Explicit Iteration Methods for Solving Variational Inequalities in Banach Spaces (original) (raw)
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Let X be a uniformly convex and smooth Banach space which admits a weakly sequentially continuous duality mapping, C a nonempty bounded closed convex subset of X. Let S = {T(s) : 0 ≤ 0 < ∞} be a nonexpansive semigroup on C such that F(S ) = ∅ and f : C → C is a contraction mapping with coefficient α ∈ (0, 1), A a strongly positive linear bounded operator with coefficientγ > 0. We prove that the sequences {x t } and {x n } are generated by the following iterative algorithms, respectively Corresponding author: poom.kum@kmutt.ac.th (Poom Kumam) Manuscript
Mathematical and Computer Modelling, 2011
In this paper, we introduce a composite explicit viscosity iteration method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces. We prove strong convergence theorems of the composite iterative schemes which solve some variational inequalities under some appropriate conditions. Our result extends and improves those announced by Li et al [General iterative methods for a one-parameter nonexpansive semigroup in Hilbert spaces, Nonlinear Anal. 70 3065-3071], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Modelling 48 (2008) 279-286], Plubtieng and Wangkeeree [S. Plubtieng, R. Wangkeeree, A general viscosity approximation method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Bull. Korean Math. Soc. 45 (4) (2008) 717-728] and many others.
Hybrid Algorithms of Nonexpansive Semigroups for Variational Inequalities
Journal of Applied Mathematics, 2012
Two hybrid algorithms for the variational inequalities over the common fixed points set of nonexpansive semigroups are presented. Strong convergence results of these two hybrid algorithms have been obtained in Hilbert spaces. The results improve and extend some corresponding results in the literature.
A Generalized Hybrid Steepest-Descent Method for Variational Inequalities in Banach Spaces
Fixed Point Theory and Applications, 2011
The hybrid steepest-descent method introduced by Yamada 2001 is an algorithmic solution to the variational inequality problem over the fixed point set of nonlinear mapping and applicable to a broad range of convexly constrained nonlinear inverse problems in real Hilbert spaces. Lehdili and Moudafi 1996 introduced the new prox-Tikhonov regularization method for proximal point algorithm to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in Hilbert spaces. In this paper, motivated by Yamada's hybrid steepest-descent and Lehdili and Moudafi's algorithms, a generalized hybrid steepest-descent algorithm for computing the solutions of the variational inequality problem over the common fixed point set of sequence of nonexpansive-type mappings in the framework of Banach space is proposed. The strong convergence for the proposed algorithm to the solution is guaranteed under some assumptions. Our strong convergence theorems extend and improve certain corresponding results in the recent literature.
On an iterative algorithm for variational inequalities in Banach spaces
In this paper, we suggest and analyze a new iterative method for solving some variational inequality involving an accretive operator in Banach spaces. We prove the strong convergence of the proposed iterative method under certain conditions. As a special of the proposed algorithm, we proved that the algorithm converges strongly to the minimum norm solution of some variational inequality.
A hybrid method for solving variational inequality problems
Applied Mathematics-A Journal of Chinese Universities, 2000
In this paper, we introduce a hybrid method, a combination of the steepest-descent method and the Krasnosel'skii-Mann one, for solving a variational inequality over the set of common fixed points of an infinite family of nonexpansive mappings in Banach spaces under two different conditions on the Banach space, either a uniformly smooth Banach space or a reflexive and strictly convex one with a uniformly Gâteaux differentiable norm, without imposing the sequential weak continuity of the normalized duality mapping. The method is an improvement and extension of some other published results. We also give a numerical example to illustrate the convergence analysis of the proposed method.