Hybrid Steepest-Descent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Mon
- PDF / 283,538 Bytes
- 17 Pages / 600.05 x 792 pts Page_size
- 106 Downloads / 202 Views
Research Article Hybrid Steepest-Descent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators Songnian He and Xiao-Lan Liang College of Science, Civil Aviation University of China, Tianjin 300300, China Correspondence should be addressed to Songnian He, [email protected] Received 30 September 2009; Accepted 13 January 2010 Academic Editor: Tomonari Suzuki Copyright q 2010 S. He and X.-L. Liang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let H be a real Hilbert space and let F : H → H be a boundedly Lipschitzian and strongly monotone operator. We design three hybrid steepest descent algorithms for solving variational inequality VIC, F of finding a point x∗ ∈ C such that Fx∗ , x − x∗ ≥ 0, for all x ∈ C, where C is the set of fixed points of a strict pseudocontraction, or the set of common fixed points of finite strict pseudocontractions. Strong convergence of the algorithms is proved.
1. Introduction Let H be a real Hilbert space with the inner product ·, · and the norm · , let C be a nonempty closed convex subset of H, and let F : C → H be a nonlinear operator. We consider the problem of finding a point x∗ ∈ C such that
Fx∗ , x − x∗ ≥ 0,
∀x ∈ C.
1.1
This is known as the variational inequality problem i.e., VIC, F, initially introduced and studied by Stampacchia 1 in 1964. In the recent years, variational inequality problems have been extended to study a large variety of problems arising in structural analysis, economics, optimization, operations research, and engineering sciences; see 1–6 and the references therein.
2
Fixed Point Theory and Applications
Yamada 7 proposed hybrid methods to solve VIC, F, where C is composed of fixed points of a nonexpansive mapping; that is, C is of the form C ≡ FixT : {x ∈ H : T x x},
1.2
where T : H → H is a nonexpansive mapping i.e., T x − T y ≤ x − y for all x, y ∈ H, F : H → H is Lipschitzian and strongly monotone. He and Xu 8 proved that VIC, F has a unique solution and iterative algorithms can be devised to approximate this solution if F is a boundedly Lipschitzian and strongly monotone operator and C is a closed convex subset of H. In the case where C is the set of fixed points of a nonexpansive mapping, they invented a hybrid iterative algorithm to approximate the unique solution of VIC, F and this extended the Yamada’s results. The main purpose of this paper is to continue our research in 8. We assume that F is a boundedly Lipschitzian and strongly monotone operator as in 8, but C is the set of fixed points of a strict pseudo-contraction T : H → H, or the set of common fixed points of finite strict pseudo-contractions Ti : H → H i 1, . . . , N. For the two cases of C, we will design the hybrid iterative algorithms for solving VIC, F and prove their strong convergence, res
Data Loading...
