On a functional equation related to two-variable weighted quasi-arithmetic means
Abstract
In this paper, we are going to describe the solutions of the functional equation (x+y2)(f(x)+f(y))=(x)f(x)+(y)f(y) concerning the unknown functions and f defined on an open interval. In our main result only the continuity of the function and a regularity property of the set of zeroes of f are assumed. As application, we determine the solutions of the functional equation G(g(u)-g(v))=H(h(u)+h(v))+F(u)+F(v) under monotonicity and differentiability conditions on the unknown functions F,G,H,g,h.
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