On Hyers-Ulam-Rassias stability of the Pexider equation
Let \((G,+)\) be an abelian group, \((X,\|\cdot\|)\) be a Banach space and \(f,g,h:G\rightarrow X\) be mappings. An equation \(f(x+y)=g(x)+h(y)\) is called a Pexider functional equation. In the paper the stability of that equation in the spirit of Hyers-Ulam-Rassias is considered. The main theorem is the following: Let \(\varphi:G\times G\rightarrow[0,\infty)\) be such that \(\varepsilon(x):=\sum_{j=1}^{\infty}2^{-j}(\varphi(2^{j-1}x,0)+\varphi(0,2^{j-1}x)+\varphi(2^{j-1}x,2^{j-1}x))<\infty\) and \(\lim_{n\rightarrow\infty}2^{-n} \varphi(2^nx,2^ny)=0\) for all \(x,y\in G\). Assume that \(\|f(x+y)-g(x)-h(y)\|\leq\varphi(x,y)\) for all \(x,y\in G\). Then there exists a unique additive mapping \(T:G\rightarrow X\) such that \(\|f(x)-T(x)\|\leq\|g(0)\|+\|h(0)\|+\varepsilon(x)\), \(\|g(x)-T(x)\|\leq\|g(0)\|+2\|h(0)\|+\varphi(x,0)+\varepsilon(x)\) and \(\|h(x)-T(x)\|\leq 2\|g(0)\|+\|h(0)\|+\varphi(0,x)+\varepsilon(x)\) for all \(x,y\in G\). Some possible applications of that theorem are presented.
scientific article; zbMATH DE number 1389232