ON A HYERS-ULAM-AOKI-RASSIAS TYPE STABILITY AND A FIXED POINT THEOREM

Let X be a set with a binary operation circle and (Y, d) a complete metric space with a binary operation circle. Take a nonnegative function epsilon on X x X, a nonnegative function 6 on X and two mappings f, g : X -> Y. With the aid of Banach's fixed point theorem, we establish two general settings on which the following holds: If d(f (x circle x'), g(x) circle g(x')) <= epsilon(x, x') and d(f (x), g(x)) <= delta(x) for all x, x' is an element of X, then there exists a unique mapping f(infinity) : X -> Y such that f(infinity)(x circle x') = f(infinity)(x) circle f(infinity)(x'), d(f (x), f(infinity)(x)) <= A epsilon(x,x) + B delta(x) and d(g(x), f(infinity)(x)) <= A epsilon(x,x) C delta(x) for all x, x' is an element of X and some finite constants A, B and C. Moreover, we describe various concrete settings to which the above results are applicable. Some of them are the known results.