Abstract
We introduce a new composite iterative scheme by the viscosity approximation method for nonexpansive mappings and monotone mappings in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of set of fixed points of nonexpansive mapping and the set of solutions of variational inequality for an inverse-strongly monotone mappings, which is a solution of a certain variational inequality. Our results substantially develop and improve the corresponding results of [Chen et al. 2007 and Iiduka and Takahashi 2005]. Essentially a new approach for finding the fixed points of nonexpansive mappings and solutions of variational inequalities for monotone mappings is provided.
Highlights
Let H be a real Hilbert space and C a nonempty closed convex subset of H
Recall that a mapping f : C → C is a contraction on C if there exists a constant k ∈ 0, 1 such that f x −f y ≤ k x−y, x, y ∈ C
For an α-inverse-strongly-monotone mapping A of C to H and a nonexpansive mapping S of C into itself such that F S ∩ VI C, A / ∅, f ∈ ΣC, x0 ∈ C, {αn} ⊂ 0, 1, and {λn} ⊂ 0, 2α, xn 1 αnf xn 1 − αn SPC xn − λnAxn, n ≥ 0, 1.6 and showed that the sequence {xn} generated by 1.6 converges strongly to a point in F S ∩ VI C, A under condition 1.4 on {αn} and {λn}, which is a solution of a certain variational inequality
Summary
Let H be a real Hilbert space and C a nonempty closed convex subset of H. For an α-inverse-strongly-monotone mapping A of C to H and a nonexpansive mapping S of C into itself such that F S ∩ VI C, A / ∅, f ∈ ΣC, x0 ∈ C, {αn} ⊂ 0, 1 , and {λn} ⊂ 0, 2α , xn 1 αnf xn 1 − αn SPC xn − λnAxn , n ≥ 0, 1.6 and showed that the sequence {xn} generated by 1.6 converges strongly to a point in F S ∩ VI C, A under condition 1.4 on {αn} and {λn}, which is a solution of a certain variational inequality. We point out that the iterative scheme 1.7 is a new approach for finding the fixed points of nonexpansive mappings and solutions of variational inequalities for monotone mappings
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