Stability of the multi-Jensen equation
Let \(V\) and \(W\) denote normed spaces. A function \(f:V^m\to W\) is called multi-Jensen, if it satisfies the Jensen functional equation in each of its variables. It is well-known [see \textit{W. Prager} and \textit{J. Schwaiger}, Bull. Korean Math. Soc. 45, No.~1, 133--142 (2008; Zbl 1151.39023)] that multi-Jensen functions are characterized by the functional equation \[ J_f(x,y):=f\left(\frac{x+y}{2}\right) -\frac{1}{2^m}\sum_{S\subset\mathbf{m}}f(x_S+y_{\mathbf{m}\setminus S})=0 \quad (x,y\in V^m), \] where \(\mathbf{m}:=\{1,\dots,m\}\) and, for \(S\subset\mathbf{m}\) and an \(m\)-tuple \(z=(z_1,\dots,z_m)\in V^m\), the \(i\)th coordinate of \(z_S\in V^m\) is defined to be \(z_i\) if \(i\in\mathbf{m}\) and to be zero if \(i\not\in\mathbf{m}\). The stability problems of this functional equation were considered in several recent papers. Prager and Schwaiger [loc. cit.] proved the stability in the Hyers-Ulam sense, while the author [ibid. 45, No.~4, 729--737 (2008; Zbl 1172.39033)] investigated the stability in Gavruta's sense. The main result of this paper concerns the stability of the above functional equation in the Rassias sense, i.e., when the stability estimate for \(J_f\) is expressed in terms of the sum of \(p\)th power of the coordinates of \(x\) and \(y\).
scientific article; zbMATH DE number 5644924
scientific article