A Modified Subgradient Extragradeint Method for Variational Inequality Problems and Fixed Point Problems in Real Banach Spaces

A Modified Subgradient Extragradeint Method for Variational Inequality Problems and Fixed Point Problems in Real Banach Spaces.

ABSTRACT

Let E be a 2-uniformly and uniformly real Banach with dual space E ∗. Let A: C → E ∗ be a and Lipschitz continuous mapping and U: C → C be relatively nonexpansive.

An algorithm for approximating the common elements of the set of fixed points of a relatively nonexpansive map U and the set of solutions of a variational inequality problem for the monotone and Lipschitz continuous map A in E is constructed and proved to strongly. 

TABLE OF CONTENTS

Certification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i
Approval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iv
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 Background of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Variational inequality problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Fixed Point Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Literature Review 4
2.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Nonexpansive Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Theory and Methods 9
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Metric Projection Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.1 Calculating the projection onto a closed convex set in Hilbert spaces . . . . . 19
4 Main Result 23
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Application 29
5.1 Strong Convergence Theorem for a Countable Family of Relatively Nonexpansive
Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32

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INTRODUCTION  

The notion of monotone operators was introduced by Zarantonello [Zarantonello, 1960], Minty [Minty, 1962] and Kac˘urovskii [Kac˘urovskii, 1960]. Monotonicity conditions in the context of variational methods for nonlinear operator equations were also used by Vainberg and Kac˘urovskii [Vainberg et al., 1959].

A map A: D(A) ⊂ H → H is monotone if hAx − Ay, x − yi ≥ 0 ∀x,y ∈ H. Consider the problem of finding the equilibrium states of the system described by du dt + Au = 0, (1.1) where A is a monotone-type mapping on a real Hilbert space. This equation describes the evolution of many physical phenomena which generate energy over time.

It is known that many physically significant problems in different areas of research can be transformed into an equation of the form Au = 0. (1.2) At equilibrium state, equation (1.1) reduces to equation (1.2) whose solutions, in this case, correspond to the equilibrium state of the system described by equation (1.1). 

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BIBLIOGRAPHY

[Alber, 1996] Ya. Alber, (1996) Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (A. G. Kartsatos, Ed.), Marcel Dekker, New York, pp. 15-50.
[Alber et al., 2001] Ya. Alber and S. Guerre-Delabriere (2001), On the projection methods for fixed
point problems, Analysis (Munich), vol. 21, no. 1, pp. 17-39.
[Baiocchi et al., 1984] C. Baiocchi, A. Capelo, Variational and Quasi-Variational Inequalities, Wiley, New York.
[Banach, 1922] S. Banach (1922). Sur les oprations dans les ensembles abstraits et leur application aux e`quations inte`grales. Fundamenta Mathematicae, 3:133-181, 1922.
[Bertsekas et al., 1982] D.P. Bertsekas, E.M. Gafni (1982), Projection methods for variational inequalities with applications to the traffic assignment problem, Math. Prog. Study 17, 139-159.
[Browder, 1965] Browder, FE, (1965): Nonexpansive nonlinear operators in a Banach space. Proc.
Natl. Acad. Sci. USA 54, 1041-1044.
[Censor et al., 2011] Censor, Y., Gibali, A. & Reich, S. (2011), The subgradient extragradient
method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148, 318-335.
[Chidume, 1981] C.E. Chidume (1981), On the approximation of fixed points of nonexpansive mapping, Houston J. math., 345-554.
[Chidume et al., 2005] C. E. Chidume and J. Li (2005), Projection methods for approximating
fixed points of Lipschitz suppressive operators, Panamer. Math. J. 15, no. 1, 29-39.

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