Abstract
AbstractThe 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.
Highlights
Let H be a real Hilbert space with inner product ·, · and norm ·, respectively
We consider the following general variational inequality problem over the fixed point set of nonlinear mapping in the framework of Banach space
Problem 1.1. general variational inequality problem over the fixed point set of nonlinear mapping
Summary
Let H be a real Hilbert space with inner product ·, · and norm · , respectively. Let C be a nonempty closed convex subset of H and D a nonempty closed convex subset of C. We consider the following general variational inequality problem over the fixed point set of nonlinear mapping in the framework of Banach space. Does sequence {xn}, defined by 1.6 , converges strongly a solution to a general variational inequality problem in the Banach space setting, that is, Problem 1.1 in a case where T : C → C is given as such a nonexpansive mapping?. Our strong convergence theorems extend and improve corresponding results of Ceng et al ; Ceng et al ; Lehdili and Moudafi 19 ; Sahu 9 ; and Yamada 6
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