Abstract
In this paper, we unify the system of functional equations defining a multi-Jensen-quadratic mapping to obtain a single equation. We also prove, using the fixed point method, the generalized Hyers–Ulam stability of this equation both in Banach spaces and in complete non-Archimedean normed spaces.
Highlights
It is well known that among functional equations, the Jensen equation ( )f x + y = f (x) + f (y)(which is closely connected with the notion of convex function) and the Jordan–von Neumann equation q(x + y) + q(x − y) = 2q(x) + 2q(y)(which is useful in some characterizations of inner product spaces) play a prominent role
We reduce the system of n equations defining the multiJensen-quadratic mapping to a single functional equation and we prove the generalized
We reduce the system of n equations defining the k-Jensen and n − kquadratic mapping to obtain a single functional equation
Summary
Gavruta) Hyers–Ulam stability of this equation both in Banach spaces and in complete non-Archimedean normed spaces. The first work on the Hyers–Ulam stability of functional equations in complete non-Archimedean normed spaces (some particular cases were considered earlier; see [5] for details) is [25]. After it a lot of papers (see for instance [8, 16, 30] and the references therein) on the stability of other equations in such spaces have been published. For more information about this method we refer the reader to [2, 5, 6, 15, 17] and the references therein
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