A discussion on a generalized Geraghty multi-valued mappings and applications

Abstract

This research intends to investigate the existence results for both coincidence points and common fixed point of generalized Geraghty multi-valued mappings endowed with a directed graph. The proven results are supported by an example. We also consider fractional integral equations as an application.

1 Introduction and preliminaries

The metric fixed point theory, which has wide common application potential in distinct quantitative sciences, develops itself in parallel with the developments in applied areas [8, 14–16]. One of the examples of this trend was given by Echenique [12] who considered metric fixed point theory in the framework of graphs. Later, Espinola and Kirk [13] applied some well-known fixed point results and techniques to the graph theory. On the other hand, in 2008, Jachymski [17] proposed a new contraction, the concept graph contraction, in order to generalize the distinguished Banach contraction principle.

In this paper, we shall consider generalized Geraghty multi-valued mappings in the context of complete metric spaces endowed with a graph. After investigating the existence and uniqueness of a fixed point for such mappings, we shall consider a fractional integral equation [1–7, 9, 10, 18–21, 23–25, 27] and we solve this equation via our obtained results.

Throughout the paper, we presume that all considered sets and subsets are non-empty. The pair $(M ,d)$ denotes a metric space and $(M^{*},d)$ represent complete metric space. Let Δ be the diagonal of $M \times M$. Assume that Γ is a directed graph with vertex set $V(\varGamma )$ which is coincided with M and edge set $E(\varGamma )$ which contains Δ. Suppose there is not any parallel edges in G. We shall say that $g:M \rightarrow M$ is a Banach G-contraction if:

• (i) $(x,y)\in E(\varGamma )$ implies $(g(x),g(y))\in E(\varGamma )$, for all $x,y\in M$,

• (ii) $\exists 0<\alpha <1$ such that, for $x,y\in M$,

  $$ (x,y)\in E(\varGamma )\quad \Rightarrow \quad d\bigl(g(x),g(y)\bigr)\le \alpha d(x,y). $$

Assume that $(M ,d)$ be a metric space and $P_{b,cl}(M )$ be the family of bounded and closed sets in M. For $x\in M$ and $M _{1}, M _{2}\in P_{b,cl}(M )$,

$$\begin{aligned}& D(x, M _{1})=\inf_{a\in M _{1}} d(x,a), \\& D(M _{1},M _{2})=\sup_{a\in M _{1}} D(M _{1},M _{2}). \end{aligned}$$

Define $H: P_{b,cl}(M ) \times P_{b,cl}(M ) \to [0,\infty )$ with

$$ H(M _{1},M _{2})=\max \Bigl\lbrace \sup _{x\in M _{1}} d(x,M _{2}), \sup_{y \in M _{2}}d(y,M _{2})\Bigr\rbrace , $$

for $M _{1},M _{2} \in P_{b,cl}(M )$. H is the famous Hausdorff metric.

Lemma 1.1

([11])

Let$(M ,d)$be a metric space. For$M _{1},M _{2}\in P_{b,cl}(M )$and$x,y\in M$, the following relations hold:

• (1) $D(x,M _{2})\leq d(x,b)$, $b\in M _{2}$,

• (2) $D(x,M _{2})\leq H(M _{1},M _{2})$,

• (3) $D(x,M _{1})\leq d(x,y)+D(y,M _{1})$,

• (4) for all$\omega \in M _{1}$, there exists some$\varphi \in M _{2}$such that$d(\omega ,\varphi )\leq q H(M _{1},M _{2})$, where$q>1$.

Definition 1.2

([28])

For a set M, let $\varGamma =(V(\varGamma ),E(\varGamma ))$ be a graph with $V(\varGamma )=M$. Then:

• (i) a mapping $T:M \rightarrow P_{b,cl}(M )$ is said to be graph preserving if it preserves the edges, i.e.,

  $$ \mbox{if } (x,y)\in E(\varGamma ), \mbox{then } (u,v)\in E( \varGamma ) \mbox{ for all } u\in Tx \mbox{ and } v\in Ty, $$

• (ii) mappings $S,T:M \rightarrow P_{b,cl}(M )$ are said to be mixed graph preserving respect to $h_{1}$, $h_{2}$ if the preserve the edges, i.e., for $x,y\in M$, if $(h_{1}(x), h_{2}(y))\in E(\varGamma )$, then $(x,\tau )\in E(\varGamma )$ for all $x\in Tx$ and $\tau \in Sy$ and

  $$ \mbox{if } \bigl(h_{2}(x), h_{1}(y)\bigr)\in E(\varGamma ) , \mbox{then } (b,r)\in E(\varGamma ) \mbox{ for all } b\in Sx \mbox{ and } r\in Ty . $$

Consider the class

$$\begin{aligned} \varPsi :=&\bigl\{ \psi : [0,\infty ) \to [0,\infty )\text{, increasing, continuous, and } \psi (ct) \leq c\psi (t), \\ & \text{for all }c>1, \text{with } \psi (0)=0\bigr\} . \end{aligned}$$

Let $\mathcal{F}$ denote the family of all functions $\beta : [0,\infty ) \to [0,1)$.

2 Main results

We shall start this section by introducing the notion of the generalized Geraghty-type G-multi-valued mapping.

Definition 2.1

On a metric space $(M ,d)$, let $\varGamma =(V(\varGamma ),E(\varGamma ))$ be a graph with vertex set $V(\varGamma )=M$ and the set $E(\varGamma )$ of its edges such that $E(\varGamma )\supseteq \Delta$. For $h_{1},h_{2}:M \rightarrow M$ and $S, T:M \rightarrow P_{b,cl}(M )$; S, T are called a generalized $h_{1}$, $h_{2}$-Geraghty-type G-multi-valued mapping provided that

• (i) S, T are mixed graph preserving respect to $h_{1}$, $h_{2}$;

• (ii) for $x,y\in M$ with $(h_{1}(x),h_{2}(y))\in E(\varGamma )$, there exists $L\geq 0$ such that, for

  $$\begin{aligned}& \begin{aligned}[b] P\bigl(h_{1}(x),h_{2}(y)\bigr)={}& \max \biggl\{ d\bigl(h_{1}(x),h_{2}(y)\bigr),D \bigl(h_{1}(x),Tx\bigr),D\bigl(h_{2}(y),Sy\bigr), \\ & \frac{D(h_{1}(x),Tx)+D(h_{2}(y),Sy)}{2}\biggr\} \quad\text{and} \end{aligned} \end{aligned}$$

  (1)
  $$\begin{aligned}& Q\bigl(h_{1}(x),h_{2}(y)\bigr)=\min \bigl\{ D\bigl(h_{1}(x),Sy\bigr),D\bigl(h_{2}(y),Tx\bigr) \bigr\} , \end{aligned}$$

  (2)

  we have

  $$\begin{aligned} \psi \bigl(H(Sx,Ty)\bigr) \leq& \gamma \bigl(\psi \bigl(P\bigl(h_{1}(x),h_{2}(y) \bigr)\bigr)\bigr)\psi \bigl(P\bigl(h_{1}(x),h_{2}(y) \bigr)\bigr) \\ &{}+L \phi \bigl(Q\bigl(h_{1}(x),h_{2}(y)\bigr) \bigr), \end{aligned}$$

  (3)

  and if $(h_{2}(x),h_{1}(y))\in E(\varGamma )$ then

  $$\begin{aligned} \psi \bigl(H(Sx,Ty)\bigr) \leq& \gamma \bigl(\psi \bigl(P\bigl(h_{2}(x),h_{1}(y) \bigr)\bigr)\bigr)\psi \bigl(P\bigl(h_{2}(x),h_{1}(y) \bigr)\bigr) \\ &{}+L \phi \bigl(Q\bigl(h_{2}(x),h_{1}(y)\bigr) \bigr), \end{aligned}$$

  (4)

  where $\gamma \in \mathcal{F}$ and $\psi ,\phi \in \psi$.

Theorem 2.2

Let$(M ,d)$be a complete metric space endowed with a graph$\varGamma =(V(\varGamma ), E(\varGamma ))$, $h_{1},h_{2}:M \rightarrow M$are surjective and$S,T:M \rightarrow P_{b,cl}(M )$generalized$h_{1}$, $h_{2}$-Geraghty-typeG-multi-valued mapping in$(M ,d)$. Suppose

• (i) ∃ $x_{0}\in M$such that$(h_{1}(x_{0}),u)\in E(\varGamma )$for some$u\in Tx_{0}$,

• (ii) if$(h_{1}(x), h_{2}(y))\in E(\varGamma )$, then$(e,f)\in E(\varGamma )$for all$e\in Tx$and$f\in Sy$and if$(h_{2}(x), h_{1}(y))\in E(\varGamma )$, then$(w,r)\in E(\varGamma )$for all$w\in Sx$and$r\in Ty$,

• (iii) for$\{x_{n}\}_{n\in \mathbb{N}}$inM, if$x_{n}\rightarrow x$and$(x_{n},x_{n+1})\in E(\varGamma )$for$n\in \mathbb{N}$, then there is a subsequence$\{x_{n_{k}}\}_{n_{k}\in \mathbb{N}}$such that$(x_{n_{k}},x)\in E(\varGamma )$for$n_{k}\in \mathbb{N}$.

Then there exist$u,v\in M$such that$h_{1}(u)\in Tu$or$h_{2}(v)\in Sv$.

Proof

Regarding that $h_{2}$ is surjective, one can find $x_{1}\in M$ such that $h_{2}(x_{1})\in Tx_{0}$ and $(h_{1}(x_{0}), h_{2}(x_{1}))\in E(\varGamma )$. Let $q=\frac{1}{\sqrt{\gamma (\psi (d(h_{1}(x_{0}),h_{2}(x_{1}))))}}$. Then $q>1$, so

$$ 0< D\bigl(h_{2}(x_{1}),Sx_{1}\bigr)\leq H(Tx_{0},Sx_{1})< qH(Tx_{0},Sx_{1}). $$

By Lemma 1.1 and (ii), $h_{1}$ is surjective, which implies that there exists $x_{2}\in M$ with $h_{1}(x_{2})\in Sx_{1}$ and $(h_{2}(x_{1}), h_{1}(x_{2}))\in E(\varGamma )$, hence

$$\begin{aligned} \psi \bigl(d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr)\bigr) < &\psi \bigl(qH(Tx_{0},Sx_{1})\bigr)\leq q \psi \bigl(H(Tx_{0},Sx_{1})\bigr) \\ \leq& q\gamma \bigl(\psi \bigl(P\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr)\bigr)\psi \bigl(P\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr) \\ &{}+qL \phi \bigl(Q\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr), \end{aligned}$$

(5)

where

$$\begin{aligned} P\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr) =&\max \biggl\{ d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr),D\bigl(h_{1}(x_{0}),Tx_{0}\bigr),D \bigl(h_{2}(x_{1}),Sx_{1}\bigr), \\ &\frac{D(h_{1}(x_{0}),Tx_{0})+D(h_{2}(x_{1}),Sx_{1})}{2}\biggr\} \\ =& \max \bigl\{ d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr),d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr) \bigr\} \end{aligned}$$

(6)

and

$$\begin{aligned} Q\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr) =&\min \bigl\{ D\bigl(h_{1}(x_{0}),Sx_{0} \bigr),D\bigl(h_{2}(x_{1}),Tx_{0}\bigr) \bigr\} \\ \leq& \min \bigl\{ d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr),d\bigl(h_{2}(x_{1}),h_{2}(x_{1}) \bigr) \bigr\} =0. \end{aligned}$$

(7)

If we have

$$ \max \bigl\{ d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr),d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr)\bigr\} =d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr), $$

then by (5) we get

$$\begin{aligned} \psi \bigl(d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr)\bigr) < &\psi \bigl(qH(Tx_{0},Sx_{1})\bigr)\leq q \psi \bigl(H(Tx_{0},Sx_{1})\bigr) \\ \leq& q\gamma \bigl(\psi \bigl(P\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr)\bigr)\psi \bigl(P\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr) \\ &{}+qL \phi \bigl(Q\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr) \\ =&q\gamma \bigl(\psi \bigl(d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr)\bigr)\bigr)\psi \bigl(d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr)\bigr) \\ =&\sqrt{\gamma \bigl(\psi \bigl(d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr)\bigr)\bigr)}\psi \bigl(d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr)\bigr) \\ < & \psi \bigl(d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr)\bigr), \end{aligned}$$

a contradiction.

Hence, we obtain $\max \{d(h_{1}(x_{0}),h_{2}(x_{1})),d(h_{2}(x_{1}),h_{1}(x_{2}))\}=d(h_{1}(x_{0}),h_{2}(x_{1}))$ and so by (5)

$$ \psi \bigl(d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr)\bigr) \leq \sqrt{\gamma \bigl(\psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr)\bigr)} \psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr). $$

(8)

Keeping $\psi \in \psi$, in mind together with $\sqrt{\gamma (\psi (d(h_{1}(x_{0}),h_{2}(x_{1}))))}<1$, we get

$$ \begin{aligned}[b] &\psi \biggl( \frac{1}{\sqrt{\gamma (\psi (d(h_{1}(x_{0}),h_{2}(x_{1}))))}}d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr)\biggr) \\ &\quad \leq \frac{1}{\sqrt{\gamma (\psi (d(h_{1}(x_{0}),h_{2}(x_{1}))))}} \psi \bigl(d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr)\bigr) \\ &\quad < \psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr). \end{aligned} $$

(9)

Since ψ is increasing, we have

$$ d\bigl(h_{2}(x_{1}),h_{1}(x_{2}) \bigr)\leq \sqrt{\gamma \bigl(\psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr)\bigr)}d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr). $$

Recall that $h_{1}(x_{2})\in Sx_{1}$ and $h_{2}(x_{1})\in Tx_{0}$. Choose

$$ q_{1}=\frac{\sqrt{\gamma (\psi (d(h_{1}(x_{0}),h_{2}(x_{1}))))}\psi (d(h_{1}(x_{0}),h_{2}(x_{1}))) }{\psi (d(h_{2}(x_{1}),h_{1}(x_{2})))}. $$

By (8), we have $q_{1}>1$. If $h_{1}(x_{2})\in Tx_{2}$, then $x_{2}$ forms a coincidence point for $h_{1}$ and T. We presume that $h_{1}(x_{2})\notin Tx_{2}$. We get

$$ 0< \psi \bigl(d\bigl(h_{1}(x_{2}),Tx_{2} \bigr)\bigr)\leq \psi \bigl(H(Sx_{1},Tx_{2}) \bigr)< q_{1} \psi \bigl(H(Sx_{1},Tx_{2}) \bigr). $$

Hence, there exists $h_{2}(x_{3})\in Tx_{2}$ such that $(h_{1}(x_{2}),h_{2}(x_{3}))\in E(\varGamma )$ and

$$ \begin{aligned} \psi \bigl(d\bigl(h_{1}(x_{2}),h_{2}(x_{3}) \bigr)\bigr)& < q_{1}\psi \bigl(H(Sx_{1},Tx_{2}) \bigr) \\ &\leq q_{1}\gamma \bigl(\psi \bigl(P\bigl(h_{1}(x_{1}),h_{2}(x_{2}) \bigr)\bigr)\bigr)\psi \bigl(P\bigl(h_{1}(x_{1}),h_{2}(x_{2}) \bigr)\bigr) \\ &\quad {}+q_{1}L\phi \bigl(Q\bigl(h_{1}(x_{1}),h_{2}(x_{2}) \bigr)\bigr). \end{aligned} $$

Similarly, $P(h_{1}(x_{1}),h_{2}(x_{2}))\leq d(h_{1}(x_{1}),h_{2}(x_{2}))$ and $Q(h_{1}(x_{1}),h_{2}(x_{2}))=0$. By (8) and a property of $(\gamma )$, we have

$$ \begin{aligned}[b] &\psi \bigl(d \bigl(h_{1}(x_{2}),h_{2}(x_{3}) \bigr)\bigr) \\ &\quad \leq \sqrt{\gamma \bigl(\psi \bigl(d\bigl(h_{1}(x_{1}),h_{2}(x_{2}) \bigr)\bigr)\bigr)} \psi \bigl(d\bigl(h_{1}(x_{1}),h_{2}(x_{2}) \bigr)\bigr) \\ &\quad \leq \sqrt{\gamma \bigl(\psi \bigl(d\bigl(h_{1}(x_{1}),h_{2}(x_{2}) \bigr)\bigr)\bigr)}\sqrt{ \gamma \bigl(\psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr)\bigr)}\psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr). \end{aligned} $$

(10)

By (8) and $\sqrt{\gamma (\psi (d(h_{1}(x_{0}),h_{2}(x_{1}))))}<1$, we have

$$ \psi \bigl(d\bigl(h_{1}(x_{1}),h_{2}(x_{2}) \bigr)\bigr)\leq \psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr). $$

The function γ is increasing, by (10), we obtain

$$ \psi \bigl(d\bigl(h_{1}(x_{2}),h_{2}(x_{3}) \bigr)\bigr)\leq \bigl(\sqrt{\gamma \bigl(\psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr)\bigr)}\bigr)^{2} \psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr). $$

(11)

Again, by (9),

$$\begin{aligned} d\bigl(h_{1}(x_{2}),h_{2}(x_{3}) \bigr)\leq \bigl(\sqrt{\gamma \bigl(\psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr)\bigr)}\bigr)^{2}d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr) \end{aligned}$$

manifestly, $h_{1}(x_{2})\neq h_{2}(x_{3})$. Take

$$ q_{2}= \frac{(\sqrt{\gamma (\psi (d(h_{1}(x_{0}),h_{2}(x_{1}))))})^{2} \psi (d(h_{1}(x_{0}),h_{2}(x_{1})))}{\psi (d(h_{1}(x_{2}),h_{2}(x_{3})))}. $$

Then $q_{2}>1$. If $h_{2}(x_{3})\in Sx_{3}$, then $x_{3}$ is a coincidence point of $h_{2}$ and S. Assume that $h_{2}(x_{3})\notin Sx_{3}$. Then

$$ 0< \psi \bigl(d\bigl(h_{2}(x_{3}),Sx_{3} \bigr)\bigr)\leq \psi \bigl(H(Tx_{2},Sx_{3})\bigr) < q_{2} \psi \bigl(H(Tx_{2},Sx_{3})\bigr). $$

Thus there exists $h_{1}(x_{4})\in Sx_{3}$ such that $(h_{2}(x_{3}),h_{1}(x_{4}))\in E(\varGamma )$ and

$$\begin{aligned} \psi \bigl(d\bigl(h_{2}(x_{3}),h_{1}(x_{4}) \bigr)\bigr) < &q_{2}\psi \bigl(H(Tx_{2},Sx_{3}) \bigr) \\ \leq& q_{2}\gamma \bigl(\psi \bigl(P\bigl(h_{1}(x_{2}),h_{2}(x_{3}) \bigr)\bigr)\bigr) \psi \bigl(P\bigl(h_{1}(x_{2}),h_{2}(x_{3}) \bigr)\bigr) \\ &{}+q_{2}L\phi \bigl(Q\bigl(h_{1}(x_{2}),h_{2}(x_{3}) \bigr)\bigr). \end{aligned}$$

(12)

Similarly, $P(h_{1}(x_{2}),h_{2}(x_{3}))\leq d(h_{1}(x_{2}),h_{2}(x_{3}))$ and $Q(h_{1}(x_{2}),h_{2}(x_{3}))=0$.

So, by (12),

$$\begin{aligned} &\psi \bigl(d\bigl(h_{2}(x_{3}),h_{1}(x_{4}) \bigr)\bigr) \\ &\quad \leq \sqrt{\gamma \bigl(\psi \bigl(d\bigl(h_{1}(x_{2}),h_{2}(x_{3}) \bigr)\bigr)\bigr)} \psi \bigl(d\bigl(h_{1}(x_{2}),h_{2}(x_{3}) \bigr)\bigr) \\ &\quad \leq \sqrt{\gamma \bigl(\psi \bigl(d\bigl(h_{1}(x_{2}),h_{2}(x_{3}) \bigr)\bigr)\bigr)}\bigl( \sqrt{\gamma \bigl(\psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr)\bigr)}\bigr)^{2} \\ &\qquad {}\times\psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr). \end{aligned}$$

(13)

By (11) and $\sqrt{\gamma (\psi (d(h_{1}(x_{0}),h_{2}(x_{1}))))}^{2}<1$, we have

$$ \psi \bigl(d\bigl(h_{1}(x_{2}),h_{2}(x_{3}) \bigr)\bigr)\leq \psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr). $$

Again, γ is increasing, so using (13),

$$ d\bigl(h_{2}(x_{3}),h_{1}(x_{4}) \bigr)\leq \bigl(\sqrt{\gamma \bigl(\psi \bigl(d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr)\bigr)\bigr)}\bigr)^{3} d\bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr). $$

It is clear that $h_{2}(x_{3})\neq h_{1}(x_{2})$. Put

$$ q_{3}= \frac{(\sqrt{\gamma (\psi (d(h_{1}(x_{0}),h_{2}(x_{1}))))})^{3}\psi (d(h_{1}(x_{0}),h_{2}(x_{1})))}{\psi (d(h_{1}(x_{2}),h_{2}(x_{3})))}. $$

Then $q_{3}>1$. Continuing this process, we construct a sequence $\{h(x_{n})\}$ in M such that

$$\begin{aligned}& h_{1}(x_{2n})\in Sx_{2n-1}, \qquad h_{2}(x_{2n-1})\in T(x_{2n-2}) \\& \bigl(h_{1}(x_{2n-2}),h_{1}(x_{2n-1}) \bigr),\bigl(h_{1}(x_{2n-1}),h_{1}(x_{2n}) \bigr) \in E(\varGamma ). \end{aligned}$$

Define the sequence $\{h(x_{n})\}$ as follows:

$$ h(x_{n})= \textstyle\begin{cases} h_{1}(x_{n}),& n \mbox{ is even}, \\ h_{2}(x_{n}),& n \mbox{ is odd}. \end{cases} $$

Let $t=\sqrt{\gamma (\psi (d(h_{1}(x_{0}),h_{2}(x_{1}))))}$, then $0< t<1$. We have

$$ d\bigl(h(x_{n}),h(x_{n+1})\bigr)\leq t^{n}d \bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr). $$

Since $0< t<1$, we have

$$ \sum_{n=0}^{\infty } \bigl(d \bigl(h(x_{n}),h(x_{n+1})\bigr)\bigr)\leq d \bigl(h_{1}(x_{0}),h_{2}(x_{1}) \bigr) \sum_{n=0}^{\infty }t^{n}< \infty . $$

Thus, the sequence $\{h(x_{n})\}$ is Cauchy in $(M ,d)$. Since M is complete, the sequence ${h(x_{n})}$ converge to point w for some $w\in M$. Let $u,v\in M$ with $h_{1}(u)=w=h_{2}(v)$. By (iii), there exists $\{h(x_{n_{k}})\}$ such that $(h(x_{n_{k}}),h_{1}(u))\in E(\varGamma )$ for any $n\in \mathbb{N}$. We assert that $h_{1}(u)\in Tu$ or $h_{2}(v)\in Sv$.

Let ${\mathcal{A}}=\{n_{k}|n_{k} \mbox{ is even}\}$ and ${\mathcal{B}} = \{n_{k} | n_{k} \mbox{ is odd}\}$. Then obviously ${\mathcal{A}}\cup {\mathcal{B}}$ is infinite and so at least ${\mathcal{A}}$ or ${\mathcal{B}}$ must be infinite. In the case that ${\mathcal{A}}$ is infinite, for each $h_{2}(x_{n_{k}+1})$, $n_{k}\in {\mathcal{A}}$, we have

$$\begin{aligned} D\bigl(h_{2}(v),S(v)\bigr) \leq& d\bigl(h_{2}(v),h_{2}(x_{n_{k}+1}) \bigr)+D\bigl(h_{2}(x_{n_{k}+1}),S(v)\bigr) \\ \leq& d\bigl(h_{2}(v),h_{2}(x_{n_{k}+1})\bigr)+H \bigl(T(x_{n_{k}}),S(v)\bigr) \\ \leq& d\bigl(h_{2}(v),h_{2}(x_{n_{k}+1})\bigr)+ \gamma \bigl(\psi \bigl(P\bigl(h_{1}(x_{n_{k}}),h_{2}(v) \bigr)\bigr)\bigr) \psi \bigl(P\bigl(h_{1}(x_{n_{k}}),h_{2}(v) \bigr)\bigr) \\ &{}+L\phi \bigl(Q\bigl(h_{1}(x_{n_{k}}),h_{2}(v) \bigr)\bigr) \\ =&d\bigl(h_{2}(v),h_{2}(x_{n_{k}+1})\bigr)+ \gamma \bigl(\psi \bigl(d\bigl(h_{1}(x_{n_{k}}),h_{1}(v) \bigr)\bigr)\bigr) \psi \bigl(d\bigl(h_{1}(x_{n_{k}}),h_{2}(v) \bigr)\bigr), \end{aligned}$$

where

$$\begin{aligned}& P\bigl(h_{1}(x_{n_{k}}),h_{2}(v)\bigr)=\max \biggl\{ d\bigl(h_{1}(x_{n_{k}}),h_{2}(v)\bigr),D \bigl(h_{1}(x_{n_{k}}), Tx_{n_{k}}\bigr),D \bigl(h_{2}(v),Sv\bigr), \\& \hphantom{P(h_{1}(x_{n_{k}}),h_{2}(v))={}}\frac{D(h_{1}(x_{n_{k}}),Tx_{n_{k}})+D(h_{2}(v),Sv)}{2}\biggr\} \quad \text{and} \\& Q\bigl(h_{1}(x_{n_{k}}),h_{2}(v)\bigr)=\min \bigl\{ D\bigl(h_{1}(x_{n_{k}}),Sv\bigr),D \bigl(h_{2}(v),Tx_{n_{k}}\bigr) \bigr\} . \end{aligned}$$

(14)

With $n \rightarrow \infty$, we get $Q(h_{1}(x_{n_{k}}),h_{2}(v))=0$ and with regard to (14) we obtain

$$ \psi \bigl(D\bigl(h_{2}(v),S(v)\bigr)\bigr) \leq \gamma \bigl(\psi \bigl(h_{2}(v),S(v)\bigr)\bigr)\psi \bigl(D\bigl(h_{2}(v),S(v) \bigr)\bigr)< \psi \bigl(D\bigl(h_{2}(v),S(v)\bigr)\bigr), $$

which is a contradiction, unless $D(h_{2}(v),S(v))=0$. Since Tv is closed, thus $h_{2}(v)\in S(v)$. Similarly, we can prove that $h_{1}(v)\in T(v)$ when ${\mathcal{B}}$ is infinite. This completes the proof. We notice also that if both ${\mathcal{A}}$, ${\mathcal{B}}$ are infinite, then $h_{1}(v)\in T(v)$ and $h_{2}(v)\in S(v)$. □

Example 2.3

Let $M =[0,1]$ and d be the standard metric on M. Let $\varGamma =(V(\varGamma ),E(\varGamma ))$ be a directed graph with $V(\varGamma ) =M$ and

$$ E(\varGamma )= \biggl\{ (x,x),\biggl(0,\frac{1}{4}\biggr),\biggl( \frac{1}{4},0\biggr), \biggl(0, \frac{1}{16}\biggr),\biggl( \frac{1}{16},0\biggr),\biggl(\frac{1}{4},\frac{1}{16} \biggr),\biggl( \frac{1}{16},\frac{1}{4}\biggr): x\in M \biggr\} . $$

Let $T:M \rightarrow P_{b,cl}(M )$ be defined by

$$\begin{aligned}& Tx= \textstyle\begin{cases} \{\frac{1}{4}\}& \mbox{if } x=1, \\ \{0,\frac{1}{4}\}& \mbox{if } x\in (0,1)-\{\frac{1}{4},\frac{1}{2}\}, \\ \{\frac{1}{16}\}& \mbox{if } x\in \{0, \frac{1}{4},\frac{1}{2}\}, \end{cases}\displaystyle \\& Sx= \textstyle\begin{cases} \{\frac{1}{16}\}& \mbox{if } x=0,\frac{1}{16},\frac{1}{4},1, \\ \{0,\frac{1}{16}\}& \mbox{if } x\in (0,1)-\{\frac{1}{16},\frac{1}{4}\}. \end{cases}\displaystyle \end{aligned}$$

Let $h_{1},h_{2}:M \rightarrow M$ be defined by $h_{1}(x)=x^{2}$, $h_{2}(x)=x$. Consider $\psi (t)=t$ and $\gamma (t)=\frac{t+1}{t+2}$. Then it is evident that S, T are mixed Geraghty-type G-multi-valued respect to $h_{1}$, $h_{2}$. Note that (i), (ii) and (iii) of Theorem 2.2 hold. Besides, if $(h(x),h(y))\in E(\varGamma )$, then $H(Th(x),Th(y))=0$. Hence, for all $x,y\in M$ we have $(h(x), h(y))\in E(\varGamma )$, ergo

$$ \psi \bigl(H(Sx,Ty)\bigr)\leq \gamma \bigl(\psi \bigl(d\bigl(h_{1}(x),h_{2}(y) \bigr)\bigr)\bigr)\psi \bigl(d\bigl(h_{1}(x),h_{2}(y) \bigr)\bigr), $$

for $x,y\in M$ with $(h_{1}(x),h_{2}(y))\in E(\varGamma )$.

If $x,y\in M$ with $(h_{2}(x),h_{1}(y))\in E(\varGamma )$, then

$$ \psi \bigl(H(Sx,Ty)\bigr)\leq \gamma \bigl(\psi \bigl(d\bigl(h_{2}(x),h_{1}(y) \bigr)\bigr)\bigr)\psi \bigl(d\bigl(h_{2}(x),h_{1}(y) \bigr)\bigr). $$

By Theorem 2.2, there exist $u,v\in M$ such that $h_{1}(u)\in Tu$ or $h_{2}(v)\in Sv$. In this example, $u=\frac{1}{4}$ or $u=\frac{1}{16}$.

If in Theorem 2.2, we set $h_{1}=h_{2}=h$, then we get the following corollary.

Corollary 2.4

Let$(M ,d)$be a complete metric space with the directed graphΓ, $h:M \rightarrow M$is surjective map and$S,T:M \rightarrow P_{b,cl}(M )$beh-graph preserving with

$$ \psi \bigl(H(Sx,Ty)\bigr)\leq \gamma \bigl(\psi \bigl(d\bigl(h(x),h(y)\bigr) \bigr)\bigr)\psi \bigl(d\bigl(h(x),h(y)\bigr)\bigr), $$

for all$x,y\in M$with$(h(x),h(y))\in E(\varGamma )$. Suppose

• (i) there exists$x_{0}\in M$such that$(h(x_{0}),u)\in E(\varGamma )$for some$u\in Tx_{0}$;

• (ii) for any sequence$\{x_{n}\}_{n\in \mathbb{N}}$inM, if$x_{n}\rightarrow x$and$(x_{n},x_{n+1})\in E(\varGamma )$for$n\in \mathbb{N}$, then there is a subsequence$\{x_{n_{k}}\}_{n_{k}\in \mathbb{N}}$such that$(x_{n_{k}},x)\in E(\varGamma )$for$n_{k}\in \mathbb{N}$.

Then there exist$u,v\in M$such that$h(u)\in Tu$or$h(v)\in Sv$.

Definition 2.5

Let $(M ,d)$ be a metric space endowed with a partial order ≤. For each $M _{1},M _{2}\in M$, $M _{1}\preceq M _{2}$ if $\omega _{1}\leq \omega _{2}$ for any $\omega _{1}\in M _{1}$, $\omega _{2}\in M _{2}$, $h:M \rightarrow M$ a surjective map, and $T:M \rightarrow P_{b,cl}(M )$. T is said to be h-increasing if for any $x,y\in M$, $h(x)\leq h(y)$ implies $Tx\preceq Ty$.

Theorem 2.6

Let$(M ,d)$be a complete metric space with partially order ≤, $h:M \rightarrow M$be a surjective map and$T:M \rightarrow P_{b,cl}(M )$be a multi-valued mapping. Suppose that

• (i) Tish-increasing;

• (ii) there exist$x_{0}\in M$and$u\in Tx_{0}$such that$h(x_{0}) \leq u$;

• (iii) for each sequence${x_{k}}$such that$h(x_{k})\leq h(x_{k+1})$, $k\in \mathbb{N}$and$h(x_{k})$converges to$h(x)$for some$x\in M$, then$h(x_{k})\leq h(x)$;

• (iv) for$x,y\in M$with$h(x)\leq h(y)$, we have

  $$ \psi \bigl(H(Tx,Ty)\bigr)\leq \gamma \bigl(\psi \bigl(d\bigl(h(x),h(y)\bigr) \bigr)\bigr)\psi \bigl(d\bigl(h(x),h(y)\bigr)\bigr), $$

  where$\gamma \in \mathcal{F}$and$\psi ,\phi \in \psi$.

Then there is$u\in M$so that$h(u)\in Tu$. In addition, ifhis injective, then there is a unique$u\in M$such that$h(u)\in Tu$.

Proof

Let $\varGamma = (V(\varGamma ), E(\varGamma ))$, be a graph with $V(\varGamma ) =M$ and

$$ E(\varGamma ) = \bigl\{ (x, y) | x \leq y\bigr\} , $$

let $(h(x), h(y))\in E(\varGamma )$, then $h(x) \leq h(y)$ and by (i), $Tx\preceq Ty$. For each $u\in Tx$, $v\in Ty$, we have $u \leq v$, thus $(u, v)\in E(\varGamma )$. That is, T is h-graph preserving. By (ii), there exist $x_{0}\in M$ and $u\in Tx_{0}$ such that $h(x_{0}) \leq u$. So $(h(x),u)\in E(\varGamma )$ and hence the property (i) in Corollary 2.4 is satisfied. Moreover, we obtain the property (ii) of Corollary 2.4 from the assumption (iii). Set $S=T$, then the S, T are h-graph preserving mappings and fulfill

$$ \psi \bigl(H(Tx,Ty)\bigr)\leq \gamma \bigl(\psi \bigl(d\bigl(h(x),h(y)\bigr) \bigr)\bigr)\psi \bigl(P\bigl(h(x),h(y)\bigr)\bigr), $$

for all $x, y\in M$ with $(h(x), h(y))\in E(\varGamma )$. By Corollary 2.4 we get $h(u)\in Tu$ for some $u\in M$.

Now, in addition, we suppose that h is injective. Let $u, v\in M$ be such that $h(u)\in Tu$ and $h(v)\in Tv$. Suppose, on the contrary, that $h(u)\neq h(v)$. We assume, without loss of generality, that $h(u) < h(v)$. Since $h(u)\in Tu$ and $h(v)\in Tv$, it yields $D(h(u), Tu) = D(h(v),Tv)$ and hence

$$\begin{aligned} \psi \bigl(d\bigl(h(u),h(v)\bigr)\bigr) \leq& \psi \bigl(H(Tu,Tv)\bigr)\leq \gamma \bigl(\psi \bigl(d\bigl(h(u),h(v)\bigr)\bigr)\bigr) \psi \bigl(d \bigl(h(u),h(v)\bigr)\bigr), \\ < &\psi \bigl(d\bigl(h(u),h(v)\bigr)\bigr). \end{aligned}$$

This leads to a contradiction. Thus $h(u) = h(v)$. Since h is injective, we have $u = v$. □

Corollary 2.7

Let$(M ,d)$be a complete metric space endowed with a graph$\varGamma =(V(\varGamma ), E(\varGamma ))$, $S,T:M \rightarrow M$be generalized mappings such that

$$ \psi \bigl(d(Sx,Ty)\bigr)\leq \gamma \bigl(\psi \bigl(d(x,y) \bigr)\bigr)\psi \bigl(d(x,y)\bigr), $$

(15)

mapping in$(M ,d)$. Suppose that

• (i) $\exists x_{0}\in M$such that$(x_{0},u)\in E(\varGamma )$for some$u\in Tx_{0}$,

• (ii) if$(x,y)\in E(\varGamma )$, then$(e,f)\in E(\varGamma )$for all$e\in Tx$and$f\in Sy$and if$(x,y)\in E(\varGamma )$, then$(w,r)\in E(\varGamma )$for all$w\in Sx$and$r\in Ty$,

• (iii) for$\{x_{n}\}_{n\in \mathbb{N}}$inM, if$x_{n}\rightarrow x$and$(x_{n},x_{n+1})\in E(\varGamma )$for$n\in \mathbb{N}$, then there is a subsequence$\{x_{n_{k}}\}_{n_{k}\in \mathbb{N}}$such that$(x_{n_{k}},x)\in E(\varGamma )$for$n_{k}\in \mathbb{N}$.

Then there exist$u,v\in M$such that$u\in Tu$or$v\in Sv$.

3 Application

In this section, we apply our theorem for a solution of the following integral system:

$$ \textstyle\begin{cases} x(t)=\int _{a}^{b}K(t,s)f(t,s,x(s),y(s))\,ds+h(t), \\ y(t)=\int _{a}^{b}K(t,s)f(t,s,y(s),x(s))\,ds+h(t), \quad t\in [a,b], \end{cases} $$

(16)

where $X:= C([a,b],\mathbb{R})$ with $\| x \| _{\infty }=\sup_{t\in [a,b]}|x(t)|$, for $x\in C([a,b],\mathbb{R})$.

Let G be a graph, defined by $V(\varGamma )=X$, and

$$ E(\varGamma )=\bigl\{ (x,y)\in X\times X:x(t)\leq y(t) \mbox{ or } y(t)\leq x(t) \mbox{ for } t\in [a,b]\bigr\} . $$

Let d be the metric induced by the norm. It follows that $(X,d)$ is a complete metric space endowed with a directed graph Γ. Define mappings $S,T: X\rightarrow X$ by

$$ \textstyle\begin{cases} Sx(t)=\int _{a}^{b}K(t,s)f(t,s,x(s),y(s))\,ds+h(t), \\ Ty(t)=\int _{a}^{b}K(t,s)f(t,s,y(s),x(s))\,ds+h(t), \end{cases} $$

where $x,y\in C([a,b],\mathbb{R})$.

Theorem 3.1

Consider Eq. (16) and suppose that

• 1. $f:[a,b]\times [a,b]\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$is continuous function and there exists$M>0$such that$\max_{t\in [a,b]}\int _{a}^{b}K(t,s)\,ds< M$.

• 2.
  $$ \bigl\vert f\bigl(t,s,y(s),x(s)\bigr)-f\bigl(t,s,x(s),y(s)\bigr) \bigr\vert \leq \frac{1}{\alpha } \frac{ \vert y(s)-x(s) \vert ^{2}}{1+ \vert y(s)-x(s) \vert }. $$

• 3. There exists a function$x_{0}\in X$such that

  $$ x_{0}(t)\leq \int _{a}^{b}K(t,s)f\bigl(t,s,x_{0}(s),Tx_{0}(s) \bigr)\,ds+h(t),\quad t \in [a,b]. $$

• 4. The inequality$x(s)\leq y(s)$implies$Tx(s)\leq Sy(s)$and$Sx(s)\leq Ty(s)$.

• 5. For$\{x_{n}\}_{n\in \mathbb{N}}$inM, if$x_{n}\rightarrow x$and$x_{n}(t)\leq x_{n+1}(t)$for$n\in \mathbb{N}$, then there is a subsequence$\{x_{n_{k}}\}_{n_{k}\in \mathbb{N}}$such that$x_{n_{k}}(t)\leq x(t)$for$n_{k}\in \mathbb{N}$, $t\in [a,b]$.

Then there exist$u,v\in X$such that$u(t)=\int _{a}^{b}K(t,s)f(t,s,u(s),v(s))\,ds+h(t)$or$v(t)=\int _{a}^{b}K(t,s)f(t,s,v(s),u(s))\,ds+h(t)$.

Proof

Let $x,y\in X$, using (2), we get

$$\begin{aligned} \bigl\vert Sx(t)-Ty(t) \bigr\vert =& \biggl\vert \int _{a}^{b}K(t,s)f\bigl(t,s,x(s),y(s)\bigr) \,ds- \int _{a}^{b}K(t,s)f\bigl(t,s,y(s),x(s)\bigr) \,ds \biggr\vert \\ \leq& \sup_{t\in [a,b]} \bigl\vert f\bigl(t,s,x(s),y(s)\bigr)-f \bigl(t,s,y(s),x(s)\bigr) \bigr\vert M \\ \leq& \frac{1}{M}\frac{ \vert x(s)-y(s) \vert ^{2}}{1+ \vert x(s)-y(s) \vert }M= \frac{ \vert x(s)-y(s) \vert ^{2}}{1+ \vert x(s)-y(s) \vert } \\ =&\gamma \bigl( \bigl\vert x(s)-y(s) \bigr\vert \bigr) \bigl\vert x(s)-y(s) \bigr\vert , \end{aligned}$$

where $\gamma (t)=\frac{t}{t+1}$. With setting $\psi (t)=t$, the condition 15 in Corollary 2.7 holds. With considering (3) and (4) from assumption of theorem and by definition of the graph Γ we deduce the conditions (i) and (ii) of in Corollary 2.7 hold. Also by the condition $(5)$ from assumption of theorem, the condition (iii) of Corollary 2.7 holds. As a result, we have

$$\begin{aligned}& u(t)= \int _{a}^{b}K(t,s)f\bigl(t,s,u(s),v(s)\bigr) \,ds+h(t), \quad \text{or} \\& v(t)= \int _{a}^{b}K(t,s)f\bigl(t,s,v(s),u(s)\bigr) \,ds+h(t) . \end{aligned}$$

 □

Now we use the results of our findings to solve fractional differential equations.

Definition 3.2

([22, 26])

The Riemann–Liouville fractional integral of order $\alpha >0$ of a continuous function $f:(0,+\infty )\rightarrow (-\infty , +\infty )$ is given by

$$ I_{0^{+}}^{\alpha }f(\varsigma )=\frac{1}{\varGamma (\alpha )} \int _{0}^{ \varsigma }(\varsigma -\eta )^{\alpha -1}f(\eta )\,d\eta , $$

provided the right-hand side is pointwise defined on $(0, +\infty )$.

Definition 3.3

([22, 26])

The Riemann–Liouville fractional derivative of order $\alpha >0$ of a continuous function $f:(0,+\infty )\rightarrow (-\infty , +\infty )$ is given by

$$ D_{0^{+}}^{\alpha }f(\varsigma )=\frac{1}{\varGamma (n-\alpha )} \biggl( \frac{d}{dt} \biggr)^{n} \int _{0}^{\varsigma } (\varsigma -\eta )^{n- \alpha -1} f(\eta )\,d\eta , $$

where $n=[\alpha ]+1$, $[\alpha ]$ denotes the integer part of the number α, provided that the right-hand side is pointwise defined on $(0, +\infty )$.

In this paper we discuss the local existence and uniqueness of positive solutions for the following coupled system of fractional boundary value problem subject to integral boundary conditions:

$$ \textstyle\begin{cases} D_{0^{+}}^{\alpha } u(\varsigma ) + f(\varsigma ,u(\varsigma ),v( \varsigma ))=0, \\ D_{0^{+}}^{\alpha } v(\varsigma ) + f(\varsigma ,v( \varsigma ),u(\varsigma ))=0, \quad 0< \varsigma < 1, \\ u(0)=0, \qquad u(1)=\int _{0}^{1}\phi (\varsigma )u(\varsigma )\, d \varsigma , \\ v(0)=0 ,\qquad v(1)=\int _{0}^{1}\psi (\varsigma )v( \varsigma )\,d\varsigma, \end{cases} $$

(17)

where $1<\alpha \leq 2$, $\phi ,\psi \in L^{1}[0,1]$ are nonnegative and $f\in C([0,1]\times [0,\infty ),[0,\infty ))$ and D is the standard Riemann–Liouville fractional derivative. The functions $\phi (\varsigma )$, $\psi (\varsigma )$ satisfy the following conditions:

$$ (Q) \quad \phi , \psi :[0,1] \rightarrow [0, + \infty )\quad \text{with } \phi , \psi \in L^{1}[0,1] $$

and

$$\begin{aligned}& \sigma _{1}:= \int _{0}^{1}\phi (\varsigma ) \varsigma ^{\alpha -1} \,d\varsigma ; \\& \sigma _{2}:= \int _{0}^{1} \varsigma ^{\alpha -1}(1- \varsigma ) \phi (\varsigma ) \,d\varsigma , \quad \varsigma \in (0,1). \end{aligned}$$

Lemma 3.4

([29])

If$\int _{0}^{1}\phi (\varsigma )\varsigma ^{\alpha -1}\,d\varsigma \neq 1$, then for any$\sigma \in C[0,1]$, the unique solution of the following boundary value problem:

$$ \textstyle\begin{cases} D_{0^{+}}^{\alpha } u(\varsigma ) +\sigma (\varsigma )=0,\quad 0< \varsigma < 1, \\ u(0)=0, \qquad u(1)=\int _{0}^{1}\phi (\varsigma )u(\varsigma )\,d\varsigma ,\end{cases} $$

is given by

$$ u(\varsigma ) = \int _{0}^{1}G_{1\alpha }(\varsigma ,\eta )\sigma ( \eta )\,d\eta , $$

where

$$\begin{aligned}& G_{\alpha }(\varsigma ,\eta )=G_{1\alpha }(\varsigma ,\eta )+G_{2\alpha }( \varsigma ,\eta ), \\& G_{1\alpha } (\varsigma ,\eta )=\frac{1}{\varGamma (\alpha )} \textstyle\begin{cases} \varsigma ^{\alpha -1}(1-\eta )^{\alpha -1}-(\eta -\varsigma )^{ \alpha -1}, \\ \varsigma ^{\alpha -1}(1-\eta )^{\alpha -1}, \end{cases}\displaystyle \\& G_{2\alpha } (\varsigma ,\eta )= \frac{\varsigma ^{\alpha -1}}{1-\int _{0}^{1}\phi (\varsigma )\varsigma ^{\alpha -1}\,d\varsigma } \int _{0}^{1}\phi (\varsigma )G_{1\alpha }(\varsigma ,\eta )\,d\varsigma . \end{aligned}$$

(18)

Then$G_{\alpha }(\varsigma ,\eta )$is a Green’s function.

Lemma 3.5

([29])

Let$\alpha \in (1,2]$. Assume that (Q) holds. Then the functions$G_{1\alpha }(\varsigma ,\eta )$have the following property:

$$ \frac{(\alpha -1)\sigma _{2}\eta (1-\eta )^{\alpha -1}\varsigma ^{\alpha -1}}{(1-\sigma _{1})\varGamma (\alpha )} \leq G_{\alpha }\leq \frac{(1-\eta )^{\alpha -1}\varsigma ^{\alpha -1}}{\varGamma (\alpha )(1-\sigma _{1})}. $$

Lemma 3.6

([29])

Assume that (Q) holds and$f(y,x,y)$continuous, then$(u,v)\in X\times X$is a solution of the system (17) if and only if it is a solution of the integral equations

$$ \textstyle\begin{cases} u(y)=\int _{0}^{1}G_{\alpha }(y,x)f(x,u(x),v(x))\,dx, \\ v(y)=\int _{0}^{1}G_{\alpha }(y,x)f(x,v(x),u(x))\,dx. \end{cases} $$

Define mappings $S,T: X\rightarrow X$ by

$$ \textstyle\begin{cases} Su(y)=\int _{0}^{1}G_{\alpha }(y,x)f(x,u(x),v(x))\,dx, \\ Tv(y)=\int _{0}^{1}G_{\alpha }(y,x)f(x,v(x),u(x))\,dx. \end{cases} $$

Theorem 3.7

Consider Eq. (17) and suppose that

• 1. $f:[0,1]\times [0,\infty )\times [0,\infty )\rightarrow \mathbb{R}$is continuous function such that

  $$ \bigl\vert f\bigl(t,s,y(s),x(s)\bigr)-f\bigl(t,s,x(s),y(s)\bigr) \bigr\vert \leq \frac{(1-s)^{\alpha -1}\varsigma ^{\alpha -1}}{\varGamma (\alpha )} \frac{ \vert y(s)-x(s) \vert ^{2}}{1+ \vert y(s)-x(s) \vert }. $$

• 2. There exists a function$x_{0}\in X$such that

  $$ x_{0}(t)\leq \int _{a}^{b}K(t,s)f\bigl(t,s,x_{0}(s),Tx_{0}(s) \bigr)\,ds, \quad t\in [a,b]. $$

• 3. The inequality$x(s)\leq y(s)$implies$Tx(s)\leq Sy(s)$and$Sx(s)\leq Ty(s)$.

• 4. For$\{x_{n}\}_{n\in \mathbb{N}}$inM, if$x_{n}\rightarrow x$and$x_{n}(t)\leq x_{n+1}(t)$for$n\in \mathbb{N}$, then there is a subsequence$\{x_{n_{k}}\}_{n_{k}\in \mathbb{N}}$such that$x_{n_{k}}(t)\leq x(t)$for$n_{k}\in \mathbb{N}$, $t\in [a,b]$.

Then there exist$u,v\in X$such that

$$\begin{aligned}& u(t)= \int _{a}^{b}G_{\alpha }(t,s)f \bigl(t,s,u(s),v(s)\bigr)\,ds\quad \textit{or} \\& v(t)= \int _{a}^{b}G_{\alpha }(t,s)f\bigl(t, s,v(s),u(s)\bigr)\,ds. \end{aligned}$$

Proof

Let $x,y\in X$, using (2), we get

$$\begin{aligned} \bigl\vert Sx(t)-Ty(t) \bigr\vert =& \biggl\vert \int _{a}^{b}G_{\alpha }(t,s)f \bigl(t,s,x(s),y(s)\bigr)\,ds- \int _{a}^{b}G_{\alpha }(t,s)f \bigl(t,s,y(s),x(s)\bigr)\,ds \biggr\vert \\ \leq& \sup_{t\in [a,b]} \bigl\vert f\bigl(t,s,x(s),y(s)\bigr)-f \bigl(t,s,y(s),x(s)\bigr) \bigr\vert \frac{\varGamma (\alpha )}{(1-s)^{\alpha -1}\varsigma ^{\alpha -1}} \\ \leq& \frac{(1-s)^{\alpha -1}\varsigma ^{\alpha -1}}{\varGamma (\alpha )} \frac{ \vert x(s)-y(s) \vert ^{2}}{1+ \vert x(s) -y(s) \vert } \frac{\varGamma (\alpha )}{(1-s)^{\alpha -1}\varsigma ^{\alpha -1}}= \frac{ \vert x(s)-y(s) \vert ^{2}}{1+ \vert x(s)-y(s) \vert } \\ =&\gamma \bigl( \bigl\vert x(s)-y(s) \bigr\vert \bigr) \bigl\vert x(s)-y(s) \bigr\vert , \end{aligned}$$

where $\gamma (t)=\frac{t}{t+1}$. With setting $\psi (t)=t$, the condition (15) in Corollary 2.7 holds. With considering (3) and (4) from the assumption of the theorem and by definition of the graph Γ we deduce the conditions (i) and (ii) in Corollary (2.7) hold. Also by the condition (5) from assumption of theorem, the condition (iii) of Corollary 2.7 holds. As a result, we have

$$\begin{aligned}& u(t)= \int _{a}^{b}G_{\alpha }(t,s)f \bigl(t,s,u(s),v(s)\bigr)\,ds \quad \text{or} \\& v(t)= \int _{a}^{b}G_{\alpha }(t,s)f \bigl(t,s,v(s),u(s)\bigr)\,ds. \end{aligned}$$

 □