Modular Function Spaces Research Articles

A Study of a Nonlinear Ordinary Differential Equation in Modular Function Spaces Endowed with a Graph

In this paper, we prove by means of a fixed-point theorem an existence result of the Cauchy problem associated to an ordinary differential equation in modular function spaces endowed with a reflexive convex digraph.

Multistep-type construction of fixed point for multivalued ρ-quasi-contractive-like maps in modular function spaces

PurposeIn this paper, the explicit multistep, explicit multistep-SP and implicit multistep iterative sequences are introduced in the context of modular function spaces and proven to converge to the fixed point of a multivalued map T such that PρT, an associate multivalued map, is a ρ-contractive-like mapping.Design/methodology/approachThe concepts of relative ρ-stability and weak ρ-stability are introduced, and conditions in which these multistep iterations are relatively ρ-stable, weakly ρ-stable and ρ-stable are established for the newly introduced strong ρ-quasi-contractive-like class of maps.FindingsNoor type, Ishikawa type and Mann type iterative sequences are deduced as corollaries in this study.Originality/valueThe results obtained in this work are complementary to those proved in normed and metric spaces in the literature.

Common Fixed-Point Theorems in Modular Function Spaces Endowed with Reflexive Digraph

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.

Fixed Point Theorems for a New Class of Mappings in Modular Spaces Endowed with a Graph

In this paper, we discuss a class of mappings more general than ρ-nonexpansive mapping defined on a modular space endowed with a graph. In our investigation, we prove the existence of fixed point results of these mappings. Then, we also introduce an iterative scheme for which proves the convergence to a fixed point of such mapping in a modular space with a graph.

Common Fixed Point Results for Three Multivalued \(\rho\)-Nonexpansive Mappings by Using Three Steps Iterative Scheme

The purpose of this research paper is to study the convergence and approximation of common fixed points for three multivalued \(\rho\)-nonexpansive mappings for three steps iterative scheme in modular function spaces. Further we construct a numerical example which illustrates our results.

Removability of zero modular capacity sets

We introduce a notion of modular with a corresponding modular function space in order to build a modular capacity theory. We give two different definitions of capacity, one of them of variational type, the other one through either the modular of the test functions, or the modular of their gradients. We study, in both cases, the removability of sets of zero capacity in fairly general abstract Sobolev spaces with zero boundary values. As a key tool, we establish a modular Poincare inequality. With the notion of modular function space in hands, we find a way to introduce a Banach function space, which allows to compare the zero capacity sets with respect to both notions. Thanks to this comparison, we characterize the compact sets of zero variational type capacity as removable sets. The paper is enriched with several examples, extending and unifying many results already known in literature in the settings of Musielak–Orlicz–Sobolev spaces, Lorentz–Sobolev spaces, variable exponent Sobolev spaces.

Some Fixed Point Theorems in Modular Function Spaces Endowed with a Graph

The aim of this paper is to give fixed point theorems for G-monotone ρ-nonexpansive mappings over ρ-compact or ρ-a.e. compact sets in modular function spaces endowed with a reflexive digraph not necessarily transitive. Examples are given to support our work.

Fixed points of relaxed (Ψ,φ)-weakly N-contraction mappings in modular spaces

The purpose of this paper is to study the existence and approximation of a common fixed point of a pair of mappings satisfying a relaxed (?,?)-weakly N-contractive condition and existence and approximation of a fixed point of a relaxed (?,?)-weakly N-contraction mapping in the setting of modular spaces. Our theorems improve and generalize the results in Mongkolkeha and Kumam [23] and ?zt?rk et. al [26]. To validate our results numerical examples are provided.

Fixed points results in modular vector spaces with applications to quantum operations and Markov operators

Recently, researchers are showing more interest on both modular vector spaces and modular function spaces. Looking at the number of results it is pertinent to say that, exploration in this direction especially in the area of fixed point theory and applications is still ongoing, many good results can still be unveiled. As a contribution from our part, we study some fixed point results in modular vector spaces associated with order relation. As an application, we were able to study the existence of fixed point(s) of both depolarizing quantum operation and Markov operators through modular functions/modular spaces. The awareness on the importance of quantum theory and Economics globally were the sole motivations of the application choices in our work. Our work complement the existing results. In fact, it adds to the number of application areas that modular vector/function spaces covered.

Fixed point theorems for asymptotically regular mappings in modular and metric spaces

Let $$L_\rho $$ be a modular function space where $$\rho $$ is a function modular satisfying the $$\Delta _2$$-type condition and is of s-convex type for $$s\in (0,1]$$ (see Definition 2 and Remark 3.2). Under this framework, the existence of fixed points for asymptotically regular mappings defined on some classes of subsets of $$L_\rho $$ is proved. As a consequence, some previous fixed point results for convex modulars (that is, modulars of 1-convex type) are extended. In particular, some variable Lebesgue spaces are modular function spaces where the modular is of s-convex type for some $$s\in (0,1)$$ but may fail to be convex. Furthermore, our arguments are applied to wider classes of domains than those that are just compact for the topology of the convergence locally in measure. This will be illustrated with several examples.

Fixed point theorems for new J-type mappings in modular spaces

In this paper, we introduce $rho$-altering $J$-type mappings in modular spaces. We prove some fixed point theorems for $rho$-altering and $rho$-altering $J$-type mappings in modular spaces. We also furnish illustrative examples to express relationship between these mappings. As a consequence, the results are applied to the existence of solution of an integral equation arising from an ODE system.

ON MODULATED TOPOLOGICAL VECTOR SPACES AND APPLICATIONS

We introduce a notion of modulated topological vector spaces, that generalises, among others, Banach and modular function spaces. As applications, we prove some results which extend Kirk’s and Browder’s fixed point theorems. The theory of modulated topological vector spaces provides a very minimalist framework, where powerful fixed point theorems are valid under a bare minimum of assumptions.

Cyclic $(\alpha ,\beta )$-Admissible Mappings in Modular Spaces and Applications to Integral Equations

The main concern of this study is to present a generalization of Banach's fixed point theorem in some classes of modular spaces, where the modular is convex and satisfying the $\Delta _{2}$-condition. In this work, the existence and uniqueness of fixed point for $(\alpha ,\beta )-(\psi ,\varphi )-$ contractive mapping and $\alpha -\beta -\psi -$weak rational contraction in modular spaces are proved. Some examples are supplied to support the usability of our results. As an application, the existence of a solution for an integral equation of Lipschitz type in a Musielak-Orlicz space is presented.

Approximation of fixed points of a multivalued ρ-quasi-nonexpansive mapping for newly defined hybrid iterative scheme

The purpose of this research paper is to define a hybrid iterative scheme named Picard Noor-type hybrid iterative scheme to approximate the fixed points of a multivalued ρ-quasinonexpansive mappings for newly defined iterative scheme in modular function spaces.

Approximating a common fixed point of a finite family of nonlinear mappings in modular function spaces

In this study, it is our purpose to investigate an algorithm for approximating a common fixed point of a finite family of $$\rho $$-quasi-nonexpansive mappings. In addition, we propose and analyze a scheme which estimates a common fixed point of a finite family of multivalued mappings in modular function spaces. As a consequence, we establish the $$\rho $$-convergence of the proposed algorithms under some mild conditions. In addition, some numerical examples which support our main results are presented. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

On Monotone Asymptotic Pointwise Nonexpansive Mappings in Modular Function Spaces

In this work, we investigate the existence of the fixed points of a monotone asymptotic pointwise nonexpansive mapping defined in a modular function space. Our result extends the fixed point result of Khamsi and Kozlowski.

Generalized almost convergence of double sequences in modular function spaces

‎‎‎‎This article deals with almost convergence of double sequences using a new generalization of fractional-order difference operator in modular spaces and application to the Korovkin-type approximation in the context of modular spaces for positive linear operators‎. ‎We then obtain several inclusion relations and present some examples‎, ‎include proper non-trivial extensions of the corresponding classical ones‎. ‎Further‎, ‎we extend our study to new modular forms of Korovkin-type approximation theorems‎. ‎Finally‎, ‎we give an example using bivariate Chlodowsky-Szasz-Kantorovich-Charlier-type operators and outline possible further extensions and improvements‎, ‎in order to illustrate the effectiveness of the proposed methods‎.

On Suzuki Mappings in Modular Spaces

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.

Approximation of fixed point of multivalued [formula omitted]-quasi-contractive mappings in modular function spaces

The purpose of this paper is to extend the recent results of Okeke et al. (2018) to the class of multivalued ρ-quasi-contractive mappings in modular function spaces. We approximate fixed points of this class of nonlinear multivalued mappings in modular function spaces. Moreover, we extend the concepts of T-stability, almost T-stability and summably almost T-stability to modular function spaces and give some results.

The Fibonacci–Mann iteration for monotone asymptotically pointwise nonexpansive mappings

In this work, we investigate the convergence of the Fibonacci–Mann iteration associated with a monotone asymptotic pointwise nonexpansive mapping defined in a modular function space. The first main result deals with the modular convergence of such iteration when the mapping is assumed to be compact. Relaxing the compactness assumption, we obtain a $$\rho $$ -a.e. convergence of the iteration. These two results are similar to the main conclusions of the original work of Schu.