On Hyperstability of the Cauchy Functional Equation in n-Banach Spaces

Brzdęk, Janusz; El‐hady, El‐sayed

doi:10.3390/math8111886

Cited by 8 publications

(4 citation statements)

References 29 publications

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“…Furthermore, the stability results in n-normed spaces, which can be found in [29,[31][32][33][35][36][37]53,[82][83][84][85][86], will be discussed in a future publication.…”

Section: Some Other Resultsmentioning

confidence: 99%

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

et al. 2021

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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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“…Furthermore, the stability results in n-normed spaces, which can be found in [29,[31][32][33][35][36][37]53,[82][83][84][85][86], will be discussed in a future publication.…”

Section: Some Other Resultsmentioning

confidence: 99%

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

et al. 2021

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“…If C = 0 then for the series in (15) to be convergent we take s = −1. If also a l = 0 for l ∈ n , then in Theorem 3, f satisfies the condition Funding: This research received no external funding.…”

Section: Remarkmentioning

confidence: 99%

“…, x m ∈ Y, then x = 0. The hyperstability phenomenon occurs when no deviation of a state affects that state (see, e.g., [14][15][16]). Proving our result we improve a result from [17], where the stability result was shown.…”

Section: Introductionmentioning

confidence: 99%

On Stability of a General n-Linear Functional Equation

Bahyrycz

Sikorska

2022

Symmetry

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Let X be a linear space over K∈{R,C}, Y be a real or complex Banach space and f:Xn→Y. With some fixed aji,Ci1…in∈K (j∈{1,…,n}, i,ik∈{1,2}, k∈{1,…,n}), we study, using the direct and the fixed point methods, the stability and the general stability of the equation f(a11x11+a12x12,…,an1xn1+an2xn2)=∑1≤i1,…,in≤2Ci1…inf(x1i1,…,xnin), for all xjij∈X (j∈{1,…,n},ij∈{1,2}). Our paper generalizes several known results, among others concerning equations with symmetric coefficients, such as the multi-Cauchy equation or the multi-Jensen equation as well as the multi-Cauchy–Jensen equation. We also prove the hyperstability of the above equation in m-normed spaces with m≥2.

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“…Many papers on the stability and hyperstability of functional equations were published thanks to this important achievement. For example, we refer to [1]- [10], [18]- [20], and [40]. Another point worth noting is that there were other versions of Theorem 1.3 in ultrametric space [3], in 2-Banach space [4], [18], and in n-Banach space [19] that helped to discuss many results on the stability of functional equations.…”

Section: Introductionmentioning

confidence: 99%