The Convergence of Three-Step Iterative Schemes for Generalized Φ − Hemi-Contractive Mappings and the Comparison of Their Rate of Convergence

Charles proved the convergence of Picard-type iteration for generalized Φ − accretive nonself-mappings in a real uniformly smooth Banach space. Based on the theorems of the zeros of strongly Φ − quasi-accretive mappings and fixed points of strongly Φ − hemi-contractions, we extend the results to Noor iterative process and SP iterative process for generalized Φ − hemi-contractive mappings. Finally, we analyze the rate of convergence of four iterative schemes, namely, Noor iteration, iteration of Corollary 2, SP iteration, and iteration of Corollary 4.

A Method for Solving the Variational Inequality Problem and Fixed Point Problems in Banach Spaces

The purpose of this research is to modify Halpern iteration’s process for finding a common element of the set of solutions of a variational inequality problem and the set of fixed points of a strictly pseudo contractive mapping in q-uniformly smooth Banach space. We also introduce a new technique to prove a strong convergence theorem for a finite family of strictly pseudo contractive mappings in q-uniformly smooth Banach space. Moreover, we give a numerical result to illustrate the main theorem.

The Convergence of Mann Iteration for Generalized Φ- hemi-contractive Maps

Charles[1] proved the convergence of Picard-type iterative for generalized Φ− accretive non-self maps in a real uniformly smooth Banach space. Based on the theorems of the zeros of strongly Φ − accretive and fixed points of strongly Φ− hemi-contractive we extend the results to Mann-type iterative and Mann iteration process with errors.

Complementation of the subspace of G-invariant vectors

Let [Formula: see text] be an isometric representation of a group [Formula: see text] in a Banach space [Formula: see text] over a normalizing non-discrete absolute valued division ring [Formula: see text]. If [Formula: see text] and [Formula: see text] are supportive and [Formula: see text] verifies the separation property, then [Formula: see text] is 1-complemented in [Formula: see text] along [Formula: see text]. As an immediate consequence, in an isometric representation of a group in a smooth Banach space whose dual is also smooth, the subspace of [Formula: see text]-invariant vectors is [Formula: see text]-complemented.