A fixed point theorem and its application to integral equations in modular function spaces

Because of its many diverse applications, fixed point theory has been a flourishing area of mathematical research for decades. Banach’s formulation of the contraction mapping principle in the early twentieth century signaled the advent of an intense interest in the metric related aspects of the theory. The metric fixed point theory in modular function spaces is closely related to the metric theory, in that it provides modular equivalents of norm and metric concepts. Modular spaces are extensions of the classical Lebesgue and Orlicz spaces, and in many instances, conditions cast in this framework are more natural and more easily verified than their metric analogs. In this chapter, we study the existence and construction of fixed points for monotone nonexpansive mappings acting in modular functions spaces equipped with a partial order or a graph structure.

In this paper we give a survey about recent versions of Korovkin-type theorems for modular function spaces, a class which includes $L^p$, Orlicz, Musielak-Orlicz spaces and many others. We consider various kinds of modular convergence, using certain summability processes, like triangular matrix statistical convergence, and filter convergence (which are generalizations of the statistical convergence). Finally, wwe consider an abstract axiomatic convergence which includes the previous ones and even almost convergence, which is not generated by any filter, as we show by an example.

We generalize the notion of best proximity points for cyclic contraction maps in modular function spaces about Kannan maps. We have found sufficient conditions for the existence and uniqueness of best proximity points and fixed points for cyclic Kannan maps in modular function spaces. As corollaries we get sufficient conditions for the existence and uniqueness of best proximity points and fixed points for cyclic maps in Orlicz spaces, endowed with an Orlicz function modular. We present an application of the results for solving integral equations.

This chapter introduces general notions related to the geometry of modular function spaces. We define the modular version of uniform convexity and property (R) which will equip us with powerful tools for proving the fixed point theorems in modular function spaces. The geometrical theory also provides a set of powerful techniques for proving existence of common fixed points for commutative families of mappings acting in modular function spaces, and for investigating the topological properties of the set of common fixed points.

The purpose of this work is to extend the Knaster–Tarski fixed-point theorem to the wider field of reflexive digraph. We give also a DeMarr-type common fixed-point theorem in this context. We then explore some interesting applications of the obtained results in modular function spaces.

This paper is devoted to the study of Ćirić-type non-unique fixed point results in modular metric spaces. We obtain various theorems about a fixed point and periodic points for a self-map on modular spaces which are not necessarily continuous and satisfy certain contractive conditions. Our results extend the results of Ćirić, Pachpatte, and Achari in modular metric spaces.

In this paper, we prove three common fixed point theorems for weak contraction map- pings of integral type in modular spaces. In the first theorem we prove a common fixed point of ρ−compatible mappings satisfying a (φ − Ψ)−weak contraction. The second theorem is another version of the first theorem. In the third theorem we study a common fixed point of ρ−compatible mappings in modular spaces involving altering distances of integral type. The results extended several similar results in metric and Banach spaces.

In metric spaces, there are quasi-contraction mappings and Suzuki-contraction mapping, that are generalization of contraction mappings. In this article, we give coincidence point theorems of quasi-contraction mappings and Suzuki-contraction mappings in modular spaces, that are generalizations of fixed point theorem of quasi-contractraction mappings and Suzuki-contraction theorem in modular spaces.

The existence of fixed points of single-valued mappings in modular function spaces has been studied by many authors. The approximation of fixed points in such spaces via convergence of an iterative process for single-valued mappings has also been attempted very recently by Dehaish and Kozlowski (Fixed Point Theory Appl. 2012:118, 2012). In this paper, we initiate the study of approximating fixed points by the convergence of a Mann iterative process applied on multivalued ρ-nonexpansive mappings in modular function spaces. Our results also generalize the corresponding results of (Dehaish and Kozlowski in Fixed Point Theory Appl. 2012:118, 2012) to the case of multivalued mappings. MSC:47H09, 47H10, 54C60.

In this paper, we establish the existence of a common fixed point of almost generalized contractions on modular spaces. As an application, we present some fixed and common fixed point results for such mappings on modular spaces endowed with a graph. The existence of fixed and common points of mappings satisfying generalized contractive conditions of integral type is also obtained in such spaces. Some examples are presented to support the results obtained herein. Our results generalize and extend various comparable results in the existing literature.

Inspired by Suzuki’s generalization for nonexpansive mappings, we define the ( C ) -property on modular spaces, and provide conditions concerning the fixed points of newly introduced class of mappings in this new framework. In addition, Kirk’s Lemma is extended to modular spaces. The main outcomes extend the classical results on Banach spaces. The major contribution consists of providing inspired arguments to compensate the absence of subadditivity in the case of modulars. The results herein are supported by illustrative examples.

We consider ordinary differential equations \(u'(t)+(I-T)u(t)=0\), where an unknown function takes its values in a given modular function space being a generalization of Musielak-Orlicz spaces, and \(T\) is nonlinear mapping which is nonexpansive in the modular sense. We demonstrate that under certain natural assumptions the Cauchy problem related to this equation can be solved. We also show a process for the construction of such a solution. This result is then linked to the recent results of the fixed point theory in modular function spaces.

In this paper, we introduce and study the concept of one-local retract in modular function spaces. In particular, we prove that any commutative family of ρ-nonexpansive mappings defined on a nonempty, ρ-closed and ρ-bounded subset of a modular function space has a common fixed point provided its convexity structure of admissible subsets is compact and normal.

In this paper, we present a five-step iterative scheme, at the beginning, it seems complicated and difficult to implement, but in fact it’s not. This scheme is constructed for - firmly nonexpansive mappings in modular function spaces. Two different -convergences result has been proved for double schemes under consideration. In our study there is a comparison between these cases through answering the question "which one is faster?” Finally, numerical examples are given by using MATLAB software program.

This article introduces a new type of C*-algebra valued modular G-metric spaces that is more general than both C*-algebra valued modular metric spaces and modular G-metric spaces. Some properties are also discussed with examples. A few common fixed point results in C*-algebra valued modular G-metric spaces are discussed using the “C*-class function”, along with some suitable examples to validate the results. Ulam–Hyers stability is used to check the stability of some fixed point results. As applications, the existence and uniqueness of solutions for a particular problem in dynamical programming and a system of nonlinear integral equations are provided.