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Algorithms For Approximation Of Solutions Of Equations Involving Nonlinear Monotone-Type And Multi-Valued Mappings

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– Algorithms For Approximation Of Solutions Of Equations Involving Nonlinear Monotone-Type And Multi-Valued Mappings –

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Abstract

It is well know that many physically significant problems in different areas of research can be transformed into an equation of the form Au = 0, (0.0.1) where A is a nonlinear monotone operator from a real Banach space E into its dual E∗ .

For instance, in optimization, if f : E −→ R ∪ {+∞} is a convex, Gˆateaux differentiable function and x ∗ is a minimizer of f, then f 0 (x ∗ ) = 0. This gives a criterion for obtaining a minimizer of f explicitly.

However, most of the operators that are involved in several significant optimization problems are not differentiable. For instance, the absolute value function x 7→ |x| has a minimizer, which, in fact, is 0. But, the absolute value function is not differentiable at 0.

Introduction

The contents of this thesis fall within the general area of nonlinear functional analysis, an area which has attracted the attention of prominent mathematicians due to its diverse applications in numerous fields of sciences. The contributions of this thesis concentrate mainly on the following three important topics.

Namely; • Approximation of zeros of nonlinear monotone mappings in classical Banach spaces. • Approximation of fixed points of a finite family of k-strictly pseudo-contractive mappings in CAT(0) spaces, and a countable family of k-strictly pseudocontractive maps in Hilbert spaces. •

Approximating solutions of Integral equations of Hammerstein-type with monotone operators in Banach spaces.

It is well known that many physically significant problems in different areas of research can be transformed into an equation of the form Au = 0, (1.1.1) where A is a nonlinear monotone operator defined on a real Banach space E.

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