In this article, we prove the generalized Hyers-Ulam-Rassias stability for the following composite functional equation: f(f(x) – f(y)) = f(x + y) + f(x – y) – f(x) – f(y), where f maps from a(β, p)-Banach space into itself, by using the fixed point method and the direct method. Also, the generalized Hyers-Ulam-Rassias stability for the composite s-functional inequality is discussed via our results.
There are presented conditions which guarantee the existence of a Lip- schitzian solution of the composite functional equation phi(x) = G (x,phi(H(x,phi(x)))) are presented.
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We determine all continuous functions f : (0, oo) ->(0,oo) satisfying the functional equation f(xG(f(x)))=f(x)G(f(x)) where G is continuous and strictly increasing function such that 1 e G((0, oo)).
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