Abstract
We introduce a new general iterative scheme for finding a common element of the set of solutions of variational inequality problem for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive mapping in a Hilbert space and then establish strong convergence of the sequence generated by the proposed iterative scheme to a common element of the above two sets under suitable control conditions, which is a solution of a certain optimization problem. Applications of the main result are also given.
Highlights
Let H be a real Hilbert space with inner product ·, · and induced norm ·
In this paper, motivated by the above-mentioned results, we introduce a new general composite iterative scheme for finding a common point of the set of solutions of the variational inequality problem 1.1 for an inverse-strongly monotone mapping and the set of fixed points of a nonexapansive mapping and prove that the sequence generated by the proposed iterative scheme converges strongly to a common point of the above two sets, which is a solution of a certain optimization problem
The following result for the existence of solutions of the variational inequality problem for inverse strongly-monotone mappings was given in Takahashi and Toyoda 9
Summary
Let H be a real Hilbert space with inner product ·, · and induced norm ·. In 2010, Jung 10 provided the following new composite iterative scheme for the fixed point problem and the problem 1.1 : x1 x ∈ C and yn αnf xn 1 − αn SPC xn − λnAxn , 1.3 xn 1 1 − βn yn βnSPC yn − λnAyn , n ≥ 1, where f is a contraction with constant k ∈ 0, 1 ,{αn},{βn} ∈ 0, 1 , and {λn} ⊂ 0, 2α He proved that the sequence {xn} generated by 1.3 strongly converges strongly to a point in F S ∩ VI C, A , which is the unique solution of a certain variational inequality. In 2007, related to a certain optimization problem, Marino and Xu 14 introduced the following general iterative scheme for the fixed point problem of a nonexpansive mapping: xn 1 αnγ f xn I − αnB Sxn, n ≥ 0, 1.5 where {αn} ∈ 0, 1 and γ > 0 They proved that the sequence {xn} generated by 1.5 converges strongly to the unique solution of the variational inequality. Our results improve and complement the corresponding results of Chen et al 6 , Iiduka and Takahashi 8 , Jung 10 , and others
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