On conditional Jensen equation (Q2777553)

The author extends some results of \textit{C. Alsina} and \textit{J. L. Garcia-Roig} [in `Functional analysis, approximation theory and numerical analysis', Singapore: World Scientific, 5-7 (1994; Zbl 0877.39015)], \textit{R. Ger} and \textit{J. Sikorska} [Pr. Nauk. Univ. Śląsk. Katowicach, Ann. Math. Silesianae 1665, No. 11, 89-99 (1997; Zbl 0894.39009)]; and \textit{Gy. Szabó} [Publ. Math. 42, No. 3-4, 265-271 (1993; Zbl 0807.39010)] by dealing with the conditional functional equations NEWLINE\[NEWLINE\varphi(x)= \varphi(y) \Rightarrow f\left( {x+y \over 2}\right) ={f(x)+f(y) \over 2}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\varphi(x+y) =\varphi (x-y) \Rightarrow f\left({x+y \over 2}\right) ={f(x)+f(y) \over 2}.NEWLINE\]NEWLINE Under very general assumptions on \(\varphi\) and \(f\) the author proves that \(f\) must be indeed a solution of the Jensen equation. Some results on the stability (in the sense of Hyers and Ulam) of these equations are proven.