Regular solutions of a functional equation derived from the invariance problem of Matkowski means

Abstract

The main result of the present paper is about the solutions of the functional equation Fx + y 2 + f1(x) + f2(y) = G(g1(x) + g2(y)), x, y ∈ I, derived originally, in a natural way, from the invariance problem of generalized weighted quasi-arithmetic means, where F, f1, f2, g1, g2 : I → R and G : g1(I) + g2(I) → R are the unknown functions assumed to be continuously differentiable with 0 / ∈ g ′ 1 (I) ∪ g ′ 2 (I), and the set I stands for a nonempty open subinterval of R. In addition to these, we will also touch upon solutions not necessarily regular. More precisely, we are going to solve the above equation assuming first that F is affine on I and g1 and g2 are continuous functions strictly monotone in the same sense, and secondly that g1 and g2 are invertible affine functions with a common additive part.We shall say that f is locally affine on J or locally constant on J if f is affine or constant on some subinterval U ⊆ J of positive length different from J. If f is neither affine (resp. constant) nor locally affine (resp. locally constant) on J, then it will be called nowhere affine (resp. nowhere constant) on J.We say that a two-place function M : J × J → R is a two-variable mean on J or, shortly, a mean ifElementary examples of means are the harmonic mean and the geometric mean over the set of positive reals, and the arithmetic mean. Rather than giving furter examples, we immediately introduce a general family of twovariable means, which covers the previous examples and which, incidentally, is also the object of our investigation.A mean M : J ×J → R will be called a generalized weighted quasi-arithmetic mean if one can find continuous functions f, g : J → R strictly monotone in the same sense such that (0.1)where the functions f and g are called the generators of the mean in question. If the generators differ on J in an additive constant, then we come to the notion of quasi-arithmetic means. To see that our previous examples are