Weak and Strong Convergence of an Iterative Algorithm for Lipschitz Pseudo-Contractive Maps in Hilbert Spaces
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Weak and Strong Convergence of an Iterative Algorithm for Lipschitz Pseudo-Contractive Maps in Hilbert Spaces

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– Weak and Strong Convergence of an Iterative Algorithm for Lipschitz Pseudo-Contractive Maps in Hilbert Spaces –

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Abstract

Let H be a real Hilbert space and K a nonempty, closed convex subset of H.Let T : K K be Lipschitz pseudo-contractive map with a nonempty fixed points set.

We introduce a modified Ishikawa iterative algorithm for Lipschitz pseudo-contractive maps and prove that our new iterative algorithm converges strongly to a fixed point of T in real Hilbert space.

Introduction

1.1 Background of the Study

The contribution of this thesis falls under a branch of mathematics called  Functional  Analysis.

Functional Analysis as  an  independent  mathematical  discipline  started  at  the turn of the 19th century and was finally established in 1920’s and 1930’s, on one hand under the influence of the  study of  specific classes of  linear  operators-integral  operators.

Integral equations connected with them-and on the other hand under the influence of the purely intrinsic development of modern mathematics with its desire to generalize and thus to clarify the true nature of some regular behaviour.

Quantum Mechanics also had a great influence on the development of Functional Analysis, since its basic concepts, for example energy, turned out to be linear operators on infinite dimensional spaces.

In  the  early stages of the development of Functional Analysis the  problems studied were those that could be stated and solved in terms of linear operators on elements of the space alone.

But as the concept of a space was being developed and deepened, the concept of a function was being developed and generalized. In the end, it became  necessary  to  consider  mapping (not necessary linear) from one space into another.

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