Remarks on stability of some inhomogeneous functional equations

Brzdęk, Janusz

doi:10.1007/s00010-014-0274-6

Cited by 35 publications

(21 citation statements)

References 25 publications

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“…Remark 1.4. Lemma 4.7 in [9] improves the statement of Proposition 1.3. Proposition 1.5 [5,Theorem 2].…”

Section: Introduction and Main Resultssupporting

confidence: 61%

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On Hyperstability of Generalised Linear Functional Equations in Several Variables

Zhang¹

2015

Bull. Aust. Math. Soc.

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We obtain some results on approximate solutions of the generalised linear functional equation $\sum _{i=1}^{m}L_{i}f(\sum _{j=1}^{n}a_{ij}x_{j})=0$ for functions mapping a normed space into a normed space. We show that, under suitable assumptions, the approximate solutions are in fact exact solutions. The theorems correspond to and complement recent results on the hyperstability of generalised linear functional equations.

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“…Remark 1.4. Lemma 4.7 in [9] improves the statement of Proposition 1.3. Proposition 1.5 [5,Theorem 2].…”

Section: Introduction and Main Resultssupporting

confidence: 61%

“…, m. Remark 2.3. Theorem 2.2 in [9] allows us to generalise Theorem 1.6 to the following inhomogeneous version of (1.2):…”

Section: Proof Of Theoremsmentioning

confidence: 99%

On Hyperstability of Generalised Linear Functional Equations in Several Variables

Zhang¹

2015

Bull. Aust. Math. Soc.

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“…Recently, Brzdęk [6] proved that if g : X → Y satisfies the general linear equation on X\{0}, then g satisfies the general linear equation on X. This observation and Theorem 2.2 yield the following result.…”

Section: Resultsmentioning

confidence: 75%

A Note on the Generalised Hyperstability of the General Linear Equation

Aiemsomboon

Sintunavarat

2017

Bull. Aust. Math. Soc.

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Let $X$ and $Y$ be two normed spaces over fields $\mathbb{F}$ and $\mathbb{K}$, respectively. We prove new generalised hyperstability results for the general linear equation of the form $g(ax+by)=Ag(x)+Bg(y)$, where $g:X\rightarrow Y$ is a mapping and $a,b\in \mathbb{F}$, $A,B\in \mathbb{K}\backslash \{0\}$, using a modification of the method of Brzdęk [‘Stability of additivity and fixed point methods’, Fixed Point Theory Appl.2013 (2013), Art. ID 285, 9 pages]. The hyperstability results of Piszczek [‘Hyperstability of the general linear functional equation’, Bull. Korean Math. Soc.52 (2015), 1827–1838] can be derived from our main result.

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“…For this purpose, we prove an analogon of Lemma 3.6 for the linear equation (see also [11,Lemma 4.7]). …”

Section: Then It Is a Solution Of The Fréchet Equation On Xmentioning

confidence: 97%

Hyperstability of general linear functional equation

Bahyrycz

Olko

2016

Aequat. Math.

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Abstract. Our purpose is to investigate criteria for hyperstability of linear type functional equations. We prove that a function satisfying the equation approximately in some sense, must be a solution of it. We give some conditions on coefficients of the functional equation and a control function which guarantee hyperstability. Moreover, we show how our outcomes may be used to check whether the particular functional equation is hyperstable. Some relevant examples of applications are presented.Mathematics Subject Classification. Primary 39B82, 47H14, 47J20; Secondary 39B62, 47H10.

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Remarks on stability of some inhomogeneous functional equations

Cited by 35 publications

References 25 publications

On Hyperstability of Generalised Linear Functional Equations in Several Variables

On Hyperstability of Generalised Linear Functional Equations in Several Variables

A Note on the Generalised Hyperstability of the General Linear Equation

Hyperstability of general linear functional equation

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