Solutions and stability of a generalization of wilson's equation

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• 作者：Bouikhalene BELAID^a ; ^{bbouikhalene@yahoo.fr" class="auth_mail" title="E-mail the corresponding author} ; Elqorachi ELHOUCIEN^b ; ^{elqorachi@hotamail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：d'Alembert's functional equation ; locally compact group ; involution ; character ; complex measure ; Wilson's functional equation ; Hyers-Ulam stability
• 刊名：Acta Mathematica Scientia
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：36
• 期：3
• 页码：791-801
• 全文大小：341 K

文摘

In this paper we study the solutions and stability of the generalized Wilson's functional equation ${\displaystyle {\int }_{G}f\left(xty\right)\text{d}\mu \left(t\right)}+{\displaystyle {\int }_{G}f\left(xt\sigma \left(y\right)\right)\text{d}\mu \left(t\right)=2f\left(x\right)g\left(y\right),x,y}\in G$, where G is a locally compact group, σ is a continuous involution of G and μ is an idempotent complex measure with compact support and which is σ  -invariant. We show that ${\displaystyle {\int }_{G}g\left(xty\right)\text{d}\mu \left(t\right)}+{\displaystyle {\int }_{G}g\left(xt\sigma \left(y\right)\right)\text{d}\mu \left(t\right)=2g\left(x\right)g\left(y\right)}\text{if}f\ne 0\text{and}{\displaystyle {\int }_{G}f\left(t.\right)\text{d}\mu \left(t\right)\ne 0,\text{where}\left[{\displaystyle {\int }_{G}f\left(t.\right)\text{d}\mu \left(t\right)}\right]\left(x\right)}={\displaystyle {\int }_{G}f\left(tx\right)\text{d}\mu \left(t\right)}$. We also study some stability theorems of that equation and we establish the stability on noncommutative groups of the classical Wilson's functional equation f(xy)+χ(y)f(xσ(y))=2f(x)g(y) x,y∈G$f\left(xy\right)+\chi \left(y\right)f\left(x\sigma \left(y\right)\right)=2f\left(x\right)g\left(y\right)x,y\in G$ where χ is a unitary character of G.