Fixed points of some nonlinear operators in spaces of multifunctions and the Ulam stability (original) (raw)

Ulam stability of some functional inclusions for multi-valued mappings

Filomat, 2017

We show that some multifunctions F : K ? n(Y), satisfying functional inclusions of the form ? (x,F(?1(x)),..., F(?n(x)))? F(x)G(x), admit near-selections f : K ? Y, fulfilling the functional equation ? (x,f (?1(x)),..,, f(?n(x)))= f(x), where functions G : K ? n(Y), ?: K x Yn ? Y and ?1,..., ?n ? KK are given, n is a fixed positive integer, K is a nonempty set, (Y,?) is a group and n(Y) denotes the family of all nonempty subsets of Y. Our results have been motivated by the notion of Ulam stability and some earlier outcomes. The main tool in the proofs is a very recent fixed point theorem for nonlinear operators, acting on some spaces of multifunctions.

Fixed Point Theory and the Ulam Stability

Journal of Function Spaces, 2014

The fixed point method has been applied for the first time, in proving the stability results for functional equations, by Baker (1991); he used a variant of Banach's fixed point theorem to obtain the stability of a functional equation in a single variable. However, most authors follow the approaches involving a theorem of Diaz and Margolis. The main aim of this survey is to present applications of different fixed point theorems to the theory of stability of functional equations, motivated by a problem raised by Ulam in 1940.

On a fixed point theorem in 2-Banach spaces and some of its applications

Acta Mathematica Scientia, 2018

The aim of this article is to prove a fixed point theorem in 2-Banach spaces and show its applications to the Ulam stability of functional equations. The obtained stability results concern both some single variable equations and the most important functional equation in several variables, namely, the Cauchy equation. Moreover, a few corollaries corresponding to some known hyperstability outcomes are presented.

Fixed points of a mapping and Hyers–Ulam stability

Journal of Mathematical Analysis and Applications, 2014

We show that many general results on Hyers-Ulam stability of some functional equations in a single variable follow immediately from a simple fixed point theorem. The theorem is formulated for self-maps of some subsets of the space of functions from a nonempty set into the set of reals. We also give some applications of that theorem, e.g., in investigations of solutions of some difference equations and functional inequalities.

On functions that are approximate fixed points almost everywhere and Ulam’s type stability

Journal of Fixed Point Theory and Applications, 2015

In this paper, we investigate functions that are approximate fixed points of some (possibly nonlinear) operators almost everywhere, with respect to some ideals of sets. We prove that (under suitable assumptions) there exist fixed points of the operators that are "near" those functions. The results are applied to obtain some general stability results of Ulam's type almost everywhere; in particular, for the polynomial functional equation.

Fixed-point theorem in classes of function with values in a dq-metric space

Journal of Fixed Point Theory and Applications, 2018

We prove a fixed point result for nonlinear operators, acting on some classes of functions with values in a dq-metric space, and show some applications of it. The result has been motivated by some issues arising in Ulam stability. We use a restricted form of a contraction condition.

Fixed point theorems for single-valued and multi-valued maps

Nonlinear Analysis: Theory, Methods & Applications, 2011

a b s t r a c t Coincidence and fixed point theorems for single-valued and multi-valued maps generalizing recent results of Suzuki and Kikkawa are obtained. Various applications, including the existence of common solutions of certain functional equations are presented.

Some general fixed-point theorems for nonlinear mappings connected with one Cauchy theorem

arXiv (Cornell University), 2022

In this work, using a new geometrical approach we study to the existence of the fixed-point of mappings that independence of the smoothness, and also of their single-values or multi values. This work proved the theorems that generalize in some sense the Brouwer and Schauder fixed-point theorems, and also such type results in multi-valued cases. One can reckon this approach is based on the generalization of the one theorem Cauchy and on the convexity properties of sets. As the used approach is based on the geometry of the image of the examined mappings that are independent of the topological properties of the space we could to prove the general results for almost every vector space. The general results we applied to the study of the nonlinear equations and inclusions in VTS, and also by applying these results are investigated different concrete nonlinear problems. Here provided also sufficient conditions under which the conditions of the theorems will fulfill.

A New Approach to Some Fixed Point Theorems for Multivalued Nonlinear F-Contractive Maps

2019

In this article, by introducing a new operator, we give a new generalized contraction condition for multivalued maps. Moreover, without assumption of lower semicontinuity, we prove some fixed point theorems in incomplete metric spaces. Our results are extension of the corresponding results of I. Altun et al. (Nonlinear Analysis: Modeling and control, 2016, Vol. 21, No. 2, 201–210). Also, we provide some examples to show that our main theorem is a generalization of some previous results.

Fixed Point Theorems for Multivalued Mappings Involving-Function

Three fixed point theorems for multi-valued mappings in symmetric spaces with closure operator are proved. As corollaries of these theorems we obtain the existence of fixed points of mappings in compact metric spaces. Another corollary of the last theorem extends a condition of Kannan's theorem. S(A) = Ç]{B\BdA, Bzsé}. Let F:2X ^ 2X be a multi-valued mapping and j/ be the family of all subsets A of X for which F(A) c A. Then j/ is invariant with respect to the set intersection and so it induces a closure operator which is denoted by 5^_