Fixed points of multivalued mappings in modular function spaces with a graph
- Monther Rashed Alfuraidan^{1}Email author
https://doi.org/10.1186/s13663-015-0292-7
© Alfuraidan; licensee Springer. 2015
Received: 26 October 2014
Accepted: 15 January 2015
Published: 25 March 2015
Abstract
Let \(\rho\in\Re\) (the class of all nonzero regular function modulars defined on a nonempty set Ω) and G be a directed graph defined on a subset C of \(L_{\rho}\). In this paper, we discuss the existence of fixed points of monotone G-contraction and G-nonexpansive mappings in modular function spaces. These results are the modular version of Jachymski fixed point results for mappings defined in a metric space endowed with a graph.
Keywords
MSC
1 Introduction
Fixed point theorems for monotone single-valued mappings in a metric space endowed with a partial ordering have been widely investigated. These theorems are hybrids of the two most fundamental and useful theorems in fixed point theory: Banach’s contraction principle ([1], Theorem 2.1) and Tarski’s fixed point theorem [2, 3]. Generalizing the Banach contraction principle for multivalued mappings to metric spaces, Nadler [4] obtained the following result.
Theorem 1.1
[4]
A number of extensions and generalizations of Nadler’s theorem were obtained by different authors; see, for instance, [5, 6] and the references cited therein. Tarski’s theorem was extended to multivalued mappings by different authors; see [7–9]. Investigation of the existence of fixed points for single-valued mappings in partially ordered metric spaces was initially considered by Ran and Reurings in [10] who proved the following result.
Theorem 1.2
[10]
- (1)There exists \(k\in(0,1)\) with$$ d\bigl(f(x),f(y)\bigr)\leq k d(x,y) \quad \textit{for all } x\succeq y . $$
- (2)
There exists \(x_{0} \in X\) with \(x_{0} \preceq f(x_{0})\) or \(x_{0} \succeq f(x_{0})\).
After this, different authors considered the problem of existence of a fixed point for contraction mappings in partially ordered sets; see [11–14] and the references cited therein. Nieto et al. in [14] proved the following theorem.
Theorem 1.3
[14]
- (1)
f is continuous and there exists \(x_{0}\in X\) with \(x_{0} \preceq f(x_{0})\) or \(x_{0} \succeq f(x_{0})\);
- (2)
\((X,d,\preceq)\) is such that for any nondecreasing \((x_{n})_{n\in N}\), if \(x_{n}\rightarrow x\), then \(x_{n} \preceq x\) for \(n \in N\), and there exists \(x_{0} \in X\) with \(x_{0} \preceq f(x_{0})\);
- (3)
\((X,d,\preceq)\) is such that for any nonincreasing \((x_{n})_{n\in N}\), if \(x_{n}\rightarrow x\), then \(x_{n} \succeq x\) for \(n \in N\), and there exists \(x_{0} \in X\) with \(x_{0} \succeq f(x_{0})\).
Recently, two results have appeared, giving sufficient conditions for f to be a PO, if \((X,d)\) is endowed with a graph. The first result in this direction was given by Jachymski and Lukawska [15, 16], which generalized the results of [12, 14, 17, 18] to a single-valued mapping in metric spaces with a graph instead of partial ordering. Subsequently, Beg et al. [19] tried to extend the results of [15] to multivalued mappings, but their extension was not carried correctly (see [20]). The aim of this paper is to give the correct extension by studying the existence of fixed points for multivalued mappings in modular function spaces endowed with a graph G. Recall that the fixed point theory in modular function spaces was initiated by Khamsi et al. [21]. The reader interested in fixed point theory in modular function spaces is referred to [22–25].
2 Preliminaries
Let Ω be a nonempty set and Σ be a nontrivial σ-algebra of subsets of Ω. Let \(\mathcal{P}\) be a δ-ring of subsets of Σ such that \(E \cap A \in\mathcal{P}\) for any \(E \in\mathcal{P}\) and \(A \in\Sigma\). Let us assume that there exists an increasing sequence of sets \(K_{n} \in \mathcal{P}\) such that \(\Omega= \bigcup K_{n}\). By ℰ we denote the linear space of all simple functions with supports from \(\mathcal{P}\). By \(\mathcal{M}_{\infty}\) we denote the space of all extended measurable functions, i.e., all functions \(f:\Omega\rightarrow[-\infty,\infty]\) such that there exists a sequence \(\{g_{n}\} \subset\mathcal{E}\), \(|g_{n}|\leq|f|\) and \(g_{n}(\omega) \rightarrow f(\omega)\) for all \(\omega\in\Omega\). By \(1_{A}\) we denote the characteristic function of the set A.
Definition 2.1
- (i)
\(\rho(0) = 0\);
- (ii)
ρ is monotone, i.e., \(|f(\omega)|\leq|g(\omega)|\) for all \(\omega\in\Omega\) implies \(\rho(f) \leq\rho(g)\), where \(f,g \in \mathcal{M_{\infty}} \);
- (iii)
ρ is orthogonally subadditive, i.e., \(\rho(f1_{A \cup B}) \leq\rho(f1_{A})+\rho(f1_{B})\) for any \(A,B \in\Sigma\) such that \(A \cap B \neq\emptyset\), \(f \in\mathcal{M}\);
- (iv)
ρ has the Fatou property, i.e., \(|f_{n}(\omega)| \uparrow |f(\omega)|\) for all \(\omega\in\Omega\) implies \(\rho(f_{n}) \uparrow\rho (f)\), where \(f \in\mathcal{M_{\infty}}\);
- (v)
ρ is order continuous in ℰ, i.e., \(g_{n} \in \mathcal{E}\) and \(|g_{n}(\omega)|\downarrow0\) implies \(\rho (g_{n})\downarrow0\).
Definition 2.2
- (1)
We say that ρ is a regular function semimodular if \(\rho (\alpha f) = 0\) for every \(\alpha> 0\) implies \(f=0\) ρ-a.e.;
- (2)
We say that ρ is a regular function modular if \(\rho(f) = 0\) implies \(f=0\) ρ-a.e.
Let us denote \(\rho(f,E) = \rho(f1_{E})\) for \(f \in\mathcal{M}\), \(E \in \Sigma\). It is easy to prove that \(\rho(f,E)\) is a function pseudomodular in the sense of Definition 2.1.1 in [22] (more precisely, it is a function pseudomodular with the Fatou property). Therefore, we can use all results of the standard theory of modular function spaces as per the framework defined by Kozlowski in [22, 26, 27].
Definition 2.3
- (a)A modular function space is the vector space \(L_{\rho}(\Omega,\Sigma)\), or briefly \(L_{\rho}\), defined by$$ L_{\rho} = \bigl\{ f\in\mathcal{M}; \rho(\lambda f) \rightarrow0\mbox{ as }\lambda \rightarrow0 \bigr\} . $$
- (b)The following formula defines a norm in \(L_{\rho}\) (frequently called Luxemburg norm):$$ \|f\|_{\rho} = \inf\bigl\{ \alpha>0; \rho(f/\alpha)\leq1 \bigr\} . $$
In the following theorem we recall some of the properties of modular spaces.
Theorem 2.1
- (1)
\((L_{\rho},\|\cdot\|_{\rho})\) is complete and the norm \(\|\cdot\|_{\rho}\) is monotone w.r.t. the natural order in ℳ.
- (2)
\(\|f_{n}\|_{\rho}\rightarrow0\) if and only if \(\rho(\alpha f_{n}) \rightarrow0\) for every \(\alpha> 0\).
- (3)
If \(\rho(\alpha f_{n}) \rightarrow0\) for \(\alpha> 0\), then there exists a subsequence \(\{g_{n}\}\) of \(\{f_{n}\}\) such that \(g_{n}\rightarrow0\) ρ-a.e.
- (4)
\(\rho(f) \leq\liminf_{n \rightarrow\infty} \rho(f_{n})\), whenever \(f_{n} \rightarrow f\) ρ-a.e. Note this property will be referred to as the Fatou property.
The following definition plays an important role in the theory of modular function spaces.
Definition 2.4
Let \(\rho\in\Re\). We say that ρ has the \(\Delta_{2}\)-property if \(\sup_{n} \rho(2f_{n},D_{k})\rightarrow0\) as \(k \rightarrow\infty\) whenever \(D_{k}\downarrow\emptyset\) and \(\sup_{n} \rho (f_{n},D_{k})\rightarrow0\), where \(\{f_{n}\}_{n\geq1}\subset M\) and \(D_{k}\in\Sigma\).
We have the following interesting result.
Theorem 2.2
[22]
- (a)
ρ has the \(\Delta_{2}\)-property,
- (b)
if \(\rho(f_{n}) \rightarrow0\), then \(\rho(2f_{n}) \rightarrow0\),
- (c)
if \(\rho(\alpha f_{n}) \rightarrow0\) for \(\alpha>0\), then \(\| f_{n}\|_{\rho}\rightarrow0\), i.e., the modular convergence is equivalent to the norm convergence.
Remark 2.1
We will also use another type of convergence which is situated between norm and modular convergence. It is defined, among other important terms, in the following definition.
Definition 2.5
- (a)
We say that \(\{f_{n}\}\) is ρ-convergent to f and write \(f_{n}\rightarrow f (\rho)\) if and only if \(\rho(f_{n}-f)\rightarrow0\).
- (b)
A sequence \(\{f_{n}\}\), where \(f_{n} \in L_{\rho}\), is called ρ-Cauchy if \(\rho(f_{n}-f_{m})\rightarrow0\) as \(n,m \rightarrow \infty\).
- (c)
A set \(B\subset L_{\rho}\) is called ρ-closed if for any sequence of \(f_{n} \in B\), the convergence \(f_{n} \rightarrow f (\rho)\) implies that f belongs to B.
- (d)
A set \(B\subset L_{\rho}\) is called ρ-bounded if \(\sup\{\rho (f-g); f,g \in B \} <\infty\).
- (e)
A set \(C\subset L_{\rho}\) is called ρ-a.e. closed if for any \(\{f_{n}\}\) in C which ρ-a.e. converges to some f, then we must have \(f \in C\).
- (f)
A set \(C\subset L_{\rho}\) is called ρ-a.e. compact if for any \(\{f_{n}\}\) in C, there exists a subsequence \(\{f_{n_{k}}\}\) which ρ-a.e. converges to some \(f \in C\).
Let us note that ρ-convergence does not necessarily imply ρ-Cauchy condition. Also, \(f_{n}\rightarrow f\) does not imply in general \(\lambda f_{n}\rightarrow\lambda f\), \(\lambda> 1\). Using Theorem 2.1 it is not difficult to prove the following.
Proposition 2.1
- (i)
\(L_{\rho}\) is ρ-complete.
- (ii)
\(L_{\rho}\) is a lattice, i.e., for any \(f, g \in L_{\rho}\), we have \(\max\{f,g\} \in L_{\rho}\) and \(\min\{f,g\} \in L_{\rho}\).
- (iii)
ρ-balls \(B_{\rho}(x,r) = \{y\in L_{\rho}; \rho(x-y) \leq r\} \) are ρ-closed and ρ-a.e. closed.
Using the property (3) of Theorem 2.1, we get the following result.
Theorem 2.3
Let \(\rho\in\Re\) and \(\{ f_{n}\}\) be a ρ-Cauchy sequence in \(L_{\rho}\). Assume that \(\{f_{n}\}\) is monotone increasing, i.e., \(f_{n} \leq f_{n+1}\) ρ-a.e. (resp. decreasing, i.e., \(f_{n+1} \leq f_{n}\) ρ-a.e.) for any \(n \geq1\). Then there exists \(f \in L_{\rho}\) such that \(\rho(f_{n} -f) \rightarrow 0\) and \(f_{n} \leq f\) ρ-a.e. (resp. \(f \leq f_{n}\) ρ-a.e.) for any \(n \geq1\).
It seems that the terminology of graph theory instead of partial ordering sets can give clearer pictures and yield to generalize the contraction theorems. Let us finish this section with such terminology for the modular space mapping which will be studied throughout.
In the sequel we assume that \(\rho\in\Re\) is a convex, σ-finite modular function, and C is a nonempty subset of the modular function space \(L_{\rho}\). We denote by \({\mathcal{C}}(C)\) the collection of all nonempty ρ-closed subsets of C, and by \({\mathcal{K}}(C)\) the collection of all nonempty ρ-compact subsets of C.
Definition 2.6
Definition 2.7
Our definition of monotone multivalued mappings is slightly different from the one used in [19]. Indeed in Definition 2.6 in [19], one may let α go to 0, which is very restrictive. Because of this, the proof of the main result is incorrect. In this work, we will show how our definition will give the correct proof.
Definition 2.8
[30]
The following properties were proved in [30].
Lemma 2.1
[30]
- (1)
\(\omega(\alpha) < \infty\) for any \(\alpha> 0\),
- (2)
ω is a strictly increasing function, and \(\omega(1) = 1\),
- (3)
\(\omega(\alpha\beta) \leq\omega(\alpha) \omega(\beta)\) for any \(\alpha, \beta\in(0,\infty)\),
- (4)
\(\omega^{-1}(\alpha) \omega^{-1}(\beta) \leq\omega ^{-1}(\alpha\beta)\), where \(\omega^{-1}\) is the function inverse of ω,
- (5)for any \(f \in L_{\rho}\), \(f \neq0\), we have$$\|f\|_{\rho}\leq\frac{1}{\omega^{-1} (1/\rho(f) )}. $$
The following technical lemma will be useful later on in this work.
Lemma 2.2
[30]
Note that this lemma is crucial since the main assumption on \(\{f_{n}\}\) will not be enough to imply that \(\{f_{n}\}\) is ρ-Cauchy since ρ fails the triangle inequality.
Property 2.1
For any sequence \(\{f_{n}\}_{n\in\mathbb{N}}\) in C, if \(f_{n}\) ρ-converges to f and \((f_{n} , f_{n+1})\in E(G)\) for \(n\in\mathbb {N}\), then \((f_{n} , f)\in E(G)\).
3 Main results
We begin with the following theorem that gives the existence of a fixed point for monotone multivalued mappings in modular spaces endowed with a graph. The key feature in this theorem is that the Lipschitzian condition on the nonlinear map is only assumed to hold on elements that are comparable in the natural partial order of \(L_{\rho}\).
Theorem 3.1
Let \(\rho\in\Re\) be convex. Let \(C\subset L_{\rho}\) be a nonempty and ρ-closed subset. Assume that ρ satisfies the \(\Delta_{2}\)-type condition. Let \(T:C \rightarrow{\mathcal{C}}(C)\) be a monotone increasing ρ-contraction mapping and \(C_{T}:=\{f\in C; f \leq g\ \rho\textit{-a.e. for some }g\in T(f)\}\). If \(C_{T}\neq\emptyset\), then T has a fixed point in C.
Proof
Note that the fixed point may not be unique. Indeed if we take A, any nonempty ρ-closed subset of C, then the multivalued map \(T:C \rightarrow{\mathcal{C}}(C)\) defined by \(T(f) = A\), for any \(f \in C\), is a monotone increasing ρ-contraction mapping. The set of fixed points of T is exactly the set A.
An easy consequence of Theorem 3.1 is the following result.
Proposition 3.1
Proof
Remark 3.1
We can modify slightly the above proof to show that the approximate fixed point sequence \(\{ f_{n}\}\) and its associated sequence \(\{F_{n}\}\) satisfy \(f_{n} \leq f_{n+1} \leq F_{n+1}\) ρ-a.e. Indeed, set \(\{\lambda_{n}\} = \{1/(n+1)\}_{n \geq1}\). Let \(f_{0} \in C_{T}\). Then from the above proof, there exists a fixed point \(f_{1} = \lambda_{1} f_{0} + (1-\lambda_{1})F_{1}\) with \(f_{0} \leq f_{1}\) ρ-a.e. It is easy to check that \(f_{0} \leq f_{1} \leq F_{1}\) ρ-a.e. Clearly \(f_{1} \in C_{T}\). By induction we build the sequences \(\{ f_{n}\}\) and \(\{F_{n}\}\) with \(F_{n} \in T(f_{n})\), \(f_{n+1} = \lambda_{n+1} f_{n} + (1-\lambda_{n+1})F_{n+1}\), and \(f_{n} \leq f_{n+1} \leq F_{n+1}\) for every \(n \geq1\). Since \(\lambda_{n} \rightarrow0\) as n go to ∞, we conclude that \(\{f_{n}\}\) is an approximate fixed point sequence of T.
Using the above results, we are now ready to prove the main fixed point theorem for ρ-nonexpansive monotone multivalued mappings. This theorem may be seen as the monotone version of Theorem 2.4 of [31]. Note that the authors of [31] must assume that ρ is additive to be able to have their conclusion. To avoid the additivity of ρ, we need the following property.
Definition 3.1
Theorem 3.2
Let \(\rho\in\Re\) be convex. Let \(C\subset L_{\rho}\) be a nonempty, ρ-closed, and ρ-bounded convex subset. Assume that ρ satisfies the \(\Delta_{2}\)-type condition, \(L_{\rho}\) satisfies the ρ-a.e.-Opial property, and C is ρ-a.e. compact. Then each monotone increasing ρ-nonexpansive map \(T : C \rightarrow{\mathcal {K}}(C)\) has a fixed point.
Proof
Next we give the graph versions of our results found above.
Theorem 3.3
- (1)
For any \(f\in C_{T}\), \(T|_{[f]_{\widetilde{G}}}\) has a fixed point.
- (2)
If \(f\in C\) with \((\bar{f},f)\in E(G)\), where \(\bar{f}\) is a fixed point of T, then there exists a sequence \(\{f_{n}\}\) such that \(f_{n+1}\in T(f_{n})\) for every \(n\geq0\), and \(\{f_{n}\}\) ρ-converges to \(\bar{f}\).
- (3)
If G is weakly connected, then T has a fixed point in G.
- (4)
If \(C':=\bigcup\{[f]_{\widetilde{G}_{\rho}} : f\in C_{T}\}\), then \(T|_{C'}\) has a fixed point in C.
Proof
3. Since \(C_{T}\neq\emptyset\), there exists \(f_{0}\in C_{T}\), and since G is weakly connected, then \([f_{0}]_{\widetilde{G}_{\rho}}=C\), and by 1 the mapping T has a fixed point.
4. It follows easily from 1 and 3. □
Remark 3.2
If we assume that G is such that \(E(G):=C\times C\), then clearly G is connected and our Theorem 3.3 gives Nadler’s theorem. Moreover, if T is single-valued, then we get the Banach contraction theorem and if T is multivalued, then we get the corrected version of the analogue of the main result of Beg et al. [19] in modular function spaces.
The following is a direct consequence of Theorem 3.3.
Corollary 3.1
Let \(\rho\in\Re\) be convex. Let \(C\subset L_{\rho}\) be a nonempty and ρ-closed subset that has Property 2.1. Assume that ρ satisfies the \(\Delta_{2}\)-type condition and C is ρ-bounded. Assume that G is weakly connected. Let \(T:C=V(G)\rightarrow{\mathcal {C}}(C)\) be a monotone increasing G-contraction multivalued mapping such that there exist \(f_{0}\) and \(f_{1} \in T(f_{0})\) with \((f_{0},f_{1})\in E(G)\). Then T has a fixed point.
Let us give an example which will illustrate the role of the above defined notions.
Example 3.1
- (1)
\(f_{0}\) is the step function on \([0,1]\) with partition \(0=x_{0}< x_{1}<x_{2}<\cdots<x_{n}=1\) such that \(f_{0}=c_{i}=\frac{1}{i+1}\) on \((x_{i-1},x_{i})\) and \(f_{0}(x_{i})=d_{i}\) arbitrary real number. Note that \(\| f_{0} \|_{\infty}=\max | c_{i} |= \frac{1}{2}\).
- (2)
\(f_{1}=e^{x}\) on \([0,1]\). So \(\| f_{1} \|_{\infty}= e\).
- (3)
\(f_{2}=X^{2}\) on \([0,1]\). So \(\| f_{2} \|_{\infty}= 1\).
Now, for all \((f,g)\in E(G)\), T is a G-contraction. Also all other assumptions of Theorem 3.3 are satisfied and T has a fixed point.
As an application of Theorem 3.3, we have the following result whose proof is similar to Proposition 3.1.
Proposition 3.2
Note that a similar conclusion to Theorem 3.2 in terms of graph may be found under strong properties satisfied by the graph G.
Declarations
Acknowledgements
The author acknowledges King Fahd University of Petroleum and Minerals for supporting this research.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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