On approximate generalized Lie derivations

Motivated by the notion of the Hyers-Ulam stability, we prove results that are efficient tools for the study of approximate generalized Lie derivations on Lie algebras. We also provide simple examples of applications of our outcomes. In particular, we obtain some auxiliary results on the stability of the additive Cauchy equation

Hyperstability of the Fréchet Equation and a Characterization of Inner Product Spaces

We prove some stability and hyperstability results for the well-known Fr´echet equation stemming fromone of the characterizations of the inner product spaces. As the main tool, we use a fixed point theorem for the function spaces.We finish the paper with some new inequalities characterizing the inner product spaces

Banach limit in the stability problem of a linear functional equation

We present some applications of the Banach limit in the study of the stability of the linear functional equation in a single variable

Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients

We study the Hyers-Ulam stability in a Banach space X of the system of first order linear difference equations of the form xn+1=Axn+dn for n∈N0 (nonnegative integers), where A is a given r×r matrix with real or complex coefficients, respectively, and (dn)n∈N0 is a fixed sequence in Xr. That is, we investigate the sequences (yn)n∈N0 in Xr such that δ∶=supn∈N0yn+1-Ayn-dn<∞ (with the maximum norm in Xr) and show that, in the case where all the eigenvalues of A are not of modulus 1, there is a positive real constant c (dependent only on A) such that, for each such a sequence (yn)n∈N0, there is a solution (xn)n∈N0 of the system with supn∈N0yn-xn≤cδ

Hyers-Ulam Stability of the Delay Equation y

We investigate the approximate solutions y:[−τ,∞)→R of the delay differential equation y'(t)=λy(t-τ)(t∈[0,∞)) with an initial condition, where λ>0 and τ>0 are real constants. We show that they can be “approximated” by solutions of the equation that are constant on the interval [-τ,0] and, therefore, have quite simple forms. Our results correspond to the notion of stability introduced by Ulam and Hyers

Applications of Banach limit in Ulam stability

We show how to get new results on Ulam stability of some functional equations using the Banach limit. We do this with the examples of the linear functional equation in single variable and the Cauchy equation