Strong convergence theorems for strongly monotone mappings in Banach spaces

TL;DR

This paper establishes strong convergence theorems for iterative algorithms solving strongly monotone operator equations in uniformly smooth and convex Banach spaces, using Lyapunov functions and geometric properties.

Contribution

It introduces new convergence results for strongly monotone mappings in Banach spaces, extending previous work with novel Lyapunov functions and geometric analysis.

Findings

Proves strong convergence of iterative algorithms to solutions of $Ax=0$.

Utilizes Lyapunov functions and geometric properties of Banach spaces.

Extends convergence theory for strongly monotone operators in Banach spaces.

Abstract

Let $E$ be a uniformly smooth and uniformly convex real Banach space and $E^*$ be its dual space. Suppose $A : E\rightarrow E^*$ is bounded, strongly monotone and satisfies the range condition such that $A^{-1}(0)\neq \emptyset$. Inspired by Alber [2], we introduce Lyapunov functions and use the new geometric properties of Banach spaces to show the strong convergence of an iterative algorithm to the solution of $Ax=0$.