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

The main result of the present paper is about the solutions of the functional equation F (x+y/2 )+f_1(x)+f_2(y)=G(g_1(x)+g_2(y)), x,y∈ I, derived originally, in a natural way, from the invariance problem of generalized weighted quasi-arithmetic means, where F,f_1,f_2,g_1,g_2:I→ℝ and G:g_1(I)+g_2(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 g_1 and g_2 are continuous functions strictly monotone in the same sense, and secondly that g_1 and g_2 are invertible affine functions with a common additive part.