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

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.