Abstract
The theory of Ulam stability was initiated by a problem raised in 1940 by S. Ulam and concerning approximate solutions to the equation of homomorphism in groups. It is somehow connected to various other areas of investigation such as, e.g., optimization and approximation theory. Its main issue is the error that we make when replacing functions satisfying the equation approximately with exact solutions of the equation. This article is a survey of the published so far results on Ulam stability for functional equations in 2-normed spaces. We present and discuss them, pointing to the various pitfalls they contain and showing possible simple generalizations. In this way, in particular, we demonstrate that the easily noticeable symmetry between them and the analogous results obtained for the classical metric or normed spaces is in fact only apparent.
Highlights
The theory of Ulam stability has drawn the attention of many researchers because of various possible applications
The concepts of an approximate solution and nearness of two functions can be understood in various ways. This depends on the needs of a particular situation and tools that are available for us. One of such non-classical ways of measuring a distance can be provided by the notions of 2-norms and n-norms
We provide auxiliary information on n-normed spaces, and in Section 3 we proof some simple, but general results on stability of some functional equations, which are very useful in further parts of the paper
Summary
The theory of Ulam stability has drawn the attention of many researchers because of various possible applications. We are reviewing the results on Ulam stability proved for function taking values in n-normed spaces. Much more precise results, but only for functions taking real values, have been obtained in [10] by applying the Banach limit technique (see in [11] for the application of that technique in the stability of functional equations in a single variable). We provide auxiliary information on n-normed spaces (which includes the 2-normed spaces), and in Section 3 we proof some simple, but general results on stability of some functional equations, which are very useful in further parts of the paper.
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