Continuity of the Yosida Approximants Corresponding to General Duality Mappings

TL;DR

This paper proves the continuity of Yosida approximants related to general duality mappings in locally uniformly convex Banach spaces, extending previous results and providing applications to maximal monotone operators.

Contribution

It extends the continuity results of Yosida approximants to general duality mappings in locally uniformly convex Banach spaces, beyond the normalized case.

Findings

Established a nondecreasing function ilored to local regions in Banach spaces.

Proved the continuity of Yosida approximants and resolvents for maximal monotone operators.

Discussed pseudomonotone homotopy and Browder degree in this context.

Abstract

Let $X$ be a real locally uniformly convex Banach space and $X^*$ be the dual space of $X$. Let $\varphi:\mathbf R_+\to \mathbf R_+$ be a strictly increasing and continuous function such that $\varphi(0) = 0$, $\varphi(r) \to \infty$ as $r\to\infty$, and let $J_\varphi$ be the duality mapping corresponding to $\varphi$. We will prove that for every $R>0$ and every $x_0\in X$ there exists a nondecreasing function $\psi = \psi (R, x_0) :\mathbf R_+\to \mathbf R_+$ such that $\psi(0) = 0$, $\psi(r)>0$ for $r>0$, and $\langle x^*- x_0^*, x-x_0\rangle \ge \psi(\|x-x_0\|) \|x-x_0\|$ for all $x$ satisfying $\|x-x_0\|\le R$ and all $x^*\in J_\varphi x$ and $x_0^*\in J_\varphi x_0.$ This result extends the previous results of Pr\"{u}ss and Kartsatos who studied the normalized duality mapping $J$ (with $\varphi(r)=r$) for uniformly convex and locally uniformly Banach spaces, respectively. As an application of the above result, we give a concise proof of the continuity of the Yosida approximants $A_\lambda^\varphi$ and resolvents $J_\lambda^\varphi$ of a maximal monotone operator $A:X\supset X\to 2^{X^*}$ on $(0, \infty) \times X$ for an arbitrary $\varphi$ when $X$ is reflexive and both $X$ and $X^*$ are locally uniformly convex. In addition, we discuss pseudomonotone homotopy of the Yosida approximants $A_\lambda^\varphi$ with reference to the Browder degree.