On the functional equation $$G\left( x,G\left( y,x\right) \right) = G\left( y,G\left( x,y\right) \right) $$Gx,Gy,x=Gy,Gx
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On the functional equation G (x, G (y, x)) = G (y, G (x, y)) and means Lin Li1
· Janusz Matkowski2
© Akadémiai Kiadó, Budapest, Hungary 2019
Abstract We consider the functional equation G (x, G (y, x)) = G (y, G (x, y)), posed in Jarczyk and Jarczyk (Aequ Math 72:198–200, 2006). We show that every continuous and reducible solution generates a mean resembling the weighted quasi-arithmetic mean, but no weighted quasi-arithmetic mean is a solution of this equation. This fact particularly implies that the equation is not a direct consequence of the bisymmetry equation and the reflexivity condition. The closedness of the family of solutions with respect to conjugacy is noted. Finally, the translative solutions, homogeneous solutions, and suitable iterative composite functional equation for single variable functions are discussed. Keywords Rational function · Mean · Invariant mean · Iteration Mathematics Subject Classification Primary 33B15 · 26E30
1 Introduction The functional equation G (x, G (y, x)) = G (y, G (x, y))
(1.1)
for G : R2 → R, was posed by Jarczyk and Jarczyk in [3] when they considered the following equation x + f (y + f (x)) = y + f (x + f (y)) , (1.2) which, after setting G (x, y) = x + f (y), leads to Eq. (1.1). The authors showed that there is no continuous solution f : R → R of Eq. (1.2). However, this conclusion is not true on
B
Lin Li [email protected] Janusz Matkowski [email protected]
1
College of Mathematics, Physics and Information Engineering, Jiaxing University, Zhejiang 314001, People’s Republic of China
2
Computer Science and Econometrics, Faculty of Mathematics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Gora, Poland
123
L. Li, J. Matkowski
complex plane C (see [7]). As a consequence of [3,7], Balcerowski [2] proved that every solution f : H → H of Eq. (1.2) on a abelian group H is one-to-one. Furthermore, 2 f is additive with the additional condition that f (0) = 0. Later, J. Rätz showed by an example that the property of additivity for f is not necessary in general (for more results on abelian group, see J. Rätz’s papers [5,6]). As observed in [2,5,6], injectivity with respect to each variable, called reducibility, is an important property for solutions. In this note we are mainly interested in the continuous solutions of Eq. (1.1), which was one of the open problems listed in [3]. Contrary to Eq. (1.2), there exist infinitely many continuous solutions G : R2 → R of Eq. (1.1), which are not reducible with respect to each variable. The suitable examples are given in Sect. 2. The main results in Sect. 2 show that there is a close relationship of the solutions of Eq. (1.1) and means. More precisely, every continuous and reducible solution generates a mean resembling the classical non-symmetric weighted quasi-arithmetic mean (Theorem 2.2). Additionally, if G is reflexive, then it is a non-symmetric strict mean. It can be little surprising that no weighted quasi-arithmetic mean is a solution of this equation. This fact particularly implies that Eq. (1.1) i
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