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

• Published in: Aequationes mathematicae Vol. 96; no. 5; pp. 1089 - 1124
• Main Author: Kiss, Tibor
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.10.2022; Springer Nature B.V
• Subjects: Analysis; Combinatorics; Continuity (mathematics); Functional equations; Invariance; Mathematical analysis; Mathematics; Mathematics and Statistics; Invariance of means; Invariance problem; Matkowski means; Primary 39B22; Generalized weighted quasi-arithmetic means; Secondary 26E60; 39B72
• ISSN: 0001-9054; 1420-8903
• DOI: 10.1007/s00010-022-00880-8

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.