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A fixed point approach to stability of a quadratic equation. (English) Zbl 1118.39015

The authors prove the orthogonal stability of the quadratic functional equation of Pexider type by using the fixed point alternative theorem. The following theorem is the main result of this paper: Suppose that X is a real orthogonality space with a symmetric orthogonal relation and Y is a Banach space. Let the mappings f,g,h,k:XY satisfy

f(x+y)+g(x-y)-h(x)-k(y)ε

for all x,yX with xy. There exist an orthogonally additive mapping T:XY and a constant C 1 0 such that

f(x)-T(x)C 1 ε(forallxX)

if and only if there exists a constant C 2 0 with

f(2x)-f(-2x)-4f(x)-4f(-x)C 2 ε(forallxX)·

Indeed, if f:XY satisfies

f(2x)-f(-2x)-4f(x)-4f(-x)ε(forallxX),

then there exist orthogonally additive mappings T 1 ,T 2 ,T 3 :XY such that

f(x)-f(0)-T 1 (x)140 3ε,g(x)-g(0)-T 2 (x)98 3ε,h(x)+k(x)-h(0)-k(0)-T 3 (x)256 3ε

for all xX.

In the Introduction, quoting the stability of quadratic equations of Pexider type, the authors missed citing a paper “Stability of the quadratic equation of Pexider type” [Abh. Math. Semin. Univ. Hamb. 70, 175–190 (2000; Zbl 0991.39018)] by S.-M. Jung, which contains the first result about the stability of quadratic equations of Pexider type (see Theorem 5 and Corollary 6 in the paper).


MSC:
39B82Stability, separation, extension, and related topics
39B55Orthogonal additivity and other conditional functional equations
39B52Functional equations for functions with more general domains and/or ranges