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

 – 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.

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INTRODUCTION

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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REFERENCES

Y.I. Alber, ”Metric and Generarized Projection Operators in Banach Spaces : prop- erties and applications,” in Theory and Applications of Non linear operators of Accretive and Monotone Type, vol.178 of Lecture Notes in Pure and Applied Math- ematics, pp. 15-50, Marcel Dekker, New York, NY, USA,1996.

Asplund, Positivity of Duality Mappings, Bull. Amer. Math. Soc., 73 (1967), 200-203

Banach, Sur les Operations Dans Les Emsembles Abstraits et Leur Application Aux Equations Integrals. Fundamenta Mathematicae 3, 133-181 (1922).

H.H. Bauschke, J. Borwein, On Projection Algorithms for Solving Convex Feasibility Problems. SIAM Rev. 38, 367-426 (1996).

Berinde, Generalized Contractions and Applications (Romanian), Editura cub press 22, Baiamare, 1997.

Berinde, Iterative Approximation of Fixed Points (Springer, Berlin, 2007).

Borwein and J.M. Bowein, Fixed Point Iterations for Real Functions, J.Math.Anal.Appl. 157(1991), 112-126.

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