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Volume 30, Issue 4
Approximation of the Cubic Functional Equations in Lipschitz Spaces

A. Ebadian, N. Ghobadipour, I. Nikoufar & M. Eshaghi Gordji

Anal. Theory Appl., 30 (2014), pp. 354-362.

Published online: 2014-11

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  • Abstract

Let $\mathcal{G}$ be an Abelian group and let $\rho:\mathcal{G} \times \mathcal{G} \rightarrow [0, \infty)$ be a metric on $\mathcal{G}$. Let $\varepsilon$ be a normed space. We prove that under some conditions if $f:\mathcal{G}\to\varepsilon$ is an odd function and $C_x:\mathcal{G}\to\varepsilon$ defined by $C_x(y):=2f(x+y)+2f(x-y)+12f(x)-$ $f(2x+y)-f(2x-y)$ is a cubic function for all $x\in \mathcal{G},$ then there exists a cubic function $C:\mathcal{G}\to\varepsilon$ such that $f-C$ is Lipschitz. Moreover, we investigate the stability of cubic functional equation $2f(x+y)+2f(x-y)+12f(x)-f(2x+y)$ $-f(2x-y)=0$ on Lipschitz spaces.

  • AMS Subject Headings

39B82, 39B52

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COPYRIGHT: © Global Science Press

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@Article{ATA-30-354, author = {}, title = {Approximation of the Cubic Functional Equations in Lipschitz Spaces}, journal = {Analysis in Theory and Applications}, year = {2014}, volume = {30}, number = {4}, pages = {354--362}, abstract = {

Let $\mathcal{G}$ be an Abelian group and let $\rho:\mathcal{G} \times \mathcal{G} \rightarrow [0, \infty)$ be a metric on $\mathcal{G}$. Let $\varepsilon$ be a normed space. We prove that under some conditions if $f:\mathcal{G}\to\varepsilon$ is an odd function and $C_x:\mathcal{G}\to\varepsilon$ defined by $C_x(y):=2f(x+y)+2f(x-y)+12f(x)-$ $f(2x+y)-f(2x-y)$ is a cubic function for all $x\in \mathcal{G},$ then there exists a cubic function $C:\mathcal{G}\to\varepsilon$ such that $f-C$ is Lipschitz. Moreover, we investigate the stability of cubic functional equation $2f(x+y)+2f(x-y)+12f(x)-f(2x+y)$ $-f(2x-y)=0$ on Lipschitz spaces.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2014.v30.n4.2}, url = {http://global-sci.org/intro/article_detail/ata/4499.html} }
TY - JOUR T1 - Approximation of the Cubic Functional Equations in Lipschitz Spaces JO - Analysis in Theory and Applications VL - 4 SP - 354 EP - 362 PY - 2014 DA - 2014/11 SN - 30 DO - http://doi.org/10.4208/ata.2014.v30.n4.2 UR - https://global-sci.org/intro/article_detail/ata/4499.html KW - Cubic functional equation, Lipschitz space, stability. AB -

Let $\mathcal{G}$ be an Abelian group and let $\rho:\mathcal{G} \times \mathcal{G} \rightarrow [0, \infty)$ be a metric on $\mathcal{G}$. Let $\varepsilon$ be a normed space. We prove that under some conditions if $f:\mathcal{G}\to\varepsilon$ is an odd function and $C_x:\mathcal{G}\to\varepsilon$ defined by $C_x(y):=2f(x+y)+2f(x-y)+12f(x)-$ $f(2x+y)-f(2x-y)$ is a cubic function for all $x\in \mathcal{G},$ then there exists a cubic function $C:\mathcal{G}\to\varepsilon$ such that $f-C$ is Lipschitz. Moreover, we investigate the stability of cubic functional equation $2f(x+y)+2f(x-y)+12f(x)-f(2x+y)$ $-f(2x-y)=0$ on Lipschitz spaces.

A. Ebadian, N. Ghobadipour, I. Nikoufar & M. Eshaghi Gordji. (1970). Approximation of the Cubic Functional Equations in Lipschitz Spaces. Analysis in Theory and Applications. 30 (4). 354-362. doi:10.4208/ata.2014.v30.n4.2
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