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.

Representation and Stability of General Nonic Functional Equation

“…Stability results for general linear functional equations can be found in [43][44][45][46]. Moreover, hyperstability results for general linear functional equations can be found in [25,47,48].…”

A new form of multi-cubic mappings and the stability results

“…is equivalent to equation ( 2). Here, we recall that equations ( 1)-( 4) are particular cases of the general linear functional equation which has been introduced and studied in [1] and [2]. Equation ( 4) motivates us to define a new form of multi-cubic mappings which are different from [3], [5] and [16].…”