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Abstract

Let $0<\beta<\alpha<1$ and let $p\in (0,1)$. We consider the functional equation $$ \varphi(x)=p\varphi \biggl(\frac{x-\beta}{1-\beta}\biggr) +(1-p)\varphi \biggl(\!\min\biggl\{\frac{x}{\alpha}, \frac{x(\alpha-\beta)+\beta(1-\alpha)}{\alpha(1-\beta)}\biggr\}\biggr) $$ and its solutions in two classes of functions, namely $$\eqalign{ {\cal I}&=\{\varphi\colon\mathbb R\to\mathbb R\mid \varphi\hbox{ is increasing, } \varphi|_{(-\infty,0]}=0,\,\varphi|_{[1,\infty)}=1\},\cr {\cal C}&=\{\varphi\colon\mathbb R\to\mathbb R\mid \varphi\hbox{ is continuous, } \varphi|_{(-\infty,0]}=0,\,\varphi|_{[1,\infty)}=1\}.}$$ We prove that the above equation has at most one solution in $\mathcal C$ and that for some parameters $\alpha,\beta$ and $p$ such a solution exists, and for some it does not. We also determine all solutions of the equation in $\mathcal I$ and we show the exact connection between solutions in both classes.