Generalized hyperstability of a Drygas functional equation on a restricted domain using Brzdȩk's fixed point theorem
Using a result of \textit{J. Brzdȩk} and \textit{K. Ciepliński} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 18, 6861--6867 (2011; Zbl 1237.39022)] the author proves some hyperstanbility results for the Drygas functional equation (1) \(f(x+y)+f(x-y)=2f(x)+f(y)+f(-y)\). The setting is the following. \(X\) is a normed space, \((G,+,d)\) is an abelian group together with a translation invariant metric \(d\) such that \((G,d)\) is complete. \(U\) is a non empty subset of \(X\setminus\{0\}\) such that \(-x, nx\in U\) for all \(x\in U\) and all \(n\in\mathbb{N}\), \(n\geq m_0\) for some \(m_0\). Furthermore \(h: U\to[0,\infty[\) has the property that \(s(\pm n):=\inf\{t\in[0,\infty[\,\mid\, h(nx)\leq th(x)\text{ for all }x\in U\}\) exists for all \(n\geq m_0\). Then the following result holds true. Theorem. Assume that \(s(n)\) tends to \(0\) for \(n\) tending to \(\infty\) and \(-\infty\) and that \(s(2n+1)+2s(n+1)+s(n)+s(-n)<1\) for some \(n\geq m_0\). Then any function \(f: U\to G\) satisfying \(d(f(x+y)+f(x-y)\), \(2f(x)+f(y)+f(-y)) \leq h(x)+h(y)\) for all \(x,y\in U\) such that \(x+y, x-y\in U\) satisfies (1) for all \(x,y\in U\) such that \(x+y,x-y\in U\). The assumption that \(X\) should be a normed space seem to be superfluous. Possibly it is enough to assume that \(X\) is an abelian group. When \(X\) is a normed space the author gives a corollary for \(f=\|\,\|\), \(G=\mathbb{R}\) using an additional assumption on \(h\) which says that in this case \(X\) necessarily must be an inner product space.
scientific article; zbMATH DE number 6817763
scientific article