Modular Function Spaces Research Articles

Convergence of Forked Sequence to a Fixed Point in Modular Function Spaces

Convergence of Forked Sequence to a Fixed Point in Modular Function Spaces

Some new results on generalized Hyers-Ulam stability in modular function spaces

Some new results on generalized Hyers-Ulam stability in modular function spaces

Ishikawa iteration convergence to fixed points of a multi-valued mapping in modular function spaces

Ishikawa iteration convergence to fixed points of a multi-valued mapping in modular function spaces

Examining Nonlinear Fredholm Equations in Lebesgue Spaces with Variable Exponents

We investigate the existence of solutions for the Fredholm integral equation Φ(ϑ)=G(ϑ,Φ(ϑ))+∫01F(ϑ,ζ,Φ(ζ))dζ, for ϑ∈[0,1], in the setting of the modular function spaces Lρ. We also derive an application of this research within the framework of variable exponent Lebesgue spaces Lp(·) subject to specific conditions imposed on the exponent function p(·) and the functions F and G.

Banach Contraction Principle-Type Results for Some Enriched Mappings in Modular Function Spaces

The idea of enriched mappings in normed spaces is relatively a newer idea. In this paper, we initiate the study of enriched mappings in modular function spaces. We first introduce the concepts of enriched ρ-contractions and enriched ρ-Kannan mappings in modular function spaces. We then establish some Banach Contraction Principle type theorems for the existence of fixed points of such mappings in this setting. Our results for enriched ρ-contractions are generalizations of the corresponding results from Banach spaces to modular function spaces and those from contractions to enriched ρ-contractions. We make a first ever attempt to prove existence results for enriched ρ-Kannan mappings and deduce the result for ρ-Kannan mappings. Note that even ρ-Kannan mappings in modular function spaces have not been considered yet. We validate our main results by examples.

A New Iterative Sequence of (λ, ρ)-Firmly Nonexpansive Multi-Valued Mappings in Modular Function Spaces with Applications

A New Iterative Sequence of (λ, ρ)-Firmly Nonexpansive Multi-Valued Mappings in Modular Function Spaces with Applications

Generalization of ρ-Attractive Elements in Modular Function Spaces

In this paper, we introduce two new classes of mappings called ρ-α- and ρ-α-k-non-spreading mappings to broaden the idea of ρ- attractive elements in modular function spaces (MFS). In the MFS that are put up, we also demonstrate several approximation results and existence results. Illustration examples are provided to clarify the results.

CONTRACTIVE SEMIGROUPS IN TOPOLOGICAL VECTOR SPACES, ON THE 100TH ANNIVERSARY OF STEFAN BANACH’S CONTRACTION PRINCIPLE

AbstractCelebrating 100 years of the Banach contraction principle, we prove some fixed point theorems having all ingredients of the principle, but dealing with common fixed points of a contractive semigroup of nonlinear mappings acting in a modulated topological vector space. This research follows the ideas of the author’s recent papers [‘On modulated topological vector spaces and applications’, Bull. Aust. Math. Soc.101 (2020), 325–332, and ‘Normal structure in modulated topological vector spaces’, Comment. Math.60 (2020), 1–11]. Modulated topological vector spaces generalise, among others, Banach spaces and modular function spaces. The interest in modulars reflects the fact that the notions of ‘norm like’ but ‘noneuclidean’ (and not even necessarily convex) constructs to measure a level of proximity between complex objects are frequently used in science and technology. To prove our fixed point results in this setting, we introduce a new concept of Opial sets using analogies with the norm-weak and modular versions of the Opial property. As an example, the results of this work can be applied to spaces like $L^p$ for $p> 0 $ , variable Lebesgue spaces $L^{p(\cdot )}$ where $1 \leq p(t) < + \infty $ , Orlicz and Musielak–Orlicz spaces.

Iterative approximation of fixed points of generalized $ \alpha _{m} $-nonexpansive mappings in modular spaces

<abstract><p>Our aim of this work is to approximate the fixed points of generalized $ \alpha _{m} $-nonexpansive mappings employing $ AA $-iterative scheme in the structure of modular spaces. The results of fixed points for generalized $ \alpha _{m} $-nonexpansive mappings is proven in this context. Moreover, the stability of the scheme and data dependence results are given for $ m $-contraction mappings. In order to demonstrate that the $ AA $-iterative scheme converges faster than some other schemes for generalized $ \alpha_{m} $-nonexpansive mappings, numerical examples are shown at the end.</p></abstract>

Fixed points of monotone asymptotic pointwise ρ-nonexpansive semigroups in modular spaces

In this paper, we prove common fixed point results for monotone asymptotic pointwise ?-nonexpansive semigroups in modular function spaces. Thus, utilizing the monotonicity of semigroups, our findings generalize and extend some prominent recent results of the literature in the setting of modular function spaces.

On a new nonlinear convex structure

<abstract><p>In this work we start from near vector spaces, which we endow with some additional properties that allow convex analysis. The seminormed structure used here will also be improved by adding properties such as the null condition and null equality, thus resulting in a new type of space, which is still weaker than the conventional Banach structures: pre-convex regular near-Banach space. On the newly defined structure, we introduce the concept of uniform convexity and analyze several resulting properties. The major outcomes prove a remarkable resemblance to the classical properties resulting from uniform convexity on hyperbolic metric spaces or modular function spaces, including the famous Browder-Göhde fixed point theorem.</p></abstract>

Fixed point theorems and equivalence results for classes of multivalued mappings in modular function spaces

Abstract We contribute to the development of equivalence of fixed point iterative sequences for multivalued mappings in modular function spaces, by proving the equivalence of convergence of implicit Mann, implicit Ishikawa, implicit Noor, implicit multistep iterative sequences for multivalued ρ− quasi-contractive-like mapppings in modular function spaces. An example is provided to support the applicability of the results. This work is complementary to equivalence results on normed and metric spaces in the literature.

Fixed points of $\rho$-nonexpansive mappings using MP iterative process

This research article introduces a new iterative process called MP iteration and prove some convergence and approximation results for the fixed points of $\rho$-nonexpansive mappings in modular function spaces. To demonstrate that MP iterative process converges faster than some well-known existing iterative processes for $\rho$-nonexpansive mappings, we constructed some numerical examples. In the end, the concept of summably almost T-stability for MP iterative process is discussed.

SOME COINCIDENCE POINT THEOREMS IN MODULAR 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.

Some Fixed Point Results for Commuting Families of Mappings in Modular Spaces

Assume that ρ is a convex modular satisfying the -type condition and the modular space is either α-uniformly ρ-noncompact convex or it satisfies the strong Opial condition. We prove that the fixed point set of a commuting family of asymptotically non-expansive mappings defined from a ρ-convex weakly compact set C into C is a non-empty nonexpansive retract of C. We show that our results about existence of fixed point for commuting families of nonexpansive mappings also work if we replace the weak topology by some other topologies weaker than the ρ-topology. In particular, we can obtain fixed point results when C is ρ-a.e. compact in a modular function space.

Convergence Results for Total Asymptotically Nonexpansive Monotone Mappings in Modular Function Spaces

In this article, we consider an extensive class of monotone nonexpansive mappings. We use S -iteration to approximate the fixed point for monotone total asymptotically nonexpansive mappings in the settings of modular function space.

Measures of noncompactness in modular spaces and fixed point theorems for multivalued nonexpansive mappings

This paper is devoted to state some fixed point results for multivalued mappings in modular vector spaces. For this purpose, we study the uniform noncompact convexity, a geometric property for modular spaces which is similar to nearly uniform convexity in the Banach spaces setting. Using this property, we state several new fixed point theorems for multivalued nonexpansive mappings in modular spaces.

$rho$-Attractive Elements in Modular Function Spaces

In this paper, we introduce the notion of ρ-attractive elements in modular function spaces. A new class of mappings called ρ-k-nonspreading mappings is also introduced. Making a good use of the two notions, we first prove existence results and then some approximation results in the setup of modular function spaces. An example is presented to support the results proved herein.

A survey on recent results in Korovkin’s approximation theory in modular spaces

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.

Approximation of Fixed Points of (λ, ρ)-Quasi Firmly Nonexpansive Mappings

The purpose of this paper is to introduce (λ, ρ)-quasi firmly nonexpansive mappings in context of modular function spaces and to prove some basic properties and approximation results for fixed point for these mappings. Further we discuss the concept of summably almost T-stability and also some examples are provided to support our results.