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Common fixed point theorems for a finite family of multivalued mappings in an ordered Banach space
Fixed Point Theory and Applications volume 2018, Article number: 17 (2018)
Abstract
In this paper, we prove some common fixed point theorems for a finite family of multivalued and singlevalued 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.
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 [2–6]). 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
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 monotoneclosed 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.
\(P\ne \{0\}\);

2.
For all \(a,b \in \mathbb{R}^{+}\) and \(x, y\in P\), we have \(ax+by \in P\);

3.
\(P\cap (P)=\{0\}\).
Given an order cone P on X, we can define a partial order “≤” on X by
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
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
We define the measure of noncompactness on bounded subsets of X by
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.
\(\psi (A)=0\) if and only if A is relatively compact;

2.
\(A\subseteq B\) implies \(\psi (A)\leq \psi (B)\);

3.
\(\psi (\overline{A})=\psi (A)\);

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
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.
\(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)weakly isotone increasing if, for all \(x\in M\), the relations
$$ x_{1}\in T_{1}(x),\quad\quad x_{2}\in T_{2}(x_{1}),\quad\quad \ldots, \quad\quad x_{p}\in T_{p}(x _{p1}) $$imply
$$ T_{1}(x)\leq T_{2}(x_{1})\leq \cdots \leq T_{p}(x_{p1})\leq T_{1}(x _{p}). $$ 
2.
\(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)weakly isotone decreasing if, for all \(x\in M\), the relations
$$ x_{1}\in T_{1}(x),\quad\quad x_{2}\in T_{2}(x_{1}),\quad\quad \ldots,\quad\quad x_{p}\in T _{p}(x_{p1}) $$imply
$$ T_{1}(x_{p}) \leq T_{p}(x_{p1}) \leq T_{p1}(x_{p2}) \leq \cdots \leq T_{1}(x). $$ 
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.
T is called ψcondensing 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.
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 monotoneclosed mappings satisfying:

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

2.
\(T_{1}\) is ψcondensing;

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:
for \(n=0,1,2,\ldots \) .
As we have
we obtain
and then
which gives
Let \(A_{j}=\{x_{np+j} : n\in \mathbb{N}\}\), \(j=0,1,\ldots,p1\), and \(A_{p}=A_{0}\setminus \{x_{0}\}\), we have
The set
is bounded; and since \(T_{2},T_{3},\ldots,T_{p}\) are \(1\psi \)contractive, we have
for all \(k=1,2,\ldots,p\).
Assume that \(\psi (A)\neq 0\), we have
As \(T_{1}\) is ψcondensing, we obtain
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
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.
\(T_{2},T_{3},\ldots,T_{p}\) are \(1\psi \)contractive;

2.
\(T_{1}\) is ψcondensing;

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

2.
\(T_{1}\) is ψcondensing;

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 monotoneclosed mappings (\(p\ge 2\)) satisfying:

1.
\(T_{1},T_{2},T_{3},\ldots,T_{p}\) are ψcondensing;

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
then
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.
The puplet \((T_{1},T_{2},\ldots,T_{p})\) is called weakly isotone increasing if for all \(x\in M\), we have
$$ \textstyle\begin{cases} T_{p}(x) \leq T_{1}(y) \quad \text{for all } y\in T_{p}(x), \\ \forall i\in \{1,2,\ldots,p1\},\quad T_{i}(x)\leq T_{i+1}(y) \quad \text{for all } y\in T_{i}(x). \end{cases} $$ 
2.
We say that the puplet \((T_{1},T_{2},\ldots,T_{p})\) is weakly isotone decreasing if for all \(x\in M\), we have
$$ \textstyle\begin{cases} T_{1}(y) \leq T_{p}(x) \quad \text{for all } y\in T_{p}(x), \\ \forall i\in \{1,2,\ldots,p1\},\quad T_{i+1}(y) \leq T_{i}(x) \quad \text{for all } y\in T_{i}(x). \end{cases} $$ 
3.
The puplet \((T_{1},T_{2},\ldots,T_{p})\) 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 singlevalued 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
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 monotoneclosed mappings with \(p\geq 2\) satisfying:

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

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:
Then as in the proof of Theorem 3.10, it follows that
We have
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 monotoneclosed, 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
\((X, \Vert \cdot \Vert _{\infty })\) 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 \(x\in M\),
and \(T_{3}(x)= \{ x \}\).
The multivalued mappings \(T_{1}\), \(T_{2}\), \(T_{3} \) are closed, so monotoneclosed, and satisfying the condition \(\mathcal{B}_{M}\), and \(T_{1}\) is \((T_{2},T_{3})\)weaklyisotone 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 monotonecontinuous mappings with \(p\geq 2\) satisfying:

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

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
implies A̅ 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 monotoneclosed mappings satisfying:

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

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:
Then, as in the proof of Theorem 3.10, it follows that
Let \(A=\{x_{0},x_{1},x_{2},\ldots\}\), A is countable and
or \(T_{1},T_{2},\ldots,T_{p}\) satisfy the condition \(D_{M}\), so A̅ is weakly compact. Then there exists a subsequence \(\{x_{\sigma (n)}\}\subset A\) and \(x^{*}\in \overline{A}\) such that
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 monotoneclosed, 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 singlevalued 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
\((X, \Vert \cdot \Vert _{\infty })\) 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 \(x\in M\),
and
The multivalued mappings \(T_{1}\), \(T_{2}\), \(T_{3} \) are monotoneclosed and satisfying the condition \(D_{M}\), because the mappings \(T_{1}\), \(T_{2}\), \(T _{3} \) are bounded, and \(T_{1}\) is \((T_{2},T_{3})\)weaklyisotone. 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
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
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 Evalued Bochner integrable functions on J.
Let \(C(J,E)\) denote the space of all continuous Evalued functions on J, define a norm \(\Vert \cdot \Vert _{C}\) in \(C(J,E)\) by
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
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 \(t\rightarrow d(y,F(t))=\inf \{ \Vert yx \Vert _{E}:x \in F(t)\}\) is measurable.
Let \(i\in \{1,2,\ldots,p\}\). Denote
Definition 4.2
A multivalued function \(\beta : J\times E \rightarrow 2^{E}\) is called Carathéodory if

1.
\(t\rightarrow \beta (t,x)\) is measurable for each \(x\in E\), and

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
for all \(x\in E\) with \(\Vert x \Vert _{E}\leq r\).
Denote
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
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 \(i\in \{1,2,\ldots,p\}\).
 \((H_{5})\) :
 \((H_{6})\) :

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 \(i\in \{1,2,\ldots,p1\}\), 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
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 puplet \((T_{1},T_{2},\ldots,T_{p})\) is weakly isotone on \([a,b]\). Let \(x,y\in [a,b]\), then with a similar reasoning as in [7] we get, for each \(i\in \{1,2,\ldots,p\}\),
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 puplet \((T_{1},T_{2},\ldots,T_{p})\) 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.
The cone \(P_{C}\) is normal in \(C(J,E)\).

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

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 ψcondensing 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.
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Farid, M.A., Chaira, K., Marhrani, E.M. et al. Common fixed point theorems for a finite family of multivalued mappings in an ordered Banach space. Fixed Point Theory Appl 2018, 17 (2018). https://doi.org/10.1186/s1366301806423
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DOI: https://doi.org/10.1186/s1366301806423