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Common fixed point theorems for a finite family of multivalued mappings in an ordered Banach space

Fixed Point Theory and Applications20182018:17

https://doi.org/10.1186/s13663-018-0642-3

  • Received: 29 November 2017
  • Accepted: 20 May 2018
  • Published:

Abstract

In this paper, we prove some common fixed point theorems for a finite family of multivalued and single-valued mappings operating on ordered Banach spaces. Our results extend and generalize many results in the literature on fixed point theory and lead to existence theorems for a system of integral inclusions.

Keywords

  • Condensing mapping
  • Common fixed point
  • Integral inclusion
  • Isotone mapping
  • Measure of noncompactness
  • Ordered Banach space

MSC

  • 47H10
  • 47H30

1 Introduction

A good part of the research publications on the fixed point theory was devoted to the existence of a common fixed point for pairs of single and multivalued functions in various types of spaces such as metric spaces, ordered spaces, and so on.

By using the measure of noncompactness, Dehage et al. [1] proved some common fixed point results for pairs of condensing mappings in ordered Banach spaces and they showed that these results have interesting applications in differential and integral equations. Several authors got generalizations of these results under a weaker hypothesis (see [26]). The main objective of the present paper is to generalize the results of [2, 3] by establishing some common fixed point results for a finite family of single and multivalued functions on ordered Banach spaces. Some examples will be given to support our results. As application, we will prove the existence solutions for a system of integral inclusions, which gives a generalization of results in [7].

2 Methods

Many authors studied the existence of a common fixed point for pairs of condensing mappings in an ordered Banach space with weak and strong topology.

The goal of this article is to generalize these results to the case of a finite family of multivalued condensing mapping. The main tool in this study is the notion of noncompactness measure on Banach spaces.

Our work is organized as follows: we discuss some concepts used in this paper, and we present our main results and their consequences. We give also some examples to validate our results. Then we apply the obtained results to solve a system of integral inclusions.

3 Results and discussion

Let us first give some preliminaries and notations. For a given real Banach space X, we denote by \(2^{X}\) the space of all nonempty subsets of X. Recall that a multivalued function on X is a mapping from X into \(2^{X}\) and that a point \(x^{*}\in X\) is called a fixed point of T if \(x^{*}\in T(x^{*})\).

In the following, we denote by \(T(A)\) the set \(\bigcup_{x\in A}T(x)\) for every \(A\in 2^{X}\), by
$$ \Gamma (T)= \bigl\{ (x,y)\in X\times X : y\in T(x) \bigr\} $$
the graph of T, and by \(X'\) be the topological dual space of X.

Definition 3.1

A sequence \(\{x_{n}\}\) of X is weakly convergent to \(x\in X\) if \(\lim_{n}f(x_{n})=f(x)\) for all \(f\in X'\).

In this case, we denote \(x_{n} \rightharpoonup x\).

Definition 3.2

Let X be an ordered Banach space, T is said to be monotone-closed if for each monotone sequence \(\{x_{n}\}\) in X with \(x_{n}\rightarrow x\) and for each sequence \(\{y_{n}\}\) with \(y_{n}\in T(x_{n})\) such that \(y_{n}\rightarrow y\), we have \(y\in T(x)\).

Definition 3.3

T is said to be closed if for each sequence \(\{(x_{n},y_{n})\}\) in \(\Gamma (T)\) with \((x_{n},y_{n})\rightarrow (x,y)\) strongly in \(X\times X\), we have \(y\in T(x)\).

Definition 3.4

A nonempty closed subset P of X is called an order cone if
  1. 1.

    \(P\ne \{0\}\);

     
  2. 2.

    For all \(a,b \in \mathbb{R}^{+}\) and \(x, y\in P\), we have \(ax+by \in P\);

     
  3. 3.

    \(P\cap (-P)=\{0\}\).

     
Given an order cone P on X, we can define a partial order “≤” on X by
$$ x\leq y \quad \Longleftrightarrow \quad y-x\in P. $$
An order cone P is called normal if there is a real constant \(N>0\) such that, for all \(x,y\in X\), we have
$$ 0\leq x \leq y\quad \Rightarrow\quad \Vert x \Vert \leq N \Vert y \Vert . $$

The following lemma can be found in [3] and [8].

Lemma 3.5

Let X be an ordered real Banach space with a normal order cone. Suppose that \(\{x_{n}\}\) is a monotone sequence which contains a subsequence \(\{x_{\sigma (n)}\}\) converging weakly to some \(x\in X\). Then \(\{x_{n}\}\) converges strongly to x.

We can also define a partial order on \(2^{X}\) by
$$ A\le B \quad \Longleftrightarrow \quad x\le y,\quad \forall (x,y)\in A\times B. $$
We define the measure of noncompactness on bounded subsets of X by
$$ \psi (A)=\inf \Biggl\{ \delta >0: \exists A_{1},\ldots,A_{n} \subset X, \operatorname {diam}(A_{i})\le \delta \mbox{ and }A\subset \bigcup _{i=1}^{n}A_{i} \Biggr\} , $$
where \(\operatorname {diam}(A)\) denotes the diameter of a subset A of X.

The following lemma describes some properties of the measure of noncompactness.

Lemma 3.6

Let A and B be bounded sets of X, then
  1. 1.

    \(\psi (A)=0\) if and only if A is relatively compact;

     
  2. 2.

    \(A\subseteq B\) implies \(\psi (A)\leq \psi (B)\);

     
  3. 3.

    \(\psi (\overline{A})=\psi (A)\);

     
  4. 4.

    \(\psi (A\cup B)= \max \{\psi (A), \psi (B)\}\).

     

For more properties of the measure of noncompactness, we refer to [9, 10].

Definition 3.7

A mapping \(T :M\subseteq X\rightarrow X\) is said to be locally almost nonexpansive if, for each \(x\in M\) and \(\varepsilon >0\), there exists a weak neighborhood \(U_{x}\) of x such that
$$ \bigl\Vert T(u)-T(v) \bigr\Vert \leq \Vert u-v \Vert +\varepsilon \quad \mbox{for all } u,v \in U_{x}. $$

Now, we give our main results.

As extension of the results of [6], we define the notion of \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone for a given integer \(p\ge 2\) as follows.

Definition 3.8

Let M be a nonempty subset of an ordered Banach space X, and let \(T_{1},T_{2},\ldots,T_{p}:M\rightarrow 2^{M}\) be p multivalued mappings on M; we say that
  1. 1.
    \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone increasing if, for all \(x\in M\), the relations
    $$ x_{1}\in T_{1}(x),\quad\quad x_{2}\in T_{2}(x_{1}),\quad\quad \ldots, \quad\quad x_{p}\in T_{p}(x _{p-1}) $$
    imply
    $$ T_{1}(x)\leq T_{2}(x_{1})\leq \cdots \leq T_{p}(x_{p-1})\leq T_{1}(x _{p}). $$
     
  2. 2.
    \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone decreasing if, for all \(x\in M\), the relations
    $$ x_{1}\in T_{1}(x),\quad\quad x_{2}\in T_{2}(x_{1}),\quad\quad \ldots,\quad\quad x_{p}\in T _{p}(x_{p-1}) $$
    imply
    $$ T_{1}(x_{p}) \leq T_{p}(x_{p-1}) \leq T_{p-1}(x_{p-2}) \leq \cdots \leq T_{1}(x). $$
     
  3. 3.

    \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone if it is either \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone increasing or weakly isotone decreasing.

     

The following definition will be used in the next of this paper.

Definition 3.9

([11])

Let X be a Banach space, M be a nonempty subset of X, \(T:M\rightarrow 2^{M}\) be a multivalued mapping on M, and let ψ be a measure of noncompactness on X. For \(k\in [0,1]\), we have the following definitions:
  1. 1.

    T is called ψ-condensing if \(T(M)\) is bounded and, for every nonempty bounded subset N of M with \(\psi (N)>0\), we have \(\psi (T(N))<\psi (N)\).

     
  2. 2.

    T is called \(k-\psi \)-contractive if \(T(M)\) is bounded and, for every nonempty bounded subset N of M, we have \(\psi (T(N))\leq k\psi (N)\).

     

Our first main result is the following theorem.

Theorem 3.10

Let X be an ordered Banach space, ψ be a measure of noncompactness on X and \(p\ge 2\) be an integer. Let M be a nonempty closed subset of X and \(T_{1},T_{2},\ldots,T_{p}:M\rightarrow 2^{M}\) be p monotone-closed mappings satisfying:
  1. 1.

    \(T_{2},T_{3},\ldots,T_{p}\) are \(1-\psi \)-contractive;

     
  2. 2.

    \(T_{1}\) is ψ-condensing;

     
  3. 3.

    \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone.

     
Then \(T_{1},T_{2},\ldots,T_{p}\) have a common fixed point.

Proof

Assume that \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone increasing, and let \(x\in M\). We define a sequence \(\{x_{n}\}\) in M as follows:
$$ x_{0}=x,\quad\quad x_{pn+1}\in T_{1}(x_{pn}), \quad\quad x_{pn+2}\in T_{2}(x_{pn+1}),\quad\quad \ldots, \quad\quad x_{pn+p}\in T_{p}(x_{pn+(p-1)}) $$
for \(n=0,1,2,\ldots \) .
As we have
$$ x_{1}\in T_{1}(x_{0}),\quad\quad x_{2}\in T_{2}(x_{1}),\quad\quad \ldots,\quad \quad x_{p} \in T_{p}(x_{p-1}), $$
we obtain
$$ T_{1}(x_{0})\leq T_{2}(x_{1})\leq \cdots \leq T_{p}(x_{p-1})\leq T_{1}(x _{p}), $$
and then
$$ x_{1}\leq x_{2}\leq \cdots \leq x_{p} \leq x_{p+1}, $$
which gives
$$ x_{1}\leq x_{2}\leq x_{3} \leq \cdots \leq x_{n} \leq \cdots. $$
Let \(A_{j}=\{x_{np+j} : n\in \mathbb{N}\}\), \(j=0,1,\ldots,p-1\), and \(A_{p}=A_{0}\setminus \{x_{0}\}\), we have
$$ A_{j}\subset T_{j}(A_{j-1}),\quad \text{for all } j\in \{1,2,\ldots,p\}. $$
The set
$$ A= \Biggl( \bigcup_{k=1}^{p}A_{k} \Biggr) \cup \{x_{0}\} $$
is bounded; and since \(T_{2},T_{3},\ldots,T_{p}\) are \(1-\psi \)-contractive, we have
$$ \psi (A_{k})\leq \psi (T_{k}(A_{k-1}) \leq \psi (A_{k-1})\leq \psi (T _{k-1}(A_{k-2})\leq \cdots \leq \psi \bigl(T_{1}(A_{0}) \bigr) $$
for all \(k=1,2,\ldots,p\).
Assume that \(\psi (A)\neq 0\), we have
$$\begin{aligned} \psi (A) =& \max \bigl\{ \psi (A_{k}) : k=1,2,\ldots,p \bigr\} \\ \leq & \max \bigl\{ \psi \bigl(T_{k}(A_{k-1}) \bigr) : k=1,2,\ldots,p \bigr\} \\ \leq &\max \bigl\{ \psi (A_{k-1}) : k=1,2,\ldots,p \bigr\} \\ \leq &\max \bigl\{ \psi \bigl(T_{k-1}(A_{k-2}) \bigr) : k=2, \ldots,p \bigr\} \\ \vdots & \\ \leq & \psi \bigl(T_{1}(A_{0}) \bigr). \end{aligned}$$
As \(T_{1}\) is ψ-condensing, we obtain
$$ \psi (A) \leq \psi \bigl(T_{1}(A_{0}) \bigr) \leq \psi \bigl(T_{1}(A) \bigr) < \psi (A), $$
which is contraction. Thus \(\psi (A)=0\), and then A is relatively compact. Since \(\{x_{n}\}\) is monotone increasing in A, it is convergent to some \(x^{\ast }\). Since \({x_{pn+1}\in T_{1}(x_{pn})}\) and \(T_{1}\) has a closed graph, we obtain that \(x^{*}\in T_{1}(x^{*})\). Similarly, we obtain \(x^{\ast }\in T_{k}(x^{\ast })\) for \(k=2,3,\ldots,p\); and consequently, \(x^{\ast }\) is a common fixed point for \(T_{1},T _{2},T_{3},\ldots,T_{p}\).

The case \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone decreasing is similar, which ends the proof. □

Remark 3.11

For \(p=2\), we obtain [3] Theorem 3.10.

Let M be a nonempty subset of an ordered Banach space X, and let
$$ T_{1},T_{2},\ldots,T_{p}:M\rightarrow M $$
be p mappings. As in Definition 3.8, we can define that the notion of \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone. Note that in this case the set \(T(x)\) becomes single \(\{T(x)\}\) and \(y\in \{T(x)\}\) becomes \(y=T(x)\).

By Theorem 3.10, we obtain the following result.

Corollary 3.12

Let X be an ordered Banach space and ψ be a measure of noncompactness on X. Let M be a nonempty closed subset of X, and \(T_{1},T_{2},\ldots,T_{p}:M\rightarrow M\) be p closed mappings satisfying:
  1. 1.

    \(T_{2},T_{3},\ldots,T_{p}\) are \(1-\psi \)-contractive;

     
  2. 2.

    \(T_{1}\) is ψ-condensing;

     
  3. 3.

    \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone.

     
Then \(T_{1},T_{2},\ldots,T_{p}\) have a common fixed point.

Remark 3.13

For \(p=2\), this corollary is obtained in [2].

Using Definition 3.7, we have the following corollary.

Corollary 3.14

Let M be a nonempty, closed subset of an ordered reflexive Banach space X, and ψ be a measure of noncompactness on X.

Let \(T_{1},T_{2},\ldots,T_{p}:M\rightarrow M\) be p continuous mappings satisfying:
  1. 1.

    \(T_{2},T_{3},\ldots,T_{p}\) are locally almost nonexpansive mappings;

     
  2. 2.

    \(T_{1}\) is ψ-condensing;

     
  3. 3.

    \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone.

     
Then \(T_{1},T_{2},\ldots,T_{p}\) have a common fixed point.

Proof

By [12], Lemma 1, p. 672, we have \(T_{2},T_{3},\ldots,T_{p}\) are \(1-\psi \)-contractive; hence the proof of Theorem 3.10. □

Theorem 3.15

Let X be an ordered Banach space, and ψ be a measure of noncompactness on X. Let M be a nonempty closed subset of X, and \(T_{1},T_{2},\ldots,T_{p}:M\rightarrow 2^{M}\) be p monotone-closed mappings (\(p\ge 2\)) satisfying:
  1. 1.

    \(T_{1},T_{2},T_{3},\ldots,T_{p}\) are ψ-condensing;

     
  2. 2.

    \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone.

     
Then \(T_{1},T_{2},\ldots,T_{p}\) have a common fixed point.

Proof

We use the same notations as in the proof of the previous theorem and assume that \(\psi (A)\neq 0\); we have
$$ A = \Biggl( \bigcup_{k=1}^{p}T_{k}(A_{k-1}) \Biggr) \cup \{x_{0}\}, $$
then
$$\begin{aligned} \psi (A) = & \max \bigl\{ \psi \bigl(T_{k}(A_{k-1}) \bigr); k=1,2,\ldots,p \bigr\} \\ \leq & \max \bigl\{ \psi \bigl(T_{k}(A) \bigr); k=1,2,\ldots,p \bigr\} \\ < & \psi (A). \end{aligned}$$
Then \(\psi (A)=0\), which is contradiction. Therefore A is relatively compact, which ends the proof. □

Definition 3.16

Let M be a nonempty subset of an ordered Banach space X and \(T_{1},T_{2},\ldots,T_{p}:M\rightarrow 2^{M}\) be p mappings with \(p\geq 2\).
  1. 1.
    The p-uplet \((T_{1},T_{2},\ldots,T_{p})\) is called weakly isotone increasing if for all \(x\in M\), we have
    $$ \textstyle\begin{cases} T_{p}(x) \leq T_{1}(y) \quad \text{for all } y\in T_{p}(x), \\ \forall i\in \{1,2,\ldots,p-1\},\quad T_{i}(x)\leq T_{i+1}(y) \quad \text{for all } y\in T_{i}(x). \end{cases} $$
     
  2. 2.
    We say that the p-uplet \((T_{1},T_{2},\ldots,T_{p})\) is weakly isotone decreasing if for all \(x\in M\), we have
    $$ \textstyle\begin{cases} T_{1}(y) \leq T_{p}(x) \quad \text{for all } y\in T_{p}(x), \\ \forall i\in \{1,2,\ldots,p-1\},\quad T_{i+1}(y) \leq T_{i}(x) \quad \text{for all } y\in T_{i}(x). \end{cases} $$
     
  3. 3.

    The p-uplet \((T_{1},T_{2},\ldots,T_{p})\) is weakly isotone if it is either weakly isotone increasing or weakly isotone decreasing.

     

Remark 3.17

If \(p=2\) and \(T_{1}\), \(T_{2}\) are a single-valued mappings, then Definition 3.16 coincides with [6], Definition 2.2.

Remark 3.18

If \((T_{1},T_{2},\ldots,T_{p})\) is weakly isotone increasing (resp. decreasing), then \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone increasing (resp. decreasing).

Using this remark, the statement of Theorem 3.15 remains if we replace “\(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone” with “\((T_{1},T_{2},\ldots,T_{p})\) is weakly isotone”.

Let M be a nonempty subset of an ordered Banach space X. Motivated by [3], we introduce the following definition.

Definition 3.19

p maps \(T_{1},T_{2},\ldots,T_{p}\) with \(p\geq 2\) are said to satisfy condition \(B_{M}\) if, for any monotone sequence \(\{x_{n}\}\) of M and for any fixed \(a\in M\), the condition
$$ \{x_{n}\} \subset \{a\}\cup \Biggl( \bigcup _{k=1}^{p}T_{k} \bigl( \{x_{n}\} \bigr) \Biggr) $$
implies \(\{x_{n}\}\) has a weakly convergent subsequence.

Theorem 3.20

Let X be an ordered Banach space with a normal order cone. Let M be a nonempty closed subset of X and \(T_{1},T_{2},\ldots,T_{p}:M\rightarrow 2^{M}\) be p monotone-closed mappings with \(p\geq 2\) satisfying:
  1. 1.

    \(T_{1},T_{2},\ldots,T_{p}\) satisfy condition \(B_{M}\);

     
  2. 2.

    \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone.

     
Then \(T_{1},T_{2},\ldots,T_{p}\) have a common fixed point.

Proof

Assume that \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone increasing, and let \(x\in M\) be fixed. We define a sequence \(\{x_{n}\}\) in M as follows:
$$\begin{aligned}& x_{0}=x,\quad\quad x_{pn+1}\in T_{1}(x_{pn}), \quad\quad x_{pn+2}\in T_{2}(x_{pn+1}),\quad\quad \ldots,\quad\quad x_{pn+p}\in T_{p}(x_{pn+(p-1)}), \\& \quad n=0,1,2,\ldots. \end{aligned}$$
Then as in the proof of Theorem 3.10, it follows that
$$ x_{1}\leq x_{2}\leq x_{3} \leq \cdots \leq x_{n} \leq \cdots . $$
We have
$$ \{x_{n}\} \subset \{x_{0}\}\cup \Biggl( \bigcup _{k=1}^{p}T_{k} \bigl( \{x _{n}\} \bigr) \Biggr) . $$
Since \(T_{1},T_{2},\ldots,T_{p}\) satisfy condition \(B_{M}\), there exist \(x^{*}\in M\) and a subsequence \(\{x_{\sigma (n)}\}\) of \(\{x_{n}\} \) such that \(x_{\sigma (n)} \rightharpoonup x^{*}\). Referring to Lemma 3.5, we get \(\{x_{n}\}\) strongly converges to \(x^{*}\). Now we have \(x_{pn+1}\in T_{1}(x_{pn})\) for all \(n\in \mathbb{N}\), and \(T_{1}\) is monotone-closed, we obtain that \(x^{*}\in T_{1}(x^{*})\). A similar argument yields \(x^{*}\in T_{k}(x^{*})\), \(k=2,3,\ldots,p\), and consequently, \(x^{*}\) is a common fixed point for \(T_{1},T_{2},T_{3},\ldots,T _{p}\).

The case when \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone decreasing is similar. □

Remark 3.21

For \(p=2\), Theorem 3.20 was proved in [3].

Example 3.22

Let \(X=\mathbb{R}^{2}\), \(P= \{ x=(t,t) \in \mathbb{R}^{2} : t \geq 0\}\) and
$$ M= \bigl\{ x\in X : \Vert x \Vert _{\infty } \leq 2 \bigr\} \cup \bigl\{ (t,0) \in X : t> 2 \bigr\} . $$
\((X, \Vert \cdot \Vert _{\infty })\) is an ordered Banach space with a normal order cone P, and M is a nonempty closed subset of X.
Let \(T_{1}, T_{2},T_{3} : M \rightarrow 2^{M}\) be defined by, for all \(x\in M\),
$$\begin{aligned}& T_{1}(x)= \textstyle\begin{cases} \{( \Vert x \Vert _{\infty }, \Vert x \Vert _{\infty }),(\frac{ \Vert x \Vert _{\infty }+2}{2},\frac{ \Vert x \Vert _{\infty }+2}{2})\} &\text{if } \Vert x \Vert _{\infty } \leq 2, \\ \{(2,2)\} &\text{if } \Vert x \Vert _{\infty } > 2, \end{cases}\displaystyle \\& T_{2}(x)= \textstyle\begin{cases} \{(2,2)\} &\text{if } \Vert x \Vert _{\infty } \leq 2, \\ \{ (\frac{t}{t-1},\frac{t}{t-1}) : t\in [2, \Vert x \Vert _{\infty }] \} &\text{if } \Vert x \Vert _{\infty } > 2 \end{cases}\displaystyle \end{aligned}$$
and \(T_{3}(x)= \{ x \}\).

The multivalued mappings \(T_{1}\), \(T_{2}\), \(T_{3} \) are closed, so monotone-closed, and satisfying the condition \(\mathcal{B}_{M}\), and \(T_{1}\) is \((T_{2},T_{3})\)-weakly-isotone increasing. Besides, the mappings \(T_{1}\), \(T_{2}\), \(T_{3}\) have the common fixed point \((2,2)\).

The following corollary can be obtained from Theorem 3.20.

Corollary 3.23

Let X be an ordered Banach space with a normal order cone. Let M be a nonempty closed subset of X. Let \(T_{1},T_{2},\ldots,T_{p}:M\rightarrow M\) be p monotone-continuous mappings with \(p\geq 2\) satisfying:
  1. 1.

    \(T_{1},T_{2},\ldots,T_{p}\) satisfy condition \(B_{M}\);

     
  2. 2.

    \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone.

     
Then \(T_{1},T_{2},\ldots,T_{p}\) have a common fixed point.

Let M be a nonempty subset of an ordered Banach space X, and let \(a\in M\). Motivated by [6] we introduce the following condition.

Definition 3.24

p maps \(T_{1},T_{2},\ldots,T_{p}:M\rightarrow 2^{M}\) with \(p\geq 2\) are said to satisfy the condition \(D_{M}\) if, for any countable set of M, the condition
$$ A\subset \{a\}\cup \Biggl( \bigcup_{k=1}^{p}T_{k} ( A ) \Biggr) $$
implies is weakly compact.

Theorem 3.25

Let X be an ordered Banach space with a normal cone and M be a nonempty closed subset of X. Let \(p\geq 2\) and let \(T_{1},T_{2},\ldots,T _{p}:M\rightarrow 2^{M}\) be p monotone-closed mappings satisfying:
  1. 1.

    \(T_{1},T_{2},\ldots,T_{p}\) satisfy condition \(D_{M}\);

     
  2. 2.

    \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone.

     

Then \(T_{1},T_{2},\ldots,T_{p}\) have a common fixed point.

Proof

Assume that \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone increasing, and let \(x\in M\) be fixed. We define a sequence \(\{x_{n}\}\) in M as follows:
$$\begin{aligned}& x_{0}=x,\quad\quad x_{pn+1}\in T_{1}(x_{pn}), \quad\quad x_{pn+2}\in T_{2}(x_{pn+1}),\quad\quad \ldots, \quad\quad x_{pn+p}\in T_{p}(x_{pn+(p-1)}) , \\& \quad n=0,1,2,\ldots. \end{aligned}$$
Then, as in the proof of Theorem 3.10, it follows that
$$ x_{1}\leq x_{2}\leq x_{3} \leq \cdots \leq x_{n} \leq \cdots . $$
Let \(A=\{x_{0},x_{1},x_{2},\ldots\}\), A is countable and
$$ A\subset \{x_{0}\}\cup \Biggl( \bigcup_{k=1}^{p}T_{k} ( A ) \Biggr) , $$
or \(T_{1},T_{2},\ldots,T_{p}\) satisfy the condition \(D_{M}\), so is weakly compact. Then there exists a subsequence \(\{x_{\sigma (n)}\}\subset A\) and \(x^{*}\in \overline{A}\) such that
$$ x_{\sigma (n)} \rightharpoonup x^{*}. $$
We refer to Lemma 3.5, we get \(x_{\sigma (n)} \rightarrow x ^{*}\). Since \({x_{pn+1}\in T_{1}(x_{pn})}\) for all \(n\in \mathbb{N}\) and \(T_{1}\) is monotone-closed, we obtain that \(x^{*}\in T_{1}(x^{*})\). A similar argument yields \(x^{*}\in T_{k}(x^{*})\), \(k=2,3,\ldots,p\), and consequently, \(x^{*}\) is a common fixed point for \(T_{1},T_{2},T_{3},\ldots,T _{p}\). The case when \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone decreasing is similar. □

Remark 3.26

For \(p=2\), Theorem 3.25 was proved in [3].

The following corollary is a single-valued version of Theorem 3.25.

Corollary 3.27

Let X be an ordered Banach space with a normal cone. Let M be a nonempty closed subset of X. Let \(p\geq 2\) and let \(T_{1},T_{2},\ldots,T _{p}:M\rightarrow M\) be p closed mappings satisfying the condition \(D_{M}\) and \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)-weakly isotone. Then \(T_{1},T_{2},\ldots,T_{p}\) have a common fixed point.

Example 3.28

Let \(X=\mathbb{R}^{2}\), \(P= \{ x=(t,t) \in \mathbb{R}^{2} : t \geq 0\}\) and
$$ M= \bigl\{ x\in X : \Vert x \Vert _{\infty } \leq 2 \bigr\} \cup \bigl\{ (t,t) \in X : t> 2 \bigr\} . $$
\((X, \Vert \cdot \Vert _{\infty })\) is an ordered Banach space with a normal order cone P, and M is a nonempty closed subset of X.
Let \(T_{1}, T_{2},T_{3} : M \rightarrow 2^{M}\) be defined by, for all \(x\in M\),
$$\begin{aligned}& T_{1}(x)= \textstyle\begin{cases} \{( \Vert x \Vert _{\infty }, \Vert x \Vert _{\infty })\}& \text{if } \Vert x \Vert _{\infty } \leq 2, \\ \{(2,2)\} &\text{if } \Vert x \Vert _{\infty } > 2, \end{cases}\displaystyle \\& T_{2}(x)= \textstyle\begin{cases} \{(\frac{ \Vert x \Vert _{\infty }+2}{2},\frac{ \Vert x \Vert _{\infty }+2}{2})\} &\text{if } \Vert x \Vert _{\infty } \leq 2, \\ \{ (\frac{3t}{t+1},\frac{3t}{t+1}) : t\in [2, \Vert x \Vert _{\infty }] \} &\text{if } \Vert x \Vert _{\infty } > 2 \end{cases}\displaystyle \end{aligned}$$
and
$$ T_{3}(x)= \textstyle\begin{cases} \{ (2,2) \} &\text{if } \Vert x \Vert _{\infty } \leq 2, \\ \{ (\frac{3t}{t+1},\frac{3t}{t+1}) : t\in [2, \Vert x \Vert _{\infty }] \} & \text{if } \Vert x \Vert _{\infty } > 2. \end{cases} $$
The multivalued mappings \(T_{1}\), \(T_{2}\), \(T_{3} \) are monotone-closed and satisfying the condition \(D_{M}\), because the mappings \(T_{1}\), \(T_{2}\), \(T _{3} \) are bounded, and \(T_{1}\) is \((T_{2},T_{3})\)-weakly-isotone. Besides, the mappings \(T_{1}\), \(T_{2}\), \(T_{3}\) have the common fixed point \((2,2)\).

Remark 3.29

In Example 3.22 the mappings \(T_{1}\), \(T_{2}\), \(T_{3}\) are not satisfying the condition \(D_{M}\).

4 Application

Let \(\mathbb{R}\) be the real line and \(\mathbb{N}\) be the set of positive integers, and \(p\in \mathbb{N}\) with \(p\geq 2\). Let E be a Banach space with norm \(\Vert \cdot \Vert _{E}\), and let \(C(E)\) denote the class of all nonempty closed subsets of E. Given a closed and bounded interval \(J=[0,1]\subset \mathbb{R}\), consider the system of nonlinear integral inclusions of the form
$$ x(t)\in q(t)+ \int_{0}^{\sigma (t)}K(t,s)F_{i} \bigl(s,x(s) \bigr)\,ds\quad \mbox{for all } i\in \{1,2,\ldots,p\}, $$
(1)
where \(t\in [0,1]\), and \(\sigma : [0,1] \rightarrow [0,1]\), \(q : [0,1] \rightarrow E\), \(K:[0,1] \times [0,1] \rightarrow \mathbb{R}\) are continuous, and \(F_{i}:[0,1] \times E\rightarrow C(E)\) for all \(i\in \{1,2,\ldots,p\}\).
By a common solution for the system of integral inclusions (1), we mean a continuous function \(x:J\rightarrow E\) such that
$$ x(t)=q(t)+ \int_{0}^{\sigma (t)}K(t,s)v_{i}(s)\,ds\quad \mbox{for all } i\in \{1,2,\ldots,p\}, $$
for some \(v_{i}\in B(J,E)\) satisfying \(v_{i}(t)\in F_{i}(t,x(t))\) for all \(t\in J\), where \(B(J,E)\) is the space of all E-valued Bochner integrable functions on J.
Let \(C(J,E)\) denote the space of all continuous E-valued functions on J, define a norm \(\Vert \cdot \Vert _{C}\) in \(C(J,E)\) by
$$ \Vert x \Vert _{C}=\sup \bigl\Vert x(t) \bigr\Vert _{E}. $$
Clearly, \(C(J,E)\) is a Banach space with the norm \(\Vert \cdot \Vert _{C}\). We introduce an order relation “≤” in \(C(J,E)\) with the help of the cone \(P_{C}\) in \(C(J,E)\) defined by
$$ P_{C}= \bigl\{ x \in C(J,E) : x(t) \in P_{E} \mbox{ for all } t\in J \bigr\} , $$
where \(P_{E}\) is a normal cone in E.

Definition 4.1

A multivalued map \(F:J\rightarrow 2^{E}\) is said to be measurable if for any \(y\in E\), the function \(t\rightarrow d(y,F(t))=\inf \{ \Vert y-x \Vert _{E}:x \in F(t)\}\) is measurable.

Let \(i\in \{1,2,\ldots,p\}\). Denote
$$ \bigl\Vert F_{i}(t,x) \bigr\Vert = \bigl\{ \Vert u \Vert _{E}:u\in F_{i}(t,x) \bigr\} \quad \text{and}\quad \bigl\vert \!\bigl\vert \!\bigl\vert F_{i}(t,x)\bigr\vert \!\bigr\vert \!\bigr\vert =\sup \bigl\{ \Vert u \Vert _{E}:u\in F_{i}(t,x) \bigr\} . $$

Definition 4.2

A multivalued function \(\beta : J\times E \rightarrow 2^{E}\) is called Carathéodory if
  1. 1.

    \(t\rightarrow \beta (t,x)\) is measurable for each \(x\in E\), and

     
  2. 2.

    \(x \rightarrow \beta (t,x)\) is an upper semicontinuous almost everywhere for \(t\in J\).

     

Definition 4.3

A Carathéodory multifunction \(F_{i}(t,x)\) is called \(L^{1}\)-Carathéodory if, for every real number \(r>0\), there exists a function \(h_{i,r}\in L^{1}(J,\mathbb{R})\) such
$$ \bigl\vert \!\bigl\vert \!\bigl\vert F_{i}(t,x)\bigr\vert \!\bigr\vert \!\bigr\vert \leq h_{i,r} \quad \text{a.e. } t\in J $$
for all \(x\in E\) with \(\Vert x \Vert _{E}\leq r\).
Denote
$$ S_{F_{i}}^{1}(x)= \bigl\{ v\in B(J,E):v(t)\in F_{i} \bigl(t,x(t) \bigr) \text{ a.e. } t\in J \bigr\} . $$
In the sequel, we also need the following lemmas from [13].

Lemma 4.4

If \(\dim (E)< \infty \) and \(F_{i} : J\times E \rightarrow 2^{E}\) is \(L^{1}\)-Carathéodory, then \(S_{F_{i}}^{1}(x)\neq \emptyset \) for each \(x\in C(J,E)\).

Lemma 4.5

Let E be a Banach space, F be a Carathéodory multimap with \(S_{F_{i}}^{1}\neq \emptyset \), and let \(\mathcal{L}:L^{1}(J,E) \rightarrow C(J,E)\) be a continuous linear mapping. Then the operator \(\mathcal{L} \circ S_{F_{i}}^{1} : C(J,E) \rightarrow 2^{C(J,E)}\) is a closed graph operator on \(C(J,E) \times C(J,E)\).

Now we introduce the following definition.

Definition 4.6

A multifunction \(F_{i}(t,x)\) is said to be nondecreasing in x almost everywhere for \(t\in J\) if, for any \(x,y\in E\) with \(x< y \), we have that \(F_{i}(t,x)\leq F_{i}(t,y)\) for almost everywhere \(t\in J\).

Now we have the following condition.

Condition 2

Integral inclusions of system (1) are said to satisfy Condition 2 if there exist \(a,b\in C(J,E)\) such that
$$ a(t)\leq q(t)+ \int_{0}^{\sigma (t)}K(t,s)v_{i}(s)\,ds \leq b(t) $$
for all \(v_{i}\in B(J,E)\) such that \(v_{i}\in F_{i}(t,x(t))\) with \(i\in \{1,2,\ldots,p\}\).
We refer to [7] and we consider the following set of hypotheses in the following:
\((H_{0})\)
The function K is continuous and nonnegative on \(J\times J\), with
$$ M=\sup_{t,s\in J}K(t,s). $$
\((H_{1})\)

The multivalued \(F_{i}\) is Carathéodry for all \(i\in \{1,2,\ldots,p\}\).

\((H_{2})\)

For any bounded set A of E, \(\psi (F_{i}([0,1] \times A))\leq \lambda \psi (A)\) for some reals \(\lambda >0\), with \(i\in \{1,2,\ldots,p\}\).

\((H_{3})\)

Multivalued functions \(F_{i}\) are nondecreasing in x almost everywhere for \(t\in J\), with \(i\in \{1,2,\ldots,p\}\).

\((H_{4})\)

\(S_{F_{i}}^{1}(x)\neq \emptyset \) for each \(x\in C(J,E)\) and for all \(i\in \{1,2,\ldots,p\}\).

\((H_{5})\)

Integral inclusions of system (1) satisfy Condition 2.

\((H_{6})\)
For each \(i\in \{1,2,\ldots,p\}\), the function
$$ t \rightarrow \bigl\vert \!\bigl\vert \!\bigl\vert F_{i}(t,a(t)\bigr\vert \!\bigr\vert \!\bigr\vert _{E}+\bigl\vert \!\bigl\vert \!\bigl\vert F_{i}(t,b(t)\bigr\vert \!\bigr\vert \!\bigr\vert _{E} $$
is Lebesgue integrable on J.
\((H_{7})\)
\(F_{p}(t,x)\leq F_{1}(t,y)\) for all \(v\in S_{F_{p}} ^{1}(x)\) and, for each \(i\in \{1,2,\ldots,p-1\}\), we have \(F_{i}(t,x) \leq F_{i+1}(t,y)\) for all \(v\in S_{F_{i}}^{1}(x)\), where
$$ y(t)=q(t)+ \int_{0}^{\sigma (t)}K(t,s)v(s)\,ds. $$

The next theorem has been proved in [7] for \(p=2\).

Theorem 4.7

Let \(p\in \mathbb{N}\) with \(p\geq 2\), and assume that hypotheses \((H_{0})\)\((H_{7})\) hold. If \(\lambda M<1\), then the system of integral inclusions (1) has a common solution in \(C([0,1],E)\).

Proof

Let \(X=C(J,E)\) and consider the ordered interval \([a,b]\subset X\), which will be defined in view of \((H_{5})\). Define p mappings \(T_{1},T_{2},\ldots,T_{p}:[a,b]\rightarrow X\) by
$$ T_{i}(x)= \biggl\{ u:u(t)=q(t)+ \int_{0}^{\sigma (t)}K(t,s)v(s)\,ds, v\in S_{F_{i}}^{1}(x) \biggr\} \quad \mbox{for all } i\in \{1,2, \ldots,p\}. $$
Our strategy is to show that \(T_{i}\) satisfies all the conditions of Theorem 3.15 for all \(i\in \{1,2,\ldots,p\}\).
First we show that the p-uplet \((T_{1},T_{2},\ldots,T_{p})\) is weakly isotone on \([a,b]\). Let \(x,y\in [a,b]\), then with a similar reasoning as in [7] we get, for each \(i\in \{1,2,\ldots,p\}\),
$$\begin{aligned} T_{i}(x) =& \biggl\{ u_{1}:u_{1}= q(t)+ \int_{0}^{\sigma (t)}K(t,s)v _{1}(s) \,ds,v_{1} \in S_{F_{i}}^{1}(x) \biggr\} \\ =& \biggl\{ u_{1}:u_{1}= q(t)+ \int_{0}^{\sigma (t)}K(t,s)v_{1}(s) \,ds,v_{1} \in \bigl\{ v\in B(J,E):v(t)\in F_{i} \bigl(t,x(t) \bigr) \bigr\} \biggr\} \\ \leq & \biggl\{ u_{2}:u_{2}= q(t)+ \int_{0}^{\sigma (t)}K(t,s)v_{2}(s) \,ds,v_{2} \in \bigl\{ v\in B(J,E):v(t)\in F_{i+1} \bigl(t,y(t) \bigr) \bigr\} \biggr\} \\ \le & \biggl\{ \mbox{for all } y(t)=q(t)+ \int_{0}^{\sigma (t)}K(t,s)v(s)\,ds \mbox{ and } v \in S_{F_{i}}^{1}(x) \biggr\} \\ =& \biggl\{ u_{2}:u_{2}= q(t)+ \int_{0}^{\sigma (t)}K(t,s)v_{2}(s) \,ds,v_{2} \in S_{F_{i+1}}^{1}(y) \biggr\} \\ \le & \biggl\{ \mbox{for all } y(t)=q(t)+ \int_{0}^{\sigma (t)}K(t,s)v(s)\,ds \mbox{ and } v \in S_{F_{i}}^{1}(x) \biggr\} \\ =& T_{i+1}(y), \quad \mbox{for all } y(t)=q(t)+ \int_{0}^{\sigma (t)}K(t,s)v(s)\,ds \mbox{ and } v\in S_{F_{i}}^{1}(x) \\ =& T_{i+1}(y), \quad \mbox{for all } y\in T_{i}(x)= \biggl\{ u:u= q(t)+ \int_{0}^{\sigma (t)}K(t,s)v(s)\,ds,v\in S_{F_{i}}^{1}(x) \biggr\} . \end{aligned}$$
Thus, \(T_{i}(x)\leq T_{i+1}(y)\) for all \(y\in T_{i}(x)\). Similarly, we have \(T_{p}(x)\leq T_{1}(y)\) for all \(y\in T_{p}(x)\). Hence, the p-uplet \((T_{1},T_{2},\ldots,T_{p})\) is weakly isotone increasing.

Next, let \(x\in [a,b]\), by \((H_{5})\) we have \(a\leq T_{i}(x)\leq b\) for all \(i\in \{1,2,\ldots,p\}\). Hence \(T_{1},T_{2},\ldots,T_{p}:[a,b]\rightarrow 2^{[a,b]}\).

The following results are justified in [7]. Let A be a nonempty subset in \([a,b]\).
  1. 1.

    The cone \(P_{C}\) is normal in \(C(J,E)\).

     
  2. 2.

    For each \(t\in J\) and \(i\in \{1,2,\ldots,p\}\), \(\psi (T_{i}(A(t)))\leq \lambda M\psi (A)\).

     
  3. 3.

    For each \(i\in \{1,2,\ldots,p\}\), \(T_{i}(A)\) is a uniformly bounded and equicontinuous set in \([a,b]\).

     
We obtain \(\psi (T_{i}(A))\leq \lambda M\psi (A)\), where \(\lambda M<1\). Hence, \(T_{i}\) is a ψ-condensing multivalued mapping on \([a,b]\) for all \(i\in \{1,2,\ldots,p\}\). In [7] and from Lemma 4.5, we can prove that \(T_{i}\) is a closed graph for all \(i\in \{1,2,\ldots,p\}\).

Thus, \(T_{1},T_{2},\ldots,T_{p}\) satisfy all the conditions of Theorem 3.15, and therefore an application of it yields that \(T_{1},T_{2},\ldots,T_{p}\) have a common fixed point in \([a,b]\). This further implies that system of integral inclusions (1) has a common solution on J. □

5 Conclusions

Dehage, BC proved in [14] some common fixed point theorems for pairs of weakly isotone condensing mappings in an ordered Banach space. These results will be generalized later by Hussain and Taoudi. The authors showed that these results can be obtained if we replace the strong topology by the weaker one, and they used these results to solve the existence problem for a system of integral inclusions.

In the present paper, we extend and generalize these results for a finite family of single and multivalued functions on an ordered Banach space. And we prove the existence of solutions for a system of integral inclusions.

Declarations

Acknowledgements

The authors are thankful to the editors and the anonymous referees for their valuable comments, which reasonably improved the presentation of the manuscript.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Funding

We have no funding for this article.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Laboratory of Algebra, Analysis and Applications (L3A), Faculty of Sciences Ben M’Sik, Hassan II University of Casablanca, Casablanca, Morocco
(2)
CRMEF, Rabat, Morocco

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