On a functional equation arising in mathematical biology and theory of learning

Abstract

V. Istrat¸escu [Istr ˘ at¸escu, V. I., ˘ On a functional equation, J. Math. Anal. Appl., 56 (1976), No. 1, 133–136] used the Banach contraction mapping principle to establish an existence and approximation result for the solution of the functional equation ϕ(x) = xϕ((1 − α)x + α) + (1 − x)ϕ((1 − β)x), x ∈ [0, 1], (0 < α ≤ β < 1), which is important for some mathematical models arising in biology and theory of learning. This equation has been studied by Lyubich and Shapiro [A. P. Lyubich, Yu. I. and Shapiro, A. P., On a functional equation (Russian), Teor. Funkts., Funkts. Anal. Prilozh. 17 (1973), 81–84] and subsequently, by Dmitriev and Shapiro [Dmitriev, A. A. and Shapiro, A. P., On a certain functional equation of the theory of learning (Russian), Usp. Mat. Nauk 37 (1982), No. 4 (226), 155–156]. The main aim of this note is to solve this functional equation with more general arguments for ϕ on the right hand side, by using appropriate fixed point tools.