Gâteaux differentiability and uniform monotone approximation of convex functions in Banach spaces

Abstract

We prove that if $X^{*}$ is strictly convex, a convex function $f$ is coercive and b-Lipschitzian iff there exist two convex function sequences $\{f_{n}\}_{n=1}^{\infty}$ and $\{g_{n}\}_{n=1}^{\infty}$ such that (1) $f_{n}\leq f_{n+1}\leq f$ and $f\leq g_{n+1}\leq g_{n}$ for all integers $n \geq 1$; (2) $f_{n}$ and $g_{n}$ are continuous and Gâteaux differentiable on $X$; (3) $f_n \to f$ and ${g_n \to f}$ uniformly on $X$; (4) $f_n$ and $g_n$ are coercive and b-Lipschitzian. Moreover, we also prove that if $X^{*}$ is strictly convex, then a convex function f is Lipschitzian iff conditions (1)-(3) are true and there exists $m>0$ such that $|f_{n}(x)-f_{n}(y)|\leq m\|x-y\|$ and $|g_{n}(x)-g_{n}(y)|\leq m\|x-y\|$ whenever $x, y \in X$ and $n\in N$.