On approximate isometries and application to stability of a functional equation

Abstract

In this paper, we show a generalized stability of isometries between Banach spaces. Making use of this result, we prove the corresponding stability of the functional equation

A Fixed Point Approach to the Hyers-Ulam-Rassias Stability Problem of Pexiderized Functional Equation in Modular Spaces

“…This stability problem was first raised by Ulam [36] and thereafter it was further generalized in the works of Hyers [5] and Rassias [39]. It is a general concept which is applicable in diverse frameworks of mathematics like those in problems of differential equations [25,32,37], fixed points [20], isometrics [42], etc. Particularly for the functional equations, the Hyers-Ulam-Rassias stability has been considered in a good number of papers in different types of spaces [2, 3, 7, 10, 15, 21, 26, 27, 30, 33-35, 38, 43-45].…”

Applying Fixed Point Techniques to Stability Problems in Intuitionistic Fuzzy Banach Spaces

“…Today we know such stability problems as the problems of the Hyers-Ulam-Rassias (H-U-R) stability. It has many extended forms and has been studied in several domains of mathematics including differential equations [4], functional equations [5], isometries [6], etc. Our interest is in the study of such stabilities for certain functional equations.…”

Hyers–Ulam–Rassias Stability of Set Valued Additive and Cubic Functional Equations in Several Variables

“…The basic notion of this stability can be illustrated by looking at the question on a linear equation: "Does an approximately linear equation have a linear approximation?" Today, it has many extended forms and has been studied in several domains of mathematics including differential equations [6], functional equations [7], isometries [8], etc. In the course of our investigation, we use the Hausdorff distance between two appropriate subsets of a Banach space.…”