# Continuity of Yosida Approximants Corresponding to General Duality Mappings

## Department

Mathematics

## Document Type

Article

## Publication Date

1-1-2023

## Abstract

Let X be a real locally uniformly convex Banach space and X∗ be the dual space of X. Let φ: R+ → R+ be a strictly increasing and continuous function such that φ(0) = 0, φ(r) → ∞ as r → ∞, and let Jφ be the duality mapping of X corresponding to φ. We will prove that for every R > 0 and every x0 ∈ X there exists a nondecreasing function ψ = ψ(R, x0): R+ → R+ such that ψ(0) = 0, ψ(r) > 0 for r > 0, and hx ∗ − x ∗ 0, x − x0i ≥ ψ(kx − x0k)kx − x0k for all x satisfying kx − x0k ≤ R and all x ∗ ∈ Jφx and x ∗ 0 ∈ Jφx0. This result extends the previous results of Prüß and Kartsatos who studied the normalized duality mapping J (with φ(r) = r) for uniformly convex and locally uniformly Banach spaces, respectively. As an application, we give a concise proof of the continuity of the Yosida approximants A φ λ and resolvents J φ λ of a maximal monotone operator A: X ⊃ D(A) → 2 X∗ on (0, ∞) × X for an arbitrary φ when X is reflexive and both X and X∗ are locally uniformly convex. We then present an example of pseudomonotone homotopy involving Aφλ on which the Browder degree is invariant. We also discuss examples of positively homogeneous maximal monotone operators to which the theory developed herein is applicable.

## Journal Title

Advances in Mathematical Sciences and Applications

## Journal ISSN

13434373

## Volume

32

## Issue

1

## First Page

59

## Last Page

71

## Digital Object Identifier (DOI)

NA