Convexity of Solutions for an Iterative Equation in Banach Spaces

Abstract

By applying Schauder's fixed point theorem we investigate the existence of increasing (decreasing) solutions of the iterative equation $\mathscr{H}\left(f\right)\circ f=F$ and further give conditions under which those solutions are convex or concave. As corollaries we obtain results on iterative equation $G\left(f\left(x\right),f^{n_1},\dots ,f^{n_k}\left(x\right)\right)=F(x)$ in Banach spaces, where $n_1,n_2,\dots ,n_k\ge 2$.