Strong convergence theorems for strongly monotone mappings in Banach spaces

Abstract

Let $E$ be a uniformly smooth and uniformly convex real Banach space and $E^*$ 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 \cite{b1}, 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 $Ax=0$.

On generalized Φ-strongly monotone mappings and algorithms for the solution of equations of Hammerstein type

“…Our results generalize and improve the recent and important results of Chidume and Idu [10]. Also, our results show extention and application of the main results of Aibinu and Mewomo [1,2].…”

“…Definition 2.1. Let E be a smooth real Banach space with dual space E * , the followings were introduced by Aibinu and Mewomo [2].…”

“…Lemma 2.3. Aibinu and Mewomo [2]. Let E be a smooth uniformly convex real Banach space with E * as its dual.…”

“…Lemma 2.5. Aibinu and Mewomo [2]. Let E be a reflexive strictly convex and smooth real Banach space with the dual E * .…”

Algorithm for Solutions of Nonlinear Equations of Strongly Monotone Type and Applications to Convex Minimization and Variational Inequality Problems

“…In the sequel, we shall need the lemmas whose proofs have been established (see, e.g., Alber [26] and Aibinu and Mewomo [4]).…”