Fixed points of multivalued contraction mappings in modular metric spaces
© Abdou and Khamsi; licensee Springer. 2014
Received: 1 September 2014
Accepted: 12 November 2014
Published: 22 December 2014
The purpose of this paper is to study the existence of fixed points for contractive-type multivalued maps in the setting of modular metric spaces. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we investigate the existence of fixed points of multivalued modular contractive mappings in modular metric spaces. Consequently, our results either generalize or improve fixed point results of Nadler (Pac. J. Math. 30:475-488, 1969) and Edelstein (Proc. Am. Math. Soc. 12:7-10, 1961).
MSC:47H09, 46B20, 47H10, 47E10.
The aim of this paper is to give an outline of a fixed point theory for multivalued Lipschitzian mappings defined on some subsets of modular metric spaces. Modular metric spaces were introduced in [1, 2]. The way we approached the concept of modular metric spaces is different. Indeed we look at these spaces as the nonlinear version of the classical modular spaces as introduced by Nakano  on vector spaces and modular function spaces introduced by Musielak  and Orlicz . In  the authors have defined and investigated the fixed point property in the framework of modular metric space and introduced the analog of the Banach contraction principle theorem in modular metric space.
As is well known, a fixed point theorem for multivalued contraction mappings was established by Nadler . In 1961 Edelstein  has generalized the Banach contraction principle to mappings satisfying a less restrictive Lipschitz inequality such as local contraction. This result has been generalized to a multivalued version by Nadler . On the other hand Mizoguchi and Takahashi  have improved Reich’s result  and proved the existence of fixed points for multivalued maps in the case when values of mappings are closed bounded instead of compact.
In this paper we define the Hausdorff modular metric and obtain a multivalued version of the result [, Theorem 3.1] in modular metric spaces. We also extend the results of Nadler , Mizoguchi and Takahashi  to modular metric spaces. The linear version of some of our results may be found in the work of Kutbi and Latif .
For more on metric fixed point theory, the reader may consult the book .
2 Basic definitions and properties
for all and .
if and only if , for all ;
, for all , and ;
, for all and .
are said to be modular spaces (around ).
for any . These distances will be called Luxemburg distances.
for all , and . Then ω is a modular metric on . Moreover, the distance is exactly the distance generated by the Luxemburg norm on .
The sequence in is said to be ω-convergent to if and only if , as . x will be called the ω-limit of .
The sequence in is said to be ω-Cauchy if , as .
A subset M of is said to be ω-closed if the ω-limit of a ω-convergent sequence of M always belongs to M.
A subset M of is said to be ω-complete if any ω-Cauchy sequence in M is a ω-convergent sequence and its ω-limit is in M.
- (5)A subset M of is said to be ω-bounded if we have
A subset M of is said to be ω-compact if for any in M there exists a subset sequence and such that .
- (7)ω is said to satisfy the Fatou property if and only if for any sequence in ω-convergent to x, we have
for any .
for any , , with .
Note that if ω satisfies -type condition, then ω satisfies the -condition. The above definition will allow us to introduce the growth function in the modular metric spaces as was done in the linear case.
for any .
The following properties were proved in the linear case in .
, for any ,
Ω is a strictly increasing function, and ,
, for any ,
, where is the function inverse of Ω,
- (5)for any , , we have
The inequality in (5) follows from the definition of the distance . □
The following technical lemma will be useful later on in this work.
where K is an arbitrary nonzero constant and . Then is Cauchy for both ω and .
which implies . Classical analysis on metric spaces implies that is convergent which implies that is Cauchy for . Since ω satisfies the -type condition, is Cauchy for ω. □
Note that this lemma is crucial since the main assumption on will not be enough to imply that is ω-Cauchy since ω fails the triangle inequality.
3 Multivalued mappings in modular metric spaces
- (iii)the Hausdorff modular metric is defined on by
A point is called a fixed point of T whenever . The set of fixed points of T will be denoted by .
but T is not Lipschitzian with respect to with constant 1.
- (i)an ω-contraction if there exists a constant such that for any ,
- (ii)a -ω-uniformly locally contraction if there exists a constant such that for any ,
Before we state our results, we will need the following technical lemmas  in the setting of modular metric spaces.
Lemma 3.2 Let be a modular metric space. Assume that ω satisfies -condition. Let M be a nonempty subset of . Let be a sequence of sets in , and suppose where . Then if and , it follows that .
which implies . Since ω satisfies the -condition, we have . Since is ω-closed, we have . □
4 The main results
The statement of Nadler’s fixed point result  in modular metric spaces is as follows.
Theorem 4.1 Let be a modular metric space. Assume that ω is a convex regular modular which satisfies the -condition. Let M be a nonempty ω-complete subset of . Let be an ω-contraction map. Then T has a fixed point.
we conclude that . Since , Lemma 3.2 implies , i.e. is fixed point of T. This completes the proof of Theorem 4.1. □
Edelstein  has extended the classical fixed point theorem for a contraction to the case when X is a complete ε-chainable metric space and the mapping is an -uniformly locally contraction. This result was extended by Nadler  to multivalued mappings. Here we investigate Nadler’s result in modular metric spaces. First let us introduce the ε-chainable concept in modular metric spaces. Our definition is slightly different from the one used in the classical metric spaces since the modular functions fail in general the triangle inequality.
Definition 4.1 Let be a modular metric space. A nonempty subset is said to be finitely ε-chainable (where is fixed) if and only if there exists such that for any there is an -chain from a to b (that is, a finite set of points such that , , and , for all ).
We have the following result.
Theorem 4.2 Let be a modular metric space. Assume that ω is a convex regular modular which satisfies the -type condition and the Fatou property. Let M be a nonempty ω-complete and ω-bounded subset of , which is finitely ε-chainable, for some fixed . Let be an -ω-uniformly locally contraction map. Then T has a fixed point in M.
for any , we conclude that ω-converges to z. Since is ω-closed, we get . □
The first author was supported by the Deanship of Scientific Research (DSR), King Abdoulaziz University. The author, therefore, acknowledge with thanks DSR.
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