On approximate solutions of the Pexider equation (Q1312266)

The following functional inequality is considered: \[ \bigl \| f(x + y) - g(x) - h(y) \bigr \| \leq \varepsilon \max \biggl \{ \bigl \| f(x + y) - f(0) \bigr \|, \bigl \| g(x) - g(0) + h(y) - h(0) \bigr \| \biggr\}\tag{1} \] where all the functions are unknown and map an abelian group into a normed space and \(\varepsilon\) is a given number from [0,1). It is shown that the solutions to (1) have similar properties as those to the Pexider equation \(f(x + y) = g(x) + h(y)\). Continuity and monotonicity of solutions are also investigated. The case of ``min'' in place of ``max'' in (1) is also discussed. The inequality (1) was first proposed and studied by \textit{J. Tabor} [ibid. 35, No. 2/3, 164-185 (1988; Zbl 0652.39013); ibid. 39, No. 2/3, 179-197 (1990; Zbl 0708.39005)] in the case where \(f = g = h\) (then necessarily \(f(0) = g(0) = h(0) = 0)\).