Remarks on solutions to the functional equations of the radical type

Brzdęk, Janusz

doi:10.31197/atnaa.379095

Cited by 6 publications

(5 citation statements)

References 12 publications

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“…The next theorem can be derived from ( [18], Corollary 2.3 and Proposition2.4(a)). However, for the convenience of readers we present it with a directproof.…”

Section: Resultsmentioning

confidence: 95%

Some hyperstability results of a p-radical functional equation related to Drygas mappings in non-Archimedean Banach spaces

Hryrou

Nuino

Kabbaj

2021

Proyecciones (Antofagasta)

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The aim of this paper is to introduce and solve the following pradical functional equation related to Drygas mappings f(√(p&x^p+ y^p ))+f(√(p&x^p+ y^p ))=2f(x)+f(y)+f(-y),x,y ∈R, where f is a mapping from R into a vector space X and p ≥ 3 is an odd natural number. Using an analogue version of Brzdȩk’s fixed point theorem [14], we establish some hyperstability results for the considered equation in non-Archimedean Banach spaces. Also, we give some hyperstability results for the inhomogeneous p-radical functional equation related to Drygas mappings f(√(p&x^p+ y^p ))+f(√(p&x^p+ y^p ))=2f(x)+f(y)+f(-y)+G(x,y)

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“…The next theorem can be derived from ( [18], Corollary 2.3 and Proposition2.4(a)). However, for the convenience of readers we present it with a directproof.…”

Section: Resultsmentioning

confidence: 95%

Some hyperstability results of a p-radical functional equation related to Drygas mappings in non-Archimedean Banach spaces

Hryrou

Nuino

Kabbaj

2021

Proyecciones (Antofagasta)

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“…The other main stability result in [50] is analogous to Theorem 21, but with h 1 (x 3 , z)h 2 (y 3 , z) in (88) replaced by h(x 3 , z) + h(y 3 , z), where h : R 2 → R + fulfills an analogous assumption as h 1 and h 2 . General remarks on solutions to equations similar to (89) can be found in [66].…”

Section: Theorem 15mentioning

confidence: 99%

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

El‐hady

Brzdęk

2022

Symmetry

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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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“…In this section, we give the general solution of the radical-type functional equation (12) by using some results that are reported in [10]. If we compare (1.2) with (4.3), we obtain that the function f is even.…”

Section: General Solution Of Eq (12)mentioning

confidence: 99%

A fixed point approach to the hyperstability of the general linear equation in β-Banach spaces

2018

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The purpose of this paper is first to reformulate the fixed point theorem (see Theorem 1 of [J. Brzdȩk, J. Chudziak and Z. Páles, A fixed point approach to stability of functional equations, Nonlinear Anal. 74 2011, 17, 6728–6732]) in β-Banach spaces. We also show that this theorem is a very efficient and convenient tool for proving the hyperstability results of the general linear equation in β-Banach spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it.

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Remarks on solutions to the functional equations of the radical type

Cited by 6 publications

References 12 publications

Some hyperstability results of a p-radical functional equation related to Drygas mappings in non-Archimedean Banach spaces

Some hyperstability results of a p-radical functional equation related to Drygas mappings in non-Archimedean Banach spaces

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

A fixed point approach to the hyperstability of the general linear equation in β-Banach spaces

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