On monotone contraction mappings in modular function spaces
- Monther R Alfuraidan^{1},
- Mostafa Bachar^{2}Email author and
- Mohamed A Khamsi^{3}
https://doi.org/10.1186/s13663-015-0274-9
© Alfuraidan et al.; licensee Springer. 2015
Received: 21 September 2014
Accepted: 28 January 2015
Published: 24 February 2015
Abstract
We prove the existence of fixed points of monotone-contraction mappings in modular function spaces. This is the modular version of the Ran and Reurings fixed point theorem. We also discus the extension of these results to the case of pointwise monotone-contraction mappings in modular function spaces.
Keywords
MSC
1 Introduction
The purpose of this paper is to give an outline of a fixed point theory for monotone-contraction mappings defined on some subsets of modular function spaces which are natural generalizations of both function and sequence variants of many important spaces like the Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces, and many others [1]. Recently, the authors in [2] presented a series of fixed point results for pointwise contractions and asymptotic pointwise contractions acting in modular functions spaces. The current paper operates within the same framework.
The importance for applications of mappings defined within modular function spaces consists in the richness of the structure of modular function spaces, which, besides being Banach spaces, are equipped with modular equivalents of norm or metric notions and also are equipped with almost everywhere convergence and convergence in sub-measure. In many cases, particularly in applications to integral operators, approximation and fixed point results, modular type conditions are much more natural as modular type assumptions can be more easily verified than their metric or norm counterparts; see for example [3]. From this perspective, the fixed point theory in modular function spaces should be considered as complementary to the fixed point theory in Banach linear spaces and in metric spaces.
The theory of contractions and nonexpansive mappings defined on convex subsets of Banach spaces has been well developed since the 1960s, see for example [4–7], and generalized to other metric spaces [8–10], and modular function spaces [3, 11]. The corresponding fixed point results were then extended to larger classes of mappings like asymptotic mappings [12, 13], pointwise contractions [14], and asymptotic pointwise contractions and nonexpansive mappings [2, 15–17].
In recent years, a version of the Banach contraction principle [18] was given in partially ordered metric spaces [19, 20] and in metric spaces with a graph [21]. In this work, we discuss some of these extensions to modular function spaces.
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 Ω, 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 will denote the space of all extended measurable functions, i.e. all functions \(f:\Omega\rightarrow[-\infty,\infty]\) such that there exist 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 additive, i.e. \(\rho(f1_{A \cup B}) = \rho(f1_{A})+\rho(f1_{B})\), for any \(A,B \in\Sigma\) such that \(A \cap B = \emptyset\), \(f \in\mathcal{M}\);
- (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 [1]. 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 [1, 22, 23].
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 the 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 an \(\alpha> 0\) then there exists a subsequence \(\{g_{n}\}\) of \(\{f_{n}\}\) such that \(g_{n}\rightarrow0\) ρ-a.e.
- (4)
\(\rho(f) \leq\liminf{ \rho(f_{n})}\) whenever \(f_{n} \rightarrow f \) ρ-a.e. (Note: this property is equivalent to the Fatou property.)
The following definition plays an important role in the theory of modular function spaces [1].
Definition 2.4
Let \(\rho\in\Re\). We say that ρ has the \(\Delta_{2}\)-property if \(\sup_{n} \rho(2f_{n},D_{k})\rightarrow0\), whenever \(\{ D_{k}\} \in\Sigma\) is decreasing, \(\bigcap_{k \geq1} D_{k} = \emptyset\), and \(\lim_{k \rightarrow+\infty} \sup_{n} \rho(f_{n},D_{k}) =0\).
Theorem 2.2
- (a)
ρ has \(\Delta_{2}\),
- (b)
if \(\rho(f_{n}) \rightarrow0\) then \(\rho(2f_{n}) \rightarrow0\),
- (c)
if \(\rho(\alpha f_{n}) \rightarrow0\) for an \(\alpha>0\) then \(\| f_{n}\|_{\rho}\rightarrow0\), i.e. the modular convergence is equivalent to the norm convergence.
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 2.5
- (a)
We say that \(\{f_{n}\}\) is ρ-convergent to f and write \(f_{n}\rightarrow0\ (\rho)\) 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 \(\sup\{\rho (f-g); f \in B, g \in B \} <\infty\).
- (e)
A set \(B\subset L_{\rho}\) is called strongly ρ-bounded if there exists \(\beta> 1\) such that \(\sup\{\rho (\beta(f-g) ); f \in B, g \in B \} <\infty\).
- (f)
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 \(f \in C\).
- (g)
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 \(\rho\in\Re\). Let \(\{ f_{n}\}\) 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 \(\rho(f_{n} -f) \rightarrow0\) and \(f_{n} \leq f\) ρ-a.e. (resp. \(f \leq f_{n}\) ρ-a.e.), for any \(n \geq1\).
Let us finish this section with the modular definitions of some special mappings, which will be studied throughout. The definitions are straightforward generalizations of their norm and metric equivalents [13, 15–17].
Definition 2.6
- (i)a monotone contraction if T is monotone and there exists \(k < 1\) such thatwhenever \(f, g \in C\) and \(f \leq g\) ρ-a.e.;$$\rho\bigl(T(f) - T(g)\bigr) \leq k\ \rho(f-g) $$
- (ii)a pointwise monotone contraction if T is monotone and for any \(f \in C\) there exists \(k_{f} < 1\) such thatwhenever \(g \in C\) such that \(f \leq g\) or \(g \leq f\) ρ-a.e.;$$\rho\bigl(T(f) - T(g)\bigr) \leq k_{f}\ \rho(f-g) $$
- (iii)an asymptotic pointwise monotone contraction if T is monotone and for any \(f \in C\) and \(n \geq1\), there exists a constant \(k_{f}^{n}\) such thatwith \(k_{f} = \limsup_{n \rightarrow\infty} k_{f}^{n} < 1\), whenever \(g \in C\) such that \(f \leq g\) or \(g \leq f\) ρ-a.e.$$\rho\bigl(T^{n}(f) - T^{n}(g)\bigr) \leq k^{n}_{f}\ \rho(f-g) $$
Let us give an example which will illustrate the role of the above defined notions. This example is a typical application of the methods of modular function spaces to the theory of nonlinear integral and differential equations.
Example 2.1
3 Fixed point of monotone contraction mappings
In this section we discuss an analogue of the Ran and Reurings fixed point theorem [20] in modular function spaces. The key feature in this fixed point theorem is that the Lipschitzian condition on the nonlinear map is only assumed to hold on elements that are comparable in the partial order.
Theorem 3.1
Let \(\rho\in\Re\). Let \(C\subset L_{\rho}\) be nonempty, ρ-closed and ρ-bounded. Let \(T:C\rightarrow C\) be a monotone increasing contraction mapping. Assume that there exists \(f_{0} \in C\) which is comparable to \(T(f_{0})\), i.e., \(f_{0} \leq T(f_{0})\) or \(T(f_{0}) \leq f_{0}\) ρ-a.e. Then \(\{T^{n}(f_{0})\}\) is ρ-Cauchy and ρ-converges to \(\bar{f}\) a fixed point of T. Moreover, if \(f \in C\) is comparable to \(\bar{f}\), then \(\{T^{n}(f)\}\) ρ-converges to \(\bar{f}\).
Proof
The missing information in Theorem 3.1 is the uniqueness of the fixed point. In fact we do have a partial positive answer to this question. Indeed if \(\bar{h}\) and \(\bar{g}\) are two fixed points of T such that \(\bar{h} \leq\bar{g}\) ρ-a.e., then we must have \(\bar{h} = \bar{g}\). In general T may have more than one fixed point.
Remark 3.1
The conclusion of Theorem 3.1 does not extend easily to the case of monotone decreasing contractions. Note that the mapping T in Example 2.1 is monotone decreasing. But if we assume that there exists \(f_{0} \in C\) such that \(f_{0} \leq T^{2}(f_{0}) \leq T(f_{0})\) or \(T(f_{0}) \leq T^{2}(f_{0}) \leq f_{0}\) ρ-a.e., then \(\{T^{n}(f_{0})\}\) is ρ-Cauchy and ρ-converges to \(\bar{f}\), a fixed point of T. Moreover, if \(f \in C\) is comparable to \(\bar{f}\), then \(\{T^{n}(f)\}\) ρ-converges to \(\bar{f}\).
Next we discuss the case of monotone pointwise Lipschitzian mappings and extend the recent results of [2].
Theorem 3.2
Let us assume that \(\rho\in\Re\). Let \(C \subset L_{\rho}\) be nonempty, ρ-closed and ρ-bounded. Let \(T: C \rightarrow C\) be a pointwise monotone increasing contraction or asymptotic pointwise monotone increasing contraction. Then any two comparable fixed points of T are equal. Moreover, if \(f_{0}\) is a fixed point of T, then the orbit \(\{T^{n}(f)\}\) ρ-converges to \(f_{0}\), for any \(f\in C\) which is comparable to \(f_{0}\).
Proof
Theorem 3.2 does not discuss the existence part of the fixed point. Next we investigate this problem. We will start our discussion with the existence of fixed point results in the case of uniformly continuous function modulars.
Definition 3.1
Let us mention that uniform continuity holds for a large class of function modulars. For instance, it can be proved that in Orlicz spaces over a finite atomless measure [25] or in sequence Orlicz spaces [26] the uniform continuity of the Orlicz modular is equivalent to the \(\Delta_{2}\) property.
Lemma 3.1
[2]
Before we state our next result, we need the following definition of a property that plays in the theory of modular function spaces a role similar to the reflexivity in Banach spaces (see e.g. [11]).
Definition 3.2
[11]
We say that \(L_{\rho}\) has property (R) if and only if every decreasing sequence \(\{ C_{n}\}\) of nonempty, ρ-bounded, ρ-closed, and convex subsets of \(L_{\rho}\) has a nonempty intersection.
Theorem 3.3
Assume that \(\rho\in\Re\) is uniformly continuous and has property (R). Assume that ρ satisfies the \(\Delta_{2}\)-property. Let \(K\subset L_{\rho}\) be nonempty, convex, ρ-closed, and ρ-bounded. Let \(T:K\rightarrow K\) be a pointwise monotone increasing contraction or asymptotic pointwise monotone increasing contraction. Assume there exists \(f \in K\) such that f and \(T(f)\) are comparable. Then T has a fixed point \(f_{0}\in K\).
Proof
Remark 3.2
Declarations
Acknowledgements
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this Research group No. RG-1435-079.
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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