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.

A note on stability of the linear functional equations of higher order and fixed points of an operators

International journal on fixed point theory computation and applications

We prove two general theorems, which appear to be very useful in the investigation of the Hyers-Ulam stability of a higher-order linear functional equation in single variable, with constant coefficients. We give several examples of their applications. In particular, we show that we obtain in this way several fixed point results for a particular operator. The main tool in the proofs is a complexification of a real normed (or Banach) space X, which can be described as the tensor product X⊗ℝ 2 endowed with the Taylor norm.

On some nonlinear operators, fixed-point theorems and nonlinear equations

In this article we discuss the solvability of some class of fully nonlinear equations, and equations with p-Laplacian in more general conditions by using a new approach given in [1] for studying the nonlinear continuous operator. Moreover we reduce certain general results for the continuous operators acting on Banach spaces, and investigate their image. Here we also consider the existence of a fixed-point of the continuous operators under various conditions.

Multi-valued mappings and fixed points II

In this paper, some hybrid fixed point principles for the sum of two multi-valued operators in a Banach space are proved and they are further applied to a certain integral inclu-sion of mixed type for proving the existence results under mixed Lipschitz and Carathéodory conditions.

Fixed point theorems and stability of fixed point sets of multivalued mappings

Advances in Fixed Point Theory, 2013

In this paper, we prove new fixed point theorems of multivalued mappings in partially ordered metric spaces using newly reformulated pre-order relations. As consequence, we derive fixed point theorems for single valued mappings given by Nieto and Rodriguez-Lopez [11], [12]. We also establish some results on the stability of fixed point sets of multivalued mappings in partially ordered metric spaces. General illustrative examples are also given. Essential to our results are the pre-order relations <1,<2,<3defined in [3], and newly reformulated pre-order relations namely <4,<5,<6, which are obtained by imposing a distance condition to comparable elements of two non-empty, closed and bounded sets.