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Common fixed point theorems for a finite family of multivalued mappings in an ordered Banach space
 Mohamed Amine Farid^{1},
 Karim Chaira^{2},
 El Miloudi Marhrani^{1}Email authorView ORCID ID profile and
 Mohamed Aamri^{1}
https://doi.org/10.1186/s1366301806423
© The Author(s) 2018
 Received: 29 November 2017
 Accepted: 20 May 2018
 Published: 25 June 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.
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 [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 \(x^{*}\in T(x^{*})\).
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
 1.
\(P\ne \{0\}\);
 2.
For all \(a,b \in \mathbb{R}^{+}\) and \(x, y\in P\), we have \(ax+by \in P\);
 3.
\(P\cap (P)=\{0\}\).
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.
The following lemma describes some properties of the measure of noncompactness.
Lemma 3.6
 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
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
 1.\(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)weakly isotone increasing if, for all \(x\in M\), the relationsimply$$ x_{1}\in T_{1}(x),\quad\quad x_{2}\in T_{2}(x_{1}),\quad\quad \ldots, \quad\quad x_{p}\in T_{p}(x _{p1}) $$$$ T_{1}(x)\leq T_{2}(x_{1})\leq \cdots \leq T_{p}(x_{p1})\leq T_{1}(x _{p}). $$
 2.\(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)weakly isotone decreasing if, for all \(x\in M\), the relationsimply$$ x_{1}\in T_{1}(x),\quad\quad x_{2}\in T_{2}(x_{1}),\quad\quad \ldots,\quad\quad x_{p}\in T _{p}(x_{p1}) $$$$ T_{1}(x_{p}) \leq T_{p}(x_{p1}) \leq T_{p1}(x_{p2}) \leq \cdots \leq T_{1}(x). $$
 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])
 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
 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.
Proof
The case \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)weakly isotone decreasing is similar, which ends the proof. □
By Theorem 3.10, we obtain the following result.
Corollary 3.12
 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.
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.
 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.
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
 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.
Proof
Definition 3.16
 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
Theorem 3.20
 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.
Proof
The case when \(T_{1}\) is \(( T_{k} ) _{2\leq k \leq p}\)weakly isotone decreasing is similar. □
Example 3.22
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
 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.
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
Theorem 3.25
 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
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
Remark 3.29
In Example 3.22 the mappings \(T_{1}\), \(T_{2}\), \(T_{3}\) are not satisfying the condition \(D_{M}\).
4 Application
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.
Definition 4.2
 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
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
 \((H_{0})\) :

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 \(i\in \{1,2,\ldots,p\}\).
 \((H_{6})\) :

For each \(i\in \{1,2,\ldots,p\}\), the functionis Lebesgue integrable on J.$$ t \rightarrow \bigl\vert \!\bigl\vert \!\bigl\vert F_{i}(t,a(t)\bigr\vert \!\bigr\vert \!\bigr\vert _{E}+\bigl\vert \!\bigl\vert \!\bigl\vert F_{i}(t,b(t)\bigr\vert \!\bigr\vert \!\bigr\vert _{E} $$
 \((H_{7})\) :

\(F_{p}(t,x)\leq F_{1}(t,y)\) 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
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 \(T_{1},T_{2},\ldots,T_{p}:[a,b]\rightarrow 2^{[a,b]}\).
 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 \([a,b]\).
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.
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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.
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Authors’ Affiliations
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