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The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces. (English) Zbl 1221.39036

Suppose that X is a linear space, (Z,N ' ) a fuzzy normed space, (Y,N) a fuzzy Banach space, f:XY, n2 a fixed integer.

The authors consider the stability of the functional equation

Δf(x 1 ,,x n )=0

where

Δf(x 1 ,,x n )= i=1 n fx i -1 n j=1 n x j - i=1 n f(x i )+nf1 n i=1 n x i ·

The main result reads, roughly, as follows. Suppose that f:XY satisfy f(0)=0 and

N(Δf(x 1 ,,x n ),t 1 ++t n )min{N ' (ϕ(x 1 ),t 1 ),,N ' (ϕ(x n ),t n )}

for all x 1 ,,x n X, t 1 ,,t n >0 where ϕ:X(Z,N ' ) is a control mapping satisfying ϕ(2x)=αϕ(x), xX with some |α|<2.

Then, there exists a unique quadratic function Q:XY and a unique additive function A:XY such that the mapping Q+A approximates f (in terms of the fuzzy norm N).

Moreover, if f is odd, it can be approximated by an additive function A and if f is even, a quadratic mapping Q approximates f.

MSC:
39B82Stability, separation, extension, and related topics
46S40Fuzzy functional analysis
47H10Fixed point theorems for nonlinear operators on topological linear spaces
39B52Functional equations for functions with more general domains and/or ranges