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Common fixed points in partially ordered modular function spaces
Journal of Inequalities and Applications volume 2014, Article number: 78 (2014)
Abstract
The purpose of this paper is to study the existence and uniqueness of common fixed point results in partially ordered modular function spaces.
MSC:47H10, 54H25, 54C60, 46B40.
1 Introduction
Study of modular spaces was initiated by Nakano [1] in connection with the theory of order spaces which was further generalized by Musielak and Orlicz [2]. The study of fixed points of mappings on complete metric spaces equipped with a partial ordering ⪯ was first investigated in 2004 by Ran and Reurings [3], and then by Nieto and Rodriguez-Lopez [4]. They applied their results to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions (see also [5]). The study of this theory in the context of modular function spaces was initiated by Khamsi et al. [6] (see also [7] and [8]). Kuaket and Kumam [9] and Mongkolkeha and Kumam [10–12], considered and proved some fixed point and common fixed point results for generalized contraction mappings in modular spaces. Also, Kumam [13] obtained some fixed point theorems for non-expansive mappings in arbitrary modular spaces. Recently, Kutabi and Latif [14] studied fixed points of multivalued maps in modular function spaces.
The study of common fixed points of mappings satisfying certain contractive conditions in the setup of partially ordered metric spaces can be employed to establish the existence of solutions of many types of operator equations, such as differential and integral equations. There are a few examples given in the following papers: [15–20] and references mentioned therein. The objective of this paper is to initiate the study of common fixed point results in partially ordered modular function spaces. As an application of our results, we study the property Q for mappings involved herein.
2 Preliminaries
Some basic facts and notations about modular spaces are recalled from [21].
Definition 2.1 Let X be a real (or complex) vector space. A functional is called modular if, for any x, y in X, the following hold:
(m1) if and only if .
(m2) for every scalar α with .
(m3) provided that , and .
If (m3) is replaced by if , and , then ρ is called convex modular.
The vector space given by
is called a modular space. Generally, the modular ρ is not subadditive and therefore does not behave as a norm or a distance.
Modular space can be equipped with an F-norm defined by
If ρ is convex modular, then
defines a norm on the modular space and is called the Luxemburg norm.
Define the ρ-ball, centered at with radius r, as
Definition 2.2 A function modular is said to satisfy -type condition, if there exists such that for any , we have .
Definition 2.3 ρ is said to satisfy the -condition if whenever as .
Definition 2.4 Let be a modular space. The sequence is called:
(t1) ρ-convergent to , if as .
(t2) ρ-Cauchy, if as n and .
Note that ρ-convergence does not imply ρ-Cauchy since ρ does not satisfy the triangle inequality. In fact, one can show that this will happen if and only if ρ satisfies the -type condition.
It is well known that [6, 22] under the -condition the norm convergence and modular convergence are equivalent. The same is true when we deal with the -type condition. Throughout this paper, we assume that modular function ρ is convex and satisfies the -type condition. We also state the following definition and results given in [7].
Definition 2.5 The growth function of a function modular ρ is defined as
Observe that for all .
Lemma 2.6 The growth function ω has the following properties:
(g1) , for each .
(g2) is a convex, strictly increasing function. So, it is continuous.
(g3) ; for all .
(g4) ; for all , where is the inverse function of ω.
The following lemma shows that the growth function can be used to give an upper bound for for each .
Lemma 2.7 Let ρ be a convex modular function satisfying the -type condition. Then
whenever .
Let S and T be two self-maps on a modular function space . A point is called (1) a fixed point of S if ; (2) a coincidence point of a pair if ; (3) a common fixed point of a pair if . If for some x in , then w is called a point of coincidence of S and T.
The pair is said to be compatible if as , whenever is a sequence in X such that and are ρ-convergent to .
A pair is said to be ρ-weakly compatible if S and T commute at their coincidence points.
We denote the set of fixed points of S by .
Definition 2.8 Let be a partially modular ordered space. A pair of self-maps of is said to be ρ-weakly increasing if and for all .
Definition 2.9 Let be a partially modular ordered space and , be two self-maps on . An order pair is said to be partially ρ-weakly increasing if for all .
The pair is ρ-weakly increasing if and only if the ordered pairs and are partially ρ-weakly increasing.
Definition 2.10 Let be a partially modular ordered space. A mapping is said to be ρ-weak annihilator of if for all .
Definition 2.11 Let be a partially ordered modular space. A mapping is said to be ρ-dominating if for all .
Definition 2.12 Let be a partially modular ordered space and , , be three self-maps on , such that and . We say that and are ρ-weakly increasing with respect to if and only if for all , we have for all , and , for all , where for all .
Definition 2.13 Let be a partially modular ordered space and , , be three self-maps on such that . We say that and are partially ρ-weakly increasing with respect to if for all , we have , for all .
Definition 2.14 Let X be a vector space. Then is called an ordered modular function space iff: (i) ρ is convex modular function on X and (ii) ⪯ is a partial order on X.
Let be a partial ordered set. Then are called comparable if or holds.
3 Common fixed point results
We begin with a common fixed point theorem for two pairs of partially weakly increasing functions on an ordered modular function spaces. It may regarded as the main result of this article.
Theorem 3.1 Let be a complete ordered modular function space and S, I, T, and J self-maps on such that and . Suppose that and are ρ-weakly increasing, and the dominating maps S and T are weak annihilators of J and I, respectively. If for every two comparable elements
is satisfied, then S, I, T, and J have a common fixed point provided that for a non-decreasing sequence with for all n and implies that and either
-
(a)
are ρ-compatible, S or I is ρ-continuous and are ρ-weakly compatible;
-
(b)
are ρ-compatible, T or J is ρ-continuous and are ρ-weakly compatible.
Moreover, the set of common fixed points of S, I, T, and J is well ordered if and only if S, I, T, and J have one and only one common fixed point.
Proof (a) Let . Construct sequences and in , such that and . This can be done because and . Since S is a ρ-dominating map and the pair is partially ρ-weakly increasing so . Also, S is a ρ-weak annihilator of J so . This implies that . Since T is a dominating map, . As is a pair of partially weakly increasing mappings, and . Also T is a weak annihilator of I, so we have . This implies that . Hence for all we have . Suppose that for every n. If not, then implies that , that is, for some n. Now from inequality (3.1) we have
and therefore . So and so on. Thus becomes a constant sequence and is a required common fixed point of given mappings. Assume that for each n. From (3.1), we obtain
Inductively, we have , which implies that
Using Lemma 2.7, we have
and
Employing the properties of the growth function, we obtain
which implies that
As , and so , and . This shows that is a Cauchy sequence in . There exists such that . That is, the sequence is norm convergent to . Since the -condition implies equivalence of norm and modular convergence, is modular convergent to . Therefore . Thus, we have and . Assume that I is continuous. Since are ρ-compatible, we have . As T is a ρ-dominating map, , that is, . Therefore we have
which on taking the limit as gives . That is, . As , so implies that . Since T is ρ-dominating, . Also, as implies that . So, we have
On taking the limit as , we obtain
Hence and . As , there exists a point such that . Suppose that . Since S is ρ-dominating, is partially ρ-weakly increasing, and S is a ρ-weak annihilator of J, so we have , that is, . Thus, we have
giving . Since are ρ-weakly compatible, . Thus h is a coincidence point of T and J. As S is a ρ-dominating map, . Now as implies that . Now from (3.1), we have
which, on taking the limit as , gives
or , as , so . Thus, . That is, h is a common fixed point of S, T, I, and J.
(b) Similarly the result follows when (b) holds.
Now suppose that the set of common fixed points of S, I, T, and J is well ordered. We claim that the common fixed point of S, I, T, and J is unique. Assume to the contrary that these maps have two common fixed points u and v, that is,
From inequality (3.1), we have
Thus
Hence uniqueness is proved. The converse is straightforward. □
Corollary 3.2 Let be a complete ordered modular function space and S, I, and J self-maps on X such that , and . Suppose that and are partially ρ-weakly increasing and the dominating map S is a weak annihilator of J and I. If for every two comparable elements
is satisfied, then S, I, and J have a common fixed point provided that for a non-decreasing sequence with for all n and implies that and either
-
(a)
are ρ-compatible, S or I is ρ-continuous and are ρ-weakly compatible;
-
(b)
are ρ-compatible, S or J is ρ-continuous and are ρ-weakly compatible.
Moreover, the set of the common fixed points of S, I, and J is well ordered if and only if S, I, and J have one and only one common fixed point.
Corollary 3.3 Let be a complete ordered modular function space and S, T, and J self-maps on X such that and . Suppose that and are partially ρ-weakly increasing, and the dominating maps S and T are weak annihilators of J. If for every two comparable elements
is satisfied, then S, T, and J have a common fixed point provided that for a non-decreasing sequence with for all n and implies that and either
-
(a)
are ρ-compatible, S or J is ρ-continuous and are ρ-weakly compatible;
-
(b)
are ρ-compatible, T or J is ρ-continuous and are ρ-weakly compatible.
Moreover, the set of common fixed points of S, T, and J is well ordered if and only if S, T, and J have one and only one common fixed point.
Corollary 3.4 Let be a complete ordered modular function space, S and J self-maps on X such that and . Suppose that is a partially ρ-weakly increasing and the dominating map S is a weak annihilator of J. If for every two comparable elements
is satisfied, then S and J have a common fixed point provided that for a non-decreasing sequence with for all n and it is implied that and either are ρ-compatible, S or J is ρ-continuous and are ρ-weakly compatible. Moreover, the set of common fixed points of S and J is well ordered if and only if S and J have one and only one common fixed point.
Theorem 3.5 Let be a complete ordered modular function space and S, T, I, and J continuous self-maps on . Suppose that and are ρ-compatible, and are ρ-partially weakly increasing with respect to J and I, respectively, and
holds for every for which If and Jg are comparable. Then the pairs and have a coincidence point . Moreover if Ih and Jh are comparable, then is a coincidence point of S, T, I, and J.
Proof Let be an arbitrary point in X. Construct the sequences and in X such that and . As is ρ-partially weakly increasing with respect to J, from we have
Since is ρ-partially weakly increasing with respect to I, from , we have
Hence , that is, . Following similar arguments to those given in Theorem 3.1, we obtain and is a Cauchy sequence. Now we show the existence of the coincidence point for the pairs and . To prove this, we proceed as follows: Since X is complete, there exists such that . That is, = = = = . By compatibility of and , we have
By continuity of S, T, I, and J, we have . Note that
which on taking the limit as implies that , that is, and . Similarly,
which on taking the limit as implies that , that is, and . Next we show that . Assume to the contrary , that is, . By the given assumption, we have
a contradiction. Hence , therefore, . Hence h is a coincidence point of S, I, T, and J. □
Corollary 3.6 Let be a complete ordered modular function space and S and T, continuous self-maps on , and are ρ-partially weakly increasing with respect to identity mapping on X, and
holds for every for which f and g are comparable. Then the pair has a common fixed point.
Corollary 3.7 Let be a complete ordered modular function space and S and J continuous self-maps on . Suppose that is ρ-compatible, S is ρ-partially weakly increasing with respect to J, and
holds for every for which Jf and Jg are comparable. Then the pair has a coincidence point h∈ X.
Example 3.8 Assume that , where for . For , define if and only if .
Let be defined as
Note that
That is,
Note that ρ-compatibility of follows immediately. Also S is ρ-weakly partially increasing with respect to J. Indeed, and for all . Therefore all conditions of Corollary 3.7 are satisfied. However, the pair has as a coincidence and a common fixed point.
4 Periodic point results
If S is a map which has a fixed point f, then f is also a fixed point of for every natural number n. However, the converse is false. If a map satisfies for each , where denotes a set of all fixed point of S, then it is said to have property P [23]. We shall say that S and T have property Q if . The set is called the orbit of T. The set is called the orbit of T and S.
As an application of our results in Section 2, we provide the following periodic point theorems.
Theorem 4.1 Let S be a non-decreasing self-map of a complete ordered modular function space , satisfying
for all , or (ii) with strict inequality, and for all , . If, , then S has property P provided that for any .
Proof We shall always assume that , since the statement for is trivial. Let . Then , so a non-decreasing characteristic of the mapping S implies that is a well-ordered subset of X. Suppose that S satisfies (i). Then
Now the right-hand side of the above inequality approaches zero as . Hence , and . Suppose that S satisfies (ii). If , then there is nothing to prove. Suppose, if possible, that . Then a repetition of the argument for case (i) leads to
a contradiction. Therefore, in all cases, . □
Theorem 4.2 Let be a complete ordered modular function space. Let the mappings S and T be as in Corollary 3.6. Then S and T have property Q provided that for every .
Proof From Corollary 3.6, S and T have a common fixed point in X. Let . Now,
and we have
Now the right-hand side of the above inequality approaches zero as . Hence , and . Now,
give , and . □
Remark Recently, Paknazar et al. [24] gave the existence of the solutions of the integral equations in modular function spaces. Hajji and Hanebaly [25] also applied their fixed point result to obtain the solution of perturbed integral equations in modular function spaces (see also [26]). Our results can also be employed to solve such integral equations in the framework of complete ordered modular function spaces.
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Acknowledgements
The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No. NRU56000508). The authors are also thankful to the reviewers for their suggestions and remarks, which improved the presentation of this paper.
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Abbas, M., Ali, S. & Kumam, P. Common fixed points in partially ordered modular function spaces. J Inequal Appl 2014, 78 (2014). https://doi.org/10.1186/1029-242X-2014-78
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DOI: https://doi.org/10.1186/1029-242X-2014-78