Search Results (10)

The problem of Ulam stability for equations can be stated in terms of how much the mappings satisfying the equations approximately (in a sense) differ from the exact solutions of these equations. One of the best known results in this area is the following: Let g be a mapping from a normed space V into a Banach space B. Let $\xi \ge 0$ and $t\ne 1$ be fixed real numbers and $g:V\to B$ satisfy the inequality $\parallel g(u+v)-g\left(u\right)-g\left(v\right)\parallel \le \xi (\parallel u{\parallel }^t+\parallel v{\parallel }^t)$ for $u,v\in V\setminus \left\{0\right\}$. Then, there exists a unique additive $f:V\to B$ fulfilling the inequality $\parallel g\left(u\right)-f\left(u\right)\parallel \le \xi |1-2^{t-1}{|}^{-1}{\parallel u\parallel }^t$ for $u\in V\setminus \left\{0\right\}$. There arises a natural problem if the constant, on the right hand side of the latter inequality, is the best possible. It is known as the problem of the best Ulam constant. We discuss this problem, as well as several related issues, show possible generalizations of the existing results, and indicate open problems. To make this publication more accessible to a wider audience, we limit the related information, avoid advanced generalizations, and mainly focus only on the additive Cauchy equation $f(x+y)=f\left(x\right)+f\left(y\right)$ and on the general linear difference equation $x_{n+p}=a_1{x}_{n+p-1}+\dots +a_p{x}_n+b_n$ (considered for sequences in a Banach space). In particular, we show that there is a significant symmetry between Ulam constants of several functional equations and of their inhomogeneous or radical forms. We hope that in this way we will stimulate further research in this area. Full article

Recently, the fixed-point theorem for fuzzy contractive mappings has been investigated within the framework of intuitionistic fuzzy metric-like spaces. This interesting topic was explored through the utilization of G-Cauchy sequences as defined by Grabiec. The aim of this study is to enhance the aforementioned results in a few aspects. Initially, the proof of the fixed-point theorem is simplified and condensed, allowing for potential generalization to papers focusing on similar fixed-point analyses. Furthermore, instead of G-Cauchy sequences, the classical Cauchy sequences proposed by George and Veeramani are examined, incorporating an additional condition on the fuzzy metric. Within this context, a solution to an old unresolved question posed by Gregory and Sapena is provided. The findings are reinforced by relevant examples. Finally, the introduced fuzzy metrics are applied to the field of image processing. Full article

Recently, Ji et al. established certain fixed-point results using Mann’s iterative scheme tailored to ${\mathcal{G}}_b$-metric spaces. Stimulated by the notion of the $\mathcal{F}$-contraction introduced by Wardoski, the contraction condition of Ji et al. was generalized in this research. Several fixed-point results with Mann’s iterative scheme endowed with $\mathcal{F}$-contractions in ${\mathcal{G}}_b$-metric spaces were proven. One non-trivial example was elaborated to support the main theorem. Moreover, for application purposes, the existence of the solution to an integral equation is provided by using the axioms of the proven result. The obtained results are generalizations of several existing results in the literature. Furthermore, the results of Ji. et al. are the special case of theorems provided in the present research. Full article

Partial metrics constitute a generalization of classical metrics for which self-distance may not be zero. They were introduced by S.G. Matthews in 1994 in order to provide an adequate mathematical framework for the denotational semantics of programming languages. Since then, different works were devoted to obtaining counterparts of metric fixed-point results in the more general context of partial metrics. Nevertheless, in the literature was shown that many of these generalizations are actually obtained as a corollary of their aforementioned classical counterparts. Recently, two fuzzy versions of partial metrics have been introduced in the literature. Such notions may constitute a future framework to extend already established fuzzy metric fixed point results to the partial metric context. The goal of this paper is to retrieve the conclusion drawn in the aforementioned paper by Haghia et al. to the fuzzy partial metric context. To achieve this goal, we construct a fuzzy metric from a fuzzy partial metric. The topology, Cauchy sequences, and completeness associated with this fuzzy metric are studied, and their relationships with the same notions associated to the fuzzy partial metric are provided. Moreover, this fuzzy metric helps us to show that many fixed point results stated in fuzzy metric spaces can be extended directly to the fuzzy partial metric framework. An outstanding difference between our approach and the classical technique introduced by Haghia et al. is shown. Full article

One recent and prolific direction in the development of fixed point theory is to consider an operator $T:X\to X$ defined on a metric space $(X,d)$ which is an F—contraction, i.e., T verifies a condition of type $\tau +F\left(d\right(T\left(x\right),T\left(y\right))\le F(d(x,y))$, for all $x,y\in X$, $T\left(x\right)\ne T\left(y\right)$, where $\tau >0$ and $F:(0,\infty )\to \mathbb{R}$ satisfies some suitable conditions which ensure the existence and uniqueness for the fixed point of operator T. Moreover, the notion of F-contraction over a metric space $(X,d)$ was generalized by considering the notion of $(G,H)$—contraction, i.e., a condition of type $G\left(d\right(Tx,Ty\left)\right)\le H\left(d\right(x,y\left)\right)$, for all $x,y\in X$, $Tx\ne Ty$ for some appropriate $G,H:(0,\infty )\to \mathbb{R}$ functions. Recently, the abovementioned F-contraction theory was extended to the setup of cone metric space over the topological left modules. The principal objective of this paper is to introduce the concept of vectorial dislocated metric space over a topological left module and the notion of A-Cauchy sequence, as a generalization of the classical Cauchy sequence concept. Furthermore, based on the introduced concept, a fixed point result is provided for an operator $T:X\to X$, which satisfies the condition $(G,H)$—contraction, where $G,H$ are defined on the interior of a solid cone. Full article

In this article, the concept of extended fuzzy rectangular b-metric space (EFR_bMS, for short) is initiated, and some fixed point results frequently used in the literature are generalized via α-admissibility in the setting of EFR_bMS. For the illustration of the work presented, some supporting examples and an application to the existence of solutions for a class of integral equations are also discussed. Full article

Let $(M,\delta )$ be a metric space, $f:M\to M$, and $g:M\to [0,+\infty )$. In this paper, we obtain sufficient conditions under which the set of fixed points of f is g-admissible, i.e., $\mathrm{Fix}\left(f\right)\ne \varnothing$ and $\mathrm{Fix}\left(f\right)\subset g^{-1}\left(\left\{0\right\}\right)$. Some special cases of our main results are discussed and some examples are given. Full article

The purpose of this paper is to consider various results in the context of G ${}_b$ -metric spaces that have been recently published after the paper (Aghajani, A.; Abbas, M.; Roshan, J.R. Common fixed point of generalized weak contractive mappings in partially ordered G ${}_b$ -metric spaces. Filomat 2014, 28, 1087–1101). Our new results improve, complement, unify, enrich and generalize already well known results on G ${}_b$ -metric spaces. Moreover, some coupled and tripled coincidence point results have been provided. Full article

The purpose of this paper is to obtain a sufficient condition for a G-Cauchy sequence to be an M-Cauchy sequence in fuzzy metric spaces. Our main result provides a partial answer to the open question posed by V. Gregori and A. Sapena. For application, we give a new fuzzy version of the Banach fixed point theorem. Full article