Convexity and Uniform Monotone Approximation of Differentiable Function in Banach Spaces

Abstract

In this paper, the author proves that if the dual X* of X is weakly locally uniformly convex and the convex function f is continuous on X, then there exist two sequences

\{f_n{\}}_{n=1}^{\mathrm{\infty }}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{f_{n}\}_{n=1}^{\infty}$$\end{document}

and

\{g_n{\}}_{n=1}^{\mathrm{\infty }}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{g_{n}\}_{n=1}^{\infty}$$\end{document}

of continuous functions on X** such that (1) f_n(x) ≤ f_n+1(x) ≤ f(x) ≤ g_n+1(x) ≤ g_n(x) whenever x ∈ X; (2) the two convex functions f_n and g_n are Gâteaux differentiable on X; (3) f_n → f and g_n → f uniformly on X. Moreover, if the function f is coercive on X, then (1) f_n and g_n are two w*-lower semicontinuous convex functions on X*; (2)

\text{epi}{f}_n={\overline{\text{epi}{f}_n\cap (X\times R)}}^{w^{\ast }}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{epi}\;f_{n}=\overline{\text{epi}\;f_{n}\cap(X\times R)}^{w^{\ast}}$$\end{document}

and

\text{epi}{g}_n={\overline{\text{epi}{g}_n\cap (X\times R)}}^{w^{\ast }}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{epi}\;g_{n}=\overline{\text{epi}\;g_{n}\cap(X\times R)}^{w^{\ast}}$$\end{document}

.

Keywords

Uniform monotone approximation / Gâteaux differentiable / Weakly locally uniformly convex space / w*-Lower semicontinuous convex functions

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