Abstract
The motivation behind this paper is to use hybrid method for searching a typical component of the set of fixed point of an infinite family of non expansive mapping and the set of monotone, Lipschtiz continuous variational inequality problem. The contemplated method is combination of two method one is extragradient method and the other one is DQ method. Also, we demonstrate the strong convergence of the designed iterative technique, under some warm conditions.
Highlights
Throughout this paper, let H be a real Hilbert Space with norm || · || and inner product < ·, · >
Many algorithms for solving variational inequality problem is considered and proposed ( [3], [5], [6], [7], [8], [13], [17], [16]) Let us start with one of the method which is used in our paper, i.e., Korpelevich’s extragradient method which was popularized by Korpelevich [9] in 1976 and which initiate a sequence {an} defined as: bn = PD(an − λAan) an+1 = PD(an − λAbn), n ≥ 0 where PD is the metric projection from Rn onto D, A : D → H is a monotone operator and λ is a constant
We demonstrated the strong convergence of the designed iterative technique, under some warm conditions
Summary
Throughout this paper, let H be a real Hilbert Space with norm || · || and inner product < ·, · >. Many algorithms for solving variational inequality problem is considered and proposed ( [3], [5], [6], [7], [8], [13], [17], [16]) Let us start with one of the method which is used in our paper, i.e., Korpelevich’s extragradient method which was popularized by Korpelevich [9] in 1976 and which initiate a sequence {an} defined as: bn = PD(an − λAan) an+1 = PD(an − λAbn), n ≥ 0 where PD is the metric projection from Rn onto D, A : D → H is a monotone operator and λ is a constant. Korpelevich’s extragradient technique has widely been read for the solution of finding common point which belong to the solution set of fixed points of a nonexpansive mapping and variational inequality. We demonstrated the strong convergence of the designed iterative technique, under some warm conditions
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