An Analysis on Some New Fixed Point Theorem in Generalized Modular Metric Spaces

The existence and uniqueness of fixed point theorems of singlevalued maps have been a subject of great interest since Banach (1922) proved the well-known Banach contraction principle in 1922. This result is very interesting in its own right due to its applications like in computer science, physics, image processing engineering, economics, and telecommunication. Very early on, many mathematicians tried to find a multivalued version. Nadler (1950) was the one who successfully gave this extension. His result found many applications to differential inclusions, control theory, convex optimization, and economics. This is the reason why many authors have studied Nadlers fixed point result. Reich‘s generalization of Nadler‘s fixed point result in S. Reich (1972) states that a mapping where is the family of all nonempty compact subsets of X, has a fixed point if it satisfies for all with where such that for every In fact, Reich (1974) asked whether this result holds when T takes values in CB(X) instead of where is the family of all nonempty closed and bounded subsets of X. In 1989, Mizoguchi and Takahashi pH] gave a partial answer to Reich‘s question. Recently, Chistyakov (2010) has introduced the notion of modular metric spaces. This concept is a generalization of the classical modulars over linear spaces like Orlicz spaces. Moreover, the modular type conditions are natural and easily verified then their metric or norm equivalent. In A. A. N. Abdou, M. A. Khamsi (2013,2014), the authors initiated the fixed point theory in modular metric spaces. This work extends on these results where we discuss the definition of the Reich contraction single valued and multivalued mappings defined in modular metric spaces. In particular, we investigate the conditions under which such mappings have a fixed point. The study of common fixed points of mappings satisfying certain contractive conditions has been at the center of vigorous research activity. In 1976, Jungck proved a common fixed point theorem for commuting maps such that one of them is continuous. In 1982, Sessa generalized the concept of commuting maps to weakly commutating pair of self-mappings. In 1986, Jungck generalized this idea, first to compatible mappings and then in 1996 to weakly compatible mappings. Using the weakly compatibility, several authors established coincidence points results for various classes of mappings on metric spaces with Fatou property. In 2011, Haghi et al. showed that some coincidence point and common fixed point generalizations in fixed point theory are not real generalizations as they could easily be obtained from the corresponding fixed point theorems. In 1990, the fixed point theory in modular function spaces was initiated by Khamsi, Kozlowski, and Reich (1990).Modular function spaces are a special case of the theory of modular vector spaces introduced by Nakano (1950). Modular metric spaces were introduced in Chistyakov, V.V.(2010). Fixed point theory in modular metric spaces was studied by Abdou and Khamsi (2013). Their approach was fundamentally different from the one studied in Chistyakov, V.V.(2010). In this paper, we Generalizations of standard metric spaces are interesting because they allow for some deep understanding of the classical results obtained in metric spaces. One has always to be careful when coming up with a new generalization. For example, if we relax the triangle inequality, some of the classical known facts in metric spaces may become impossible to obtain. This is the case with the generalized metric distance introduced by Jleli and Samet in Jleli, M., Samet, B.(2015). The authors showed that this generalization encompasses metric spaces, b-metric spaces, dislocated metric spaces, and modular vector spaces. In this paper, considering both a modular metric space and a generalized metric space in the sense of Jleli and Samet (2015), we introduce a new concept of generalized modular metric space. Definition : Let X be an abstract set. A function is said to be a regular generalized modular metric (GMM) on X if it satisfies the following three axioms: for some then x-y for all for all and (GMM3) There exists C > 0 such that, if with 0 for some then The pair (X,D) is said to be a generalized modular metric space (GMMS). It is easy to check that if there exist x,yX such that there exists with for some and then we must have In fact, throughout this work, we assume Let D be a GMM on X. Fix The sets are called generalized modular sets. Next, we give some examples that inspired our definition of a GMMS. Example : (Modular vector spaces (MVS) [13]) Let A be a linear vector space over the field R. A function is called regular modular if the following hold: 1. if and only if x= 0, 2. if for any . Let be regular modular defined on a vector space X. The set is called a MVS. Let be a sequence in and . If then is said to - converge to x. is said to satisfy the -condition if there exists such that for any . Moreover, is said to satisfy the Fatou property(FP) if whenever - converges to x for any . Next, we show that a MVS may be embedded with a GMM structure. Indeed, let be a MVS. Define by Then the following hold: 1. If for some and any then x=y, 2. for any and 3. If satisfies the FP, then for any and {*„) such that -converges to we have which implies for any Therefore, (X,D) satisfies all the properties of Definition as claimed. Note that the constant C which appears in the property (GMMs) is equal to 1 provided the FP is satisfied by .

In this section we discuss the existence of fixed points for mappings which are nonexpansive or contractive in the modular sense. Certainly, one can also consider mappings which are contractive with respect to the F-norm induced by the modular. It is worth to mention that, generally speaking, there is

nonexpansiveness. Once again we would like to emphasize our philosophy that all the results expressed in terms of modulars are more convenient in the sense that their assumptions are much easier to verify. Definition . Let C be a subset of a modular spaceand let be an arbitrary mapping. 1. T is a -contraction if there exists such that for all 2. T is said to be -nonexpansive if for all 3. is said to be a fixed point of T if T(f) = f. The fixed point set of T will be denoted Fix(T). C will be said to have the fixed point property if every -nonexpansive selfmap defined on C has a fixed point. An analog to banach contraction principle, can be stated as follows. Theorem . Let C be -complete -bounded subset of and be a -strict contraction. Then T has a unique fixed point ". Moreover is

Recall that a subset D of is said to be p-complete if every -Cauchy sequence from D is convergent in D. We may relax the assumption regarding the boundedness of C and assume there exists a bounded orbit instead. In this case, the uniqueness of the fixed point is dropped and replaced by if f and g are two fixed points of T such that , then

The fixed point theory is used in many different fields of mathematics such as topology, analysis, nonlinear analysis and operator theory. Moreover, it can be applied to different disciplines such as statistics, economy, engineering, etc. In literature, studies of fixed point theory cover a wide range. The most basic and famous fixed point theorem is Banach fixed point theorem which was introduced in 1922. It guarantees the existence and uniqueness of solution of a functional equation. Besides Banach, many different fixed point theorems were introduced. Chistyakov introduced the concept of modular metric spaces, which have a physical interpretation, via F-modulars in 2008 and he further developed the theory of these spaces in 2010. Then many authors made various studies on this structures. In this paper, we first give a new fixed point theorem which is main theorem of our study. After that, by using this theorem, we express some interesting results. Moreover, we characterize completeness in modular metric spaces via this theorem. Finally, we use our main theorem to show the existence of solution for a specific problem in dynamic programming. Here, we express a series of definitions of some basic concepts related to modular metric spaces. Definition 1. Let X be a linear space on . If a functional satisfies the following conditions, we call that is a modular on X: 1. 2. If and for all numbers , then x = 0: 3. for all 4. for all with and Let and Generally, a function is denoted as for all and Definition 2. Let A function which satisfies the following conditions for all is called a metric modular on X: (ml) for all (m2) for all (m3) for all If from properties of metric modular, we obtain that for all and such that are said to be modular spaces. It is known that ifis a metric modular on a nonempty set X, then the modular space can be equipped with a metric, generated by and given by for all The pair is called a modular metric space.

The following definition is useful to set new fixed point theory on GMMS. Definition 1 Let be a GMMS. 1. The sequence in XD is said to be D- convergent to if and only if as for some

Cauchy if as for some 3. A subset C of is said to be D-closed if for any from C which D-converges to 4. A subset C of XD is said to be D-complete if for any xn) D-Cauchy sequence in C such that for some there exists a point such that 5. A subset C of XD is said to be D-bounded if, for some we have In general, if for some then we may not have for all Therefore, as it is done in modular function spaces, we will say that D satisfies -condition if and only if for some implies for all Another question that comes into this setting is the concept of D-limit and its uniqueness.

Definition 1. Let be a modular metric space and M be a nonempty subset of The map is called a Reich contraction if there exists which satisfies for any such that for any distinct elements we have A point a is said to be a fixed point of T if Theorem 1. Let be a modular metric space where is a convex regular modular. Assume that satisfies the -type condition. Let C be an -complete nonempty subset of Let be a Reich contraction mapping. Then, T has a unique fixed point and -converges to x for any Proof. The definition of Reich contraction implies the existence of which satisfies for any such that for any different It is clear that T has at most one fixed point since is regular. Next we investigate the existence of a fixed point. Fix If is a fixed point of T for some then we have nothing to prove. Otherwise assume that for any Since We conclude that for any Hence the sequence of positive numbers is convergent. Set Since there exist and such that for any Then, we have for any . Lemma 2.6 implies that is -Cauchy. Using the -completeness of C, we conclude that -converges to some

we have For any Since -converges to x, we deduce that The regularity of implies that The uniqueness of the fixed point of T will imply that -converges to x for any

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Assistant Professor in Mathematics, CRSU, Jind, Haryana