Abstract
First we investigate the Hyers–Ulam stability of the Cauchy functional equation for mappings from bounded (unbounded) intervals into Banach spaces. Then we study the Hyers–Ulam stability of the functional equation f(xy)=xg(y)+h(x)y for mappings from bounded (unbounded) intervals into multi-normed spaces.
Highlights
1 Introduction The concept of stability for a functional equation (∗) arises when the functional equation (∗) is replaced by an inequality that acts as the equation perturbation
In 1940, Ulam [16] posed the first question concerning the stability of homomorphisms between groups
Hyers [4] answered the question of Ulam in the context of Banach spaces
Summary
The concept of stability for a functional equation (∗) arises when the functional equation (∗) is replaced by an inequality that acts as the equation perturbation. We use some ideas from the works [6, 9, 14] to investigate the Hyers–Ulam stability of the Cauchy functional equation and (1.1) for mappings from bounded (unbounded) intervals into multi-normed spaces. Proposition 1.3 ([2]) Let {(En, · n) : n ∈ N} be a multi-normed space. Definition 1.4 Let {(En, · n) : n ∈ N} be a multi-normed space.
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