Survey on Recent Ulam Stability Results Concerning Derivations

Abstract

This is a survey presenting the most significant results concerning approximate (generalized) derivations, motivated by the notions of Ulam and Hyers-Ulam stability. Moreover, the hyperstability and superstability issues connected with derivations are discussed. In the section before the last one we highlight some recent outcomes on stability of conditions defining (generalized) Lie derivations.

On Stability of a General n-Linear Functional Equation

“…One can observe the connections between general n-linear functional equations and the behaviors of approximate homomorphisms and derivations on Banach algebras (see, e.g., [24][25][26]), therefore it is recommended to proceed with some research in this direction.…”

Hyers-Ulam Stability and Existence of Solutions for Nigmatullin’s Fractional Diffusion Equation

“…is Hyers-Ulam-Rassias stable with respect to ( , ) = | | −3 2 /( 2 + 2 | | 2 ) and = 2, where we use (13) in Remark 4 again.…”

“…Motivated by a famous question of Ulam concerning the stability of group homomorphisms, Hyers and Rassias introduced the concepts of Hyers-Ulam and Hyers-Ulam-Rassias stability, respectively, in the case of the Cauchy functional equation in Banach spaces, which received a great influence in the development of the generalized Hyers-Ulam-Rassias stability of all kinds of functional equations. There are many interesting results on this topic in the case of functional equations, ordinary differential equations, partial differential equations, and impulsive differential equations; see, for example, [1][2][3][4][5][6][7][8][9][10][11] and the recent survey [12,13].…”

Stability of ternary antiderivation in ternary Banach algebras via fixed point theorem

“…During the last two decades, a number of articles and research monographs have been published on various generalizations and applications of the Hyers-Ulam stability to a number of functional equations and mappings, for example, Cauchy-Jensen mappings, k-additive mappings, multiplicative mappings, bounded nth differences, Euler-Lagrange functional equations, differential equations, and Navier-Stokes equations (see [1,2,4,5,19,22,25,26,27,28,29]). Also, approximate generalized Lie derivations have been already established in [6,7].…”