Modularly weighted four dimensional matrix summability with application to Korovkin type approximation theorem

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 present a fixed point theorem of Banach type in modular spaces. Also, we give some applications of this result to a nonlinear integral equation in Musielak-Orlicz space.

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.

In this research paper, we construct a new sequence of Riemann–Liouville type fractional ‐Bernstein–Kantorovich operators. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first order modulus of continuity of these operators. Further, we study Voronovskaja type theorem, quantitative Voronovskaya type theorem, Chebyshev–Grüss inequality and Grüss–Voronovskaya type theorem.

In this paper, using the power series method we obtain an abstract Korovkin type approximation theorem for a sequence of positive linear operators defined on modular spaces.

In this study, we define [N?,??]q - summability and statistical (N?,??) summability. We also establish some inclusion relation and some related results for this new summability methods. Further we apply Korovkin type approximation theorem through statistical (N?,??) summability and we apply the classical Bernstein operator to construct an example in support of our result. Furthermore, we present a rate of convergence which is uniform in Korovkin type theorem by statistical (N?,??) summability.

In this work, we introduce the Korovkin type approximation theorems on modular spaces via statistical A-summation process for double sequences of positive linear operators and we construct an example satisfying our new approximation theorem but does not satisfy the classical one.

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.

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.

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.

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.

In this paper, we investigated the notion of generalized weighted λ-statistical convergence connecting $$ \left( {\bar{N}_{\lambda } , p_{n} } \right)^{F} $$ -statistical summability of sequences of fuzzy numbers. Some inclusion relations and other related results for the new summability methods are also given. Furthermore, we give a Korovkin-type approximation theorem for fuzzy positive linear operators using the notion of generalized weighted $$ \left( {\bar{N}_{\lambda } , p_{n} } \right)^{F} $$ -statistical summability.

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.