Strong convergence theorems for strongly monotone mappings in Banach spaces

Abstract

Let $E$ be a uniformly smooth and uniformly convex real Banach space and $E^{\ast }$ be its dual space. Suppose $A:E\to E^{\ast }$ is bounded, strongly monotone and satisfies the range condition such that $A^{-1}(0)\ne \varnothing$. 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 $Ax=0$.

12 pages. Boletim da Sociedade Paranaense de Matematica, (2018)