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On some generalizations of Gołąb-Schinzel functional equation

Matkowski J., Okrzesik J.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2010

Vol. 15

81--90

Composite functional equations in several variables generalizing the Gołąb-Schinzel equation are considerd and some simple methods allowing us to determine their one-to-one solutions, bijective solutions or the solutions having exactly one zero are presented. For an arbitrarily fixed real p, the functional equation Φ([pφ(y) + (1−p)]x +[(1−p)φ(x)+p]y) = φ(x)φ(y), x,y ∈ R, being a special generalization of the Gołąb-Schinzel equation, is considered.

One-to-one solutions of generalized Gołąb-Schinzel equation

Jabłońska E.

Demonstratio Mathematica

2008

Vol. 41, nr 3

583-588

Let K be the field of real or complex numbers and let X be a nontrivial linear space over K. Assume that [...]. We give a necessary and sufficient condition for functions f and M to satisfy the equation The functional equation f(x+M(f(x))y)=f(x)f(y) is a generalization of the well-known Gołąb-Schinzel functional equation f(x+f(x)y)=f(x)f(y).