A note on second order linear functional equations in random normed spaces

TL;DR

This paper extends stability results for second order linear functional equations in random normed spaces, generalizing prior work by relaxing assumptions and applying properties of normal distributions.

Contribution

It generalizes Jung's stability results to arbitrary random normed spaces with minimal t-norms, under monotonicity conditions.

Findings

Established stability results for second order linear functional equations in random normed spaces.

Generalized previous results by relaxing lower bound assumptions.

Applied properties of normal distributions to validate main results.

Abstract

In this paper, we apply the publication of Joung (2009) to derive a stability result for for the second order linear functional equation: $f(x) = pf(x-1)-qf(x-2)$ for all $x\in\mathbb R$, where $f$ is a mapping from $\mathbb R$ into the induced random space of any Banach space. By relaxing the lower bound assumption, we also generalize the result of Jung (2009) on arbitrary random normed spaces with the minimum $t$-norm. However, we need the monotonicity of the distribution in the lower bound assumption. By the properties of normal distributions, our main result can be applied.