In this paper, we prove the Hyers-Ulam stability and the
Hyers-Ulam-Rassias stability of the following two functional equations

    \begin{equation*} \varphi (x)=x\varphi ((1-\alpha )x+\alpha )+(1-x)\varphi ((1-\beta )x),\text{ }x\in \lbrack 0,1],\text{ }0<\alpha \leq \beta <1, \end{equation*}


    \begin{equation*} \varphi (x)=x\varphi (f(x))+(1-x)\varphi (g(x)),\text{ }x\in \lbrack 0,1] \end{equation*}

which is an open problem raised by Berinde and Khan [Berinde, V. and Khan, A. R., On a functional equation
arising in mathematical biology and theory of learning, Creat. Math. Inform., 24 (2015), No. 1, 9–16].

Additional Information


Arisoy, Hakan, Kalkan, Zeynep, Şahin, Aynur