Abstract
AbstractThe main result of the present paper is about the solutions of the functional equation$$\begin{aligned} F\Big (\frac{x+y}{2}\Big )+f_1(x)+f_2(y)=G(g_1(x)+g_2(y)),\quad x,y\in I, \end{aligned}$$F(x+y2)+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$$F,f_1,f_2,g_1,g_2:I\rightarrow {\mathbb {R}}$$F,f1,f2,g1,g2:I→Rand$$G:g_1(I)+g_2(I)\rightarrow {\mathbb {R}}$$G:g1(I)+g2(I)→Rare the unknown functions assumed to be continuously differentiable with$$0\notin g'_1(I)\cup g'_2(I)$$0∉g1′(I)∪g2′(I), and the setIstands for a nonempty open subinterval of$${\mathbb {R}}$$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 thatFis affine onIand$$g_1$$g1and$$g_2$$g2are continuous functions strictly monotone in the same sense, and secondly that$$g_1$$g1and$$g_2$$g2are invertible affine functions with a common additive part.
Funder
Hungarian Scientific Research Fund
Nemzeti Kutatási Fejlesztési és Innovációs Hivatal
Nemzeti Kutatási, Fejlesztési és Innovaciós Alap
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,General Mathematics
Cited by
2 articles.
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