_n\right)-T_i\left(u,v\right)\right|<\frac{a_i^{q-1}{\epsilon }^{q-1}}{{(\mathrm{\Gamma }(\alpha +1))}^{q-1}}{\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(s,s\right)ds+\frac{\epsilon }{{\eta }_i-1}}=\frac{a_i^{q-1}{\epsilon }^{q-1}}{{(\mathrm{\Gamma }(\alpha +1))}^{q-1}}{E}_i+\frac{1}{{\eta }_i-1}\epsilon \text{.}$$

Hence

(4.5)$${||T_i\left(u_n,v_n\right)-T_i\left(u,v\right)||}_{\infty }\le \left(\frac{a_i^{q-1}{E}_i}{{(\mathrm{\Gamma }(\alpha +1))}^{q-1}}+\frac{1}{{\eta }_i-1}\right){\epsilon }^{q-1}.$$

• (2)If $q>2,$ then from Lemma 2.1, we have

$$\left|{\left(\frac{1}{\mathrm{\Gamma }(\alpha )}{\displaystyle \underset{s}{\overset{1}{\int }}{(\tau -s)}^{\alpha -1}{a}_i(\tau )f_i\left(u_n(\tau ),v(\tau )\right)d\tau }\right)}^{q-1}-{\left(\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{s}{\overset{1}{\int }}{\left(\tau -s\right)}^{\alpha -1}{a}_i\left(\tau \right)f_i\left(u\left(\tau \right),v\left(\tau \right)\right)d\tau }\right)}^{q-1}\right|\le \frac{\left(q-1\right){\left(c\right)}^{q-2}}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{s}{\overset{1}{\int }}{\left(\tau -s\right)}^{\alpha -1}{a}_i\left(\tau \right)\left|f_i\left(u_n\left(\tau \right),v\left(\tau \right)\right)-f_i\left(u\left(\tau \right),v\left(\tau \right)\right)\right|d\tau }<\frac{\left(q-1\right)c^{q-2}{a}_i}{\mathrm{\Gamma }\left(\alpha +1\right)}\epsilon \text{.}$$

So,

$$\left|T_i\left(u_n,v_n\right)-T_i\left(u,v\right)\right|<\left(\frac{\left(q-1\right)c^{q-2}{a}_i}{\mathrm{\Gamma }\left(\alpha +1\right)}{\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(s,s\right)ds+\frac{1}{{\eta }_i-1}}\right)\epsilon \text{.}$$

Hence

(4.6)$${{\displaystyle \Vert T_i\left(u_n,v_n\right)-T_i\left(u,v\right)\Vert }}_{\infty }<\left(\frac{\left(q-1\right)c^{q-2}{a}_i}{\mathrm{\Gamma }\left(\alpha +1\right)}{E}_i+\frac{1}{{\eta }_i-1}\right)\epsilon \text{.}$$

From (4.5)–(4.6) it follows that $\left|\left|T\left(u_n,v_n\right)-T\left(u,v\right)\right|\right|\to 0$ as $n\to \infty ,$ thus T is continuous.

• Step 2: The operator T is uniformly bounded on $P.$ Let $\mathrm{\Omega }$ be an open bounded set in P. Set

$$L_i=\underset{\left(u,v\right)\in \overline{\mathrm{\Omega }}}{\max f_i\left(u\left(t\right),v\left(t\right)\right)<\infty },l_i=\underset{\left(t,u,v\right)\in \left[\mathrm{0,1}\right]\times \overline{\mathrm{\Omega }}}{\max g_i\left(t,u\left(t\right),v\left(t\right)\right)}\text{.}$$

Then for $\left(t,u,v\right)\in \left[0,1\right]\times \mathrm{\Omega }\text{,}$ we have

$$\begin{array}{l}\left|T_i\left(u\left(t\right),v\left(t\right)\right)\right|\le {\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(t,s\right){\left(\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{s}{\overset{1}{\int }}{\left(\tau -s\right)}^{\alpha -1}{a}_i\left(\tau \right)f_i\left(u\left(\tau \right),v\left(\tau \right)\right)d\tau }\right)}^{q-1}ds}\\ +\frac{1}{{\eta }_i-1}{\displaystyle \underset{0}{\overset{1}{\int }}{g}_i\left(s,u\left(s\right),v\left(s\right)\right)ds}\\ \le {\left[\frac{L_i{a}_i}{\left(\mathrm{\Gamma }\left(\alpha +1\right)\right)}\right]}^{q-1}{\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(s,s\right)ds+\frac{l_i}{\left({\eta }_i-1\right)}}\\ ={\left[\frac{L_i{a}_i}{\left(\mathrm{\Gamma }\left(\alpha +1\right)\right)}\right]}^{q-1}{E}_i+\frac{l_i}{{\eta }_i-1}<\infty \end{array}$$

thus $T\left(\mathrm{\Omega }\right)$ is uniformly bounded.

Now we prove that $T\left(\mathrm{\Omega }\right)$ equicontinuous, Let $\left(u,v\right)\in \mathrm{\Omega },0\le t_1\le t_2\le 1$. We have

$$\begin{array}{l}\left|T_i\left(u\left(t_1\right),v\left(t_1\right)\right)-T_i\left(u\left(t_2\right),v\left(t_2\right)\right)\right|\\ \le {\displaystyle \underset{0}{\overset{t_1}{\int }}\left|G_i\left(t_2,s\right)-G_i\left(t_1,s\right)\right|{\left(\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{s}{\overset{1}{\int }}{\left(\tau -s\right)}^{\alpha -1}{a}_i\left(\tau \right)f_i\left(u\left(\tau \right),v\left(\tau \right)\right)d\tau }\right)}^{q-1}ds}\\ +{\displaystyle \underset{t_2}{\overset{1}{\int }}\left|G_i\left(t_2,s\right)-G_i\left(t_1,s\right)\right|{\left(\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{s}{\overset{1}{\int }}{\left(\tau -s\right)}^{\alpha -1}{a}_i\left(\tau \right)f_i\left(u\left(\tau \right),v\left(\tau \right)\right)d\tau }\right)}^{q-1}ds}\\ +{\displaystyle \underset{t_1}{\overset{t_2}{\int }}\left|G_i\left(t_2,s\right)-G_i\left(t_1,s\right)\right|{\left(\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{s}{\overset{1}{\int }}{\left(\tau -s\right)}^{\alpha -1}{a}_i\left(\tau \right)f_i\left(u\left(\tau \right),v\left(\tau \right)\right)d\tau }\right)}^{q-1}ds}\\ \le {\left[\frac{L_i{a}_i}{\left(\mathrm{\Gamma }\left(\alpha +1\right)\right)}\right]}^{q-1}\frac{{\left|t_2-t_1\right|}^{{\beta }_i}}{\mathrm{\Gamma }\left({\beta }_i+1\right)}\text{.}\end{array}$$

Consequently, $|T_i\left(u\left(t_1\right),v\left(t_1\right)\right)-T_i\left(u\left(t_2\right),v\left(t_2\right)\right)|\to 0$, when $t_2\to t_1.$ Hence $T\left(\mathrm{\Omega }\right)$ is equicontinuous. Finally, by Arzela–Ascoli's theorem, it follows that T is completely continuous mapping on $\mathrm{\Omega }.$ ▪

Theorem 4.1.

Assume that the condition $\left(H\right)$ is satisfied, then the system $\left(S\right)$ has at least one nontrivial positive solution $\left(u,v\right)$ in the cone P, in the case $A_{0,i}=0$ and $A_{\infty ,i}=\infty ,i=1,2.$

Proof. From $A_{0,i}=0,i=1,2\text{,}$ we deduce that for

$$0<\epsilon \le \underset{i=\mathrm{1,2}}{\min }\left\{{\left[\left(1-\frac{b_i}{{\eta }_i-1}{{\displaystyle \Vert c_i\Vert }}_{L^1}\right)\frac{1}{F_i}\right]}^{\frac{1}{q-1}}\right\}\text{,}$$

there exist ${\rho }_1>0\text{,}$ such that if $0<u+v\le {\rho }_1$, then

$$f_i\left(u,v\right)\le \epsilon {\left(\left|u\right|+\left|v\right|\right)}^{p-1}$$

Let ${\mathrm{\Omega }}_1=\left\{\left(u,v\right)\in X,\Vert \left(u,v\right)\Vert <{\rho }_1\right\}\text{.}$ Assume that $\left(u,v\right)\in P\cap \partial {\mathrm{\Omega }}_1\text{,}$ then

$$\begin{array}{l}{T}_i\left(u\left(t\right),v\left(t\right)\right)\le {\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(s,s\right){\left(\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{s}{\overset{1}{\int }}{\left(\tau -s\right)}^{\alpha -1}{a}_i\left(\tau \right)\epsilon {\left(\left|u\right|+\left|v\right|\right)}^{p-1}d\tau }\right)}^{q-1}ds}\\ +\frac{1}{{\eta }_i-1}{\displaystyle \underset{0}{\overset{1}{\int }}{b}_i{c}_i\left(s\right)\left(\left|u\right|+\left|v\right|\right)ds.}\\ \le {\left(\frac{\epsilon }{\mathrm{\Gamma }\left(\alpha \right)}\right)}^{q-1}{\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(s,s\right)}\\ \times {\left({\displaystyle \underset{s}{\overset{1}{\int }}{\left(\tau -s\right)}^{\alpha -1}{a}_i\left(\tau \right){\left({{\displaystyle \Vert u\Vert }}_{\infty }+{{\displaystyle \Vert v\Vert }}_{\infty }\right)}^{p-1}d\tau }\right)}^{q-1}ds\\ +\frac{b_i}{{\eta }_i-1}{\displaystyle \underset{0}{\overset{1}{\int }}{c}_i\left(s\right)\left({{\displaystyle \Vert u\Vert }}_{\infty }+{{\displaystyle \Vert v\Vert }}_{\infty }\right)ds.}\\ =\left|\left|\left(u,v\right)\right|\right|\left({\epsilon }^{q-1}{F}_i+\frac{b_i}{{\eta }_i-1}{{\displaystyle \Vert c_i\Vert }}_{L^1}\right)\\ \le \Vert \left(u,v\right)\Vert \text{.}\end{array}$$

Hence

$$\Vert T\left(u,v\right)\Vert \le \left|\left|\left(u,v\right)\right|\right|,\text{for}\left(u,v\right)\in \partial {\mathrm{\Omega }}_1\cap P$$

Since $A_{\infty ,i}=\infty ,i=1,2\text{,}$ so for

$$\mu \ge \underset{i=\mathrm{1,2}}{\max }\left\{{\left(\frac{{\eta }^2\mathrm{\Gamma }\left(\alpha \right)}{{\xi }_i}\right)}^{\frac{1}{q-1}}\right\},{\xi }_i={\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(s,s\right){\left({\displaystyle \underset{s}{\overset{1}{\int }}{\left(\tau -s\right)}^{\alpha -1}{a}_i\left(\tau \right)d\tau }\right)}^{q-1}ds\text{,}}$$

there exists $\rho >0,$ such that if $u+v\ge \rho$, then

$$f_i\left(u,v\right)\ge \mu {\left(\left|u\right|+\left|v\right|\right)}^{p-1}\text{.}$$

Let ${\rho }_2=\max (\frac{3}{2}{\rho }_1,\eta \rho ),\eta =\max \left({\eta }_1,{\eta }_2\right)\text{,}$ and set ${\mathrm{\Omega }}_2=\left\{\left(u,v\right)\in X,\Vert \left(u,v\right)\Vert <{\rho }_2\right\}\text{,}$ it is easy to see that ${\overline{\mathrm{\Omega }}}_1\subset {\mathrm{\Omega }}_2\text{.}$ Assume that $\left(u,v\right)\in P\cap \partial {\mathrm{\Omega }}_2\text{,}$ then

$$\begin{array}{l}{T}_i\left(u\left(t\right),v\left(t\right)\right)\ge \frac{1}{{\eta }_i}{\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(s,s\right){\left(\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\displaystyle \underset{s}{\overset{1}{\int }}{\left(\tau -s\right)}^{\alpha -1}{a}_i\left(\tau \right)\mu {\left(\left|u\right|+\left|v\right|\right)}^{p-1}d\tau }\right)}^{q-1}ds}\\ \ge \frac{1}{{\eta }_i}{\left(\frac{\mu }{\mathrm{\Gamma }\left(\alpha \right)}\right)}^{q-1}{\displaystyle \underset{0}{\overset{1}{\int }}{G}_i\left(s,s\right)}\\ \times {\left({\displaystyle \underset{s}{\overset{1}{\int }}{\left(\tau -s\right)}^{\alpha -1}{a}_i\left(\tau \right){\left(\frac{1}{{\eta }_1}{{\displaystyle \Vert u\Vert }}_{\infty }+\frac{1}{{\eta }_2}{{\displaystyle \Vert v\Vert }}_{\infty }\right)}^{p-1}d\tau }\right)}^{q-1}ds\\ \ge \frac{1}{{\eta }^2}{\left(\frac{\mu }{\mathrm{\Gamma }\left(\alpha \right)}\right)}^{q-1}{\xi }_i\Vert \left(u,v\right)\Vert \ge \Vert \left(u,v\right)\Vert \text{,}\end{array}$$

thus

$$\Vert T\left(u,v\right)\Vert \ge \Vert \left(u,v\right)\Vert ,\left(u,v\right)\in \partial {\mathrm{\Omega }}_2\cap P\text{.}$$

By Guo–Krasnosel'skii fixed-point theorem, we conclude that T has a fixed point $\left(u,v\right)\in P\cap \left({\overline{\mathrm{\Omega }}}_2\backslash {\mathrm{\Omega }}_1\right)\text{.}$ This means that the system $\left(S\right)$ has at least one positive solution $\left(u,v\right)$.▪

Example 4.1.

Consider the system $\left(S\right)$, with

$$\begin{array}{l}{f}_1\left(u,v\right)={\left(u+v\right)}^3, f_2\left(u,v\right)=e^{{\left(u+v\right)}^2}-1\\ a_1\left(t\right)=e^t,a_2\left(t\right)=1\\ g_1\left(t,u,v\right)=\frac{\left(1-t\right){\left(u+v\right)}^2}{3u+4v},g_2\left(t,u,v\right)=\frac{t}{9}u\end{array}$$

where $\alpha =\frac{1}{2},{\beta }_1={\beta }_2=\frac{4}{3},p=2,{\eta }_1=\frac{3}{2},{\eta }_2=\frac{5}{4}\text{.}$ We check easily that $A_{0,i}=0,A_{\infty ,i}=\infty ,i=\mathrm{1,2.}$ Clearly,

$$g_1\left(t,u,v\right)\le \frac{1-t}{3}\left(u+v\right),g_2\left(t,u,v\right)\le \frac{t}{5}\left(u+v\right)$$

So, the assumption $\left(H\right)$ hold. Thus the system $\left(S\right)$ has at least one positive solution by Theorem 4.1.