Ulam Stability of a General Linear Functional Equation in Modular Spaces

Abstract

Using the direct method, we prove the Ulam stability results for the general linear functional equation of the form ∑i=1mAi(fφi(x¯))=D(x¯) for all x¯∈Xn, where f is the unknown mapping from a linear space X over a field K∈{R,C} into a linear space Y over field K; n and m are positive integers; φ1,…,φm are linear mappings from Xn to X; A1,…,Am are continuous endomorphisms of Y; and D:Xn→Y is fixed. In this paper, the stability inequality is considered with regard to a convex modular on Y, which is lower semicontinuous and satisfies an additional condition (the Δ2-condition). Our main result generalizes many similar stability outcomes published so far for modular space. It also shows that there is some kind of symmetry between the stability results for equations in modular spaces and those in classical normed spaces.