Cauchy's Equations and Ulam's Problem

TL;DR

This paper investigates the stability and superstability of Cauchy's functional equations and their variants, providing new results, generalizations, and partial solutions to open problems in the context of complex functions on semigroups.

Contribution

It introduces new stability results for Cauchy's equations, generalizes existing theorems, and addresses open problems in the field of functional equations and their stability.

Findings

Superstability of Cauchy exponential and additive equations established.

Generalizations of Skof's and Joung's theorems provided.

Partial affirmative answers to open problems in the stability of functional equations.

Abstract

Our aim is to study the Ulam's problem for Cauchy's functional equations. First, we present some new results about the superstability and stability of Cauchy exponential functional equation and its Pexiderized for class functions on commutative semigroup to unitary complex Banach algebra. In connection with the problem of Th. M. Rassias and our results, we generalize the theorem of Baker and theorem of L. Sze'kelyhidi. Then the superstability of Cauchy additive functional equation can be prove for complex valued functions on commutative semigroup under some suitable conditions. This result is applied to the study of a superstability result for the logarithmic functional equation, and to give a partial affirmative answer to problem 18, in the thirty-first ISFE. The hyperstability and asymptotic behaviors of Cauchy additive functional equation and its Pexiderized can be study for functions on commutative semigroup to a complex normed linear space under some suitable conditions. As some consequences of our results, we give some generalizations of Skof's theorem, S.-M. Joung's theorem, and another affirmative answer to problem 18, in the thirty-first ISFE. Also we study the stability of Cauchy linear equation in general form and in connection with the problem of G. L. Forti, in the 13th ICFEI (2009), we consider some systems of homogeneous linear equations and our aim is to establish some common Hyers-Ulam-Rassias stability for these systems of functional equations and presenting some applications of these results.