Common fixed points in partially ordered modular function spaces
© Abbas et al.; licensee Springer. 2014
Received: 16 June 2013
Accepted: 31 January 2014
Published: 17 February 2014
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.
Keywordsfixed point ordered modular function space
Study of modular spaces was initiated by Nakano  in connection with the theory of order spaces which was further generalized by Musielak and Orlicz . 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 , and then by Nieto and Rodriguez-Lopez . They applied their results to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions (see also ). The study of this theory in the context of modular function spaces was initiated by Khamsi et al.  (see also  and ). Kuaket and Kumam  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  obtained some fixed point theorems for non-expansive mappings in arbitrary modular spaces. Recently, Kutabi and Latif  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.
Some basic facts and notations about modular spaces are recalled from .
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.
is called a modular space. Generally, the modular ρ is not subadditive and therefore does not behave as a norm or a distance.
defines a norm on the modular space and is called the Luxemburg norm.
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 .
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 .
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.
are ρ-compatible, S or I is ρ-continuous and are ρ-weakly compatible;
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.
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.
Hence uniqueness is proved. The converse is straightforward. □
are ρ-compatible, S or I is ρ-continuous and are ρ-weakly compatible;
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.
are ρ-compatible, S or J is ρ-continuous and are ρ-weakly compatible;
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.
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.
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.
a contradiction. Hence , therefore, . Hence h is a coincidence point of S, I, T, and J. □
holds for every for which f and g are comparable. Then the pair has a common fixed point.
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 .
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 . 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.
for all , or (ii) with strict inequality, and for all , . If, , then S has property P provided that for any .
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 .
give , and . □
Remark Recently, Paknazar et al.  gave the existence of the solutions of the integral equations in modular function spaces. Hajji and Hanebaly  also applied their fixed point result to obtain the solution of perturbed integral equations in modular function spaces (see also ). Our results can also be employed to solve such integral equations in the framework of complete ordered modular function spaces.
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.
- Nakano H: Modulared Semi-Ordered Spaces. Maruzen, Tokyo; 1950.Google Scholar
- Musielak J, Orlicz W: On modular spaces. Stud. Math. 1959, 18: 591–597.Google Scholar
- Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4MathSciNetView ArticleGoogle Scholar
- Nieto JJ, Rodriguez-Lopez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5MathSciNetView ArticleGoogle Scholar
- Nieto JJ, Rodriguez-Lopez R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. 2007,23(12):2205–2212. 10.1007/s10114-005-0769-0MathSciNetView ArticleGoogle Scholar
- Khamsi MA, Kozolowski WK, Reich S: Fixed point theory in modular function spaces. Nonlinear Anal. 1990, 14: 935–953. 10.1016/0362-546X(90)90111-SMathSciNetView ArticleGoogle Scholar
- Benavides TD, Khamsi MA, Samadi S: Asymptotically regular mappings in modular function spaces. Sci. Math. Jpn. 2001, 53: 295–304.MathSciNetGoogle Scholar
- Khamsi MA: A convexity property in modular function spaces. Math. Jpn. 1996, 44: 269–279.MathSciNetGoogle Scholar
- Kuaket K, Kumam P: Fixed points of asymptotic pointwise contractions in modular spaces. Appl. Math. Lett. 2011, 24: 1795–1798. 10.1016/j.aml.2011.04.035MathSciNetView ArticleGoogle Scholar
- Mongkolkeha C, Kumam P: Fixed point and common fixed point theorems for generalized weak contraction mappings of integral type in modular spaces. Int. J. Math. Math. Sci. 2011., 2011: Article ID 705943Google Scholar
- Mongkolkeha C, Kumam P: Common fixed points for generalized weak contraction mappings in modular spaces. Sci. Math. Jpn. 2012, e-2012: 117–127.Google Scholar
- Mongkolkeha C, Kumam P: Some fixed point results for generalized weak contraction mappings in modular spaces. Int. J. Anal. 2013., 2013: Article ID 247378 10.1155/2013/247378Google Scholar
- Kumam P: Fixed point theorem for non-expansive mappings in modular spaces. Arch. Math. 2004, 40: 345–353.MathSciNetGoogle Scholar
- Kutbi MA, Latif A: Fixed points of multivalued mappings in modular function spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 786357Google Scholar
- Abbas M, Khamsi MA, Khan AR: Common fixed point and invariant approximation in hyperbolic ordered metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 25Google Scholar
- Abbas M, Khan AR, Nemeth SZ: Complementarity problems via common fixed points in vector lattices. Fixed Point Theory Appl. 2012., 2012: Article ID 60Google Scholar
- Abbas M, Nazir T, Radenovic S: Common fixed points of four maps in partially ordered metric spaces. Appl. Math. Lett. 2011, 24: 1520–1526. 10.1016/j.aml.2011.03.038MathSciNetView ArticleGoogle Scholar
- Altun I, Damjanovic B, Djoric D: Fixed point and common fixed point theorems on ordered cone metric spaces. Appl. Math. Lett. 2010, 23: 310–316. 10.1016/j.aml.2009.09.016MathSciNetView ArticleGoogle Scholar
- Esmaily J, Vaezpour SM, Rhoades BE: Coincidence point theorem for generalized weakly contractions in ordered metric spaces. Appl. Math. Comput. 2012, 219: 1536–1548. 10.1016/j.amc.2012.07.054MathSciNetView ArticleGoogle Scholar
- Harandi AA, Emami H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 2010, 72: 2238–2242. 10.1016/j.na.2009.10.023MathSciNetView ArticleGoogle Scholar
- Kozlowski WM: Modular Function Spaces. Dekker, New York; 1988.Google Scholar
- Khamsi MA: Fixed point theory in modular function spaces. In Recent Advances on Metric Fixed Point Theory. Universidad de Sevilla, Sevilla; 1996:31–58.Google Scholar
- Jeong GS, Rhoades BE:Maps for which . Fixed Point Theory Appl. 2005, 6: 87–131.Google Scholar
- Paknazar M, Eshaghi M, Cho YJ, Vaezpour SM: A Pata-type fixed point theorem in modular spaces with application. Fixed Point Theory Appl. 2013., 2013: Article ID 239Google Scholar
- Hajji A, Hanebaly E: Perturbed integral equations in modular function spaces. Electron. J. Qual. Theory Differ. Equ. 2003., 2003: Article ID 20Google Scholar
- Taleb AA, Hanebaly E: A fixed point theorem and its application to integral equations in modular function spaces. Proc. Am. Math. Soc. 1999, 127: 2335–2342. 10.1090/S0002-9939-99-04779-6View ArticleGoogle Scholar
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