Abstract
In this paper we prove, using the fixed point method, the generalized Hyers–Ulam stability of two functional equations in complete non-Archimedean normed spaces. One of these equations characterizes multi-Cauchy–Jensen mappings, and the other gives a characterization of multi-additive-quadratic mappings.
Highlights
Throughout this paper, N stands for the set of all positive integers, N0 := N∪{0}, R+ := [0, ∞), n ∈ N and k ∈ {0, . . . , n}
(which is closely connected with the notion of a 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
Summary
Throughout this paper, N stands for the set of all positive integers, N0 := N∪{0}, R+ := [0, ∞), n ∈ N and k ∈ {0, . . . , n}. The first partial answer (in the case of Cauchy’s equation in Banach spaces) to Ulam’s question was given by Hyers After his result, a great number of papers (see, for instance, [5, 6, 9, 10, 13, 16, 17, 18] and the references therein) on the subject has been published, generalizing Ulam’s problem and Hyers’ theorem in various directions and to other ( functional) equations. 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 [21]. A lot of papers (see, for instance, [14, 27] and the references therein) on the stability of other equations in such spaces have been published
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