**A Hybrid Algorithm for Approximating a Common Element of Solutions of a Variational Inequality Problem and a Convex Feasibility Problem.**

**ABSTRACT**

In this thesis, a hybrid extragradient-like iteration algorithm for approximating a common element of the set of solutions of a variational inequality problem for a monotone.

*k*–*Lipschitz *map and common fixed points of a countable family of relatively nonexpansive maps in a uniformly smooth and 2-uniformly convex real Banach space is introduced.

A strong convergence theorem for the sequence generated by this algorithm is proved.

The theorem obtained is a significant improvement of the results of Ceng *et al. *(J. Glob. Optim. **46**(2010), 635-646).

Finally, some applications of the theorem are presented.

**TABLE OF CONTENTS **

Certi fication i

Approval ii

Abstract iv

Acknowledgment vi

Dedication viii

1 General Introduction and Literature Review 2

1.1 Background of study . . . . . . . . . 2

1.1.1 Variational Inequality . .. . 2

1.1.2 Fixed Point Theory . . . . 3

1.1.3 Variational Inequality and Fixed Point Problem . . . . 4

1.1.4 Convex Feasibility Problem . . 4

1.2 Statement of Problem . . . . . . 4

1.3 Objective of the Study . . .. . . 4

1.4 Literature Review . . . . . . .. . 4

2 Preliminaries 7

2.1 De nition of terms . . . .. . . . . 7

2.2 Results of Interest . . . . . . . 9

3 Results of Ceng et al. 12

4 Main Results 17

4.1 Applications . . . . 22

5 Bibliography 26

**INTRODUCTION **

**1.1 Background of study**

*“There is no branch of mathematics, however abstract which may not someday be applied to phe- **nomena of the real world”* Lobachevsky.

Attesting to the authenticity of Lobachevsky’s claim, the vast applicability of mathematical models whose constraints can be expressed as fixed point and (or) variational inequality problems in solving real life problems, such as in signal processing, networking, resource allocation, image recovery and so on.

Makes the field of variational inequality and fixed point theory a worthwhile area of re- search [See for example Maainge [2008] and Maainge [2010b] and the references contained in them].

In this thesis, we concentrate on approximating a common element of solutions of a variational inequality problem and common fixed point of a countable family of relatively nonexpansive maps in real Banach spaces.

Hence, the results of this thesis will form major contributions to non- linear operator theory, which falls within the general area of nonlinear functional analysis and applications.

**Variational Inequality**

It was the year 1958, in a classroom, at the *Instituto Nazionale di Alta Mathematica *Italy, that Antonio Signorini posed the problem “what will be the equilibrium configuration of a spherically shaped elastic body resting on a rigid frictionless plane?” A natural question is: what is special about this problem? Its ambiguous boundary condition. In fact, Signorini himself called it “prob- lem with ambiguous boundary condition”.

BIBLIOGRAPHY

Alber, Y. (1996). Metric and generalized projection operators in Banach spaces: Properties and applications. A.G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, in: Lecture Notes Pure Appl. Math., 178:15–50.

Antipin, A. (2000). Methods for solving variational inequalities with related constraints. Journal of Comput. Math. Math.Phys., 40:1239–1254.

Antman, S. (1983). The influence of elasticity in analysis: modern development. . Bulletin of the American Mathematical Society, 9:267–291.

Berinde, V. (2007). Iterative Approximation of Fixed points. Springer.

Buong, N. (2010). Strong convergence theorem of an iterative method for variational inequalities and fixed point problems in Hilbert spaces. Journal of Appl. Math. Comput., 217:322–329.