On Ulam’s Type Stability of the Linear Equation and Related Issues

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FUNCTIONAL EQUATIONS, DIFFERENCE INEQUALITIES AND ULAM STABILITY NOTIONS (F.U.N.)

This Ulam's volume (F.U.N.) consists of research papers containing various parts of contemporary pure and applied mathematics with emphasis to Ulam's mathematics. It contains various parts of Functional Equations and Difference Inequalities as well as related topics in Mathematical Analysis, namely: Ulam’s stability of a class of linear Cauchy functional equations; Sequential antagonistic games with an auxiliary initial phase; Some stability results for equations and inequalities connected with the exponential function; On a problem of John M. Rassias concerning the stability in Ulam sense of Euler-Lagrange equation; Hyers-Ulam- Aoki-Rassias stability and Ulam-Gavruta-Rassias stability of quadratic homomorphisms and quadratic derivations on Banach algebras; Fundamental solutions for the generalized elliptic Gellerstedt equation; Pointwise superstability and superstability of the Jordan equation; A problem with non-local conditions on the line of degeneracy and parallel characteristics for a mixed type equation with singular coefficient; On the stability of an additive functional inequality in normed modules; Cubic derivations and quartic derivations on Banach modules; Tetrahedron isometry Ulam stability problem; Hyers-Ulam stability of Cauchy type additive functional equations; Solution and Ulam stability of a mixed type cubic and additive functional equation; Stability of mappings approximately preserving orthogonality and related topics; The Frankl problem for second order nonlinear equations of mixed type with nonsmooth degenerate curve.

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On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey II

Symmetry

Ulam stability is motivated by the following issue: how much an approximate solution of an equation differs from the exact solutions to the equation. It is connected to some other areas of investigation, e.g., optimization, approximation theory and shadowing. In this paper, we present and discuss the published results on such stability for functional equations in the classes of function-taking values in 2-normed spaces. In particular, we point to several pitfalls they contain and provide possible simple improvements to some of them. Thus we show that the easily noticeable symmetry between them and the analogous results proven for normed spaces is, in fact, mainly apparent. Our article complements the earlier similar review published in Symmetry (13(11), 2200) because it concerns the outcomes that have not been discussed in this earlier publication.

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On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey

Symmetry, 2021

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.

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On Ulam’s Type Stability of the Linear Equation and Related Issues (original) (raw)

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