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CONTINUOUS ON RAYS SOLUTIONS OF A GOŁA̧B–SCHINZEL TYPE EQUATION

Published online by Cambridge University Press:  15 December 2014

JACEK CHUDZIAK
JACEK CHUDZIAK*
Affiliation:
Faculty of Mathematics and Natural Sciences, University of Rzeszów, Prof. St. Pigonia 1, 35-310 Rzeszów, Poland email chudziak@ur.edu.pl
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Abstract

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We show that if the pair $(f,g)$ of functions mapping a linear space $X$ over the field $\mathbb{K}=\mathbb{R}\text{ or }\mathbb{C}$ into $\mathbb{K}$ satisfies the composite equation

$$\begin{eqnarray}f(x+g(x)y)=f(x)f(y)\quad \text{for }x,y\in X\end{eqnarray}$$
and $f$ is nonconstant, then the continuity on rays of $f$ implies the same property for $g$. Applying this result, we determine the solutions of the equation.

Keywords

Goła̧b–Schinzel equationcontinuity on raysalgebraically interior pointexponential functionmultiplicative function

MSC classification

Primary: 39B12: Iteration theory, iterative and composite equations
Secondary: 39B22: Equations for real functions 39B32: Equations for complex functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 2 , April 2015 , pp. 273 - 277
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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