Strong Convergence of a Modified Extragradient Method to the Minimum-Norm Solution of Variational Inequalities

and Applied Analysis 3 2. Preliminaries Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖, and let C be a closed convex subset of H. It is well known that, for any u ∈ H, there exists a unique u0 ∈ C such that ‖u − u0‖ inf{‖u − x‖ : x ∈ C}. 2.1 We denote u0 by PCu, where PC is called the metric projection of H onto C. The metric projection PC ofH onto C has the following basic properties: i ‖PCx − PCy‖ ≤ ‖x − y‖ for all x, y ∈ H; ii 〈x − y, PCx − PCy〉 ≥ ‖PCx − PCy‖ for every x, y ∈ H; iii 〈x − PCx, y − PCx〉 ≤ 0 for all x ∈ H, y ∈ C. We need the following lemma for proving our main results. Lemma 2.1 see 15 . Assume that {an} is a sequence of nonnegative real numbers such that an 1 ≤ ( 1 − γn ) an δn, 2.2 where {γn} is a sequence in 0, 1 and {δn} is a sequence such that 1 ∑∞ n 1 γn ∞; 2 lim supn→∞δn/γn ≤ 0 or ∑∞ n 1 |δn| < ∞. Then limn→∞an 0. 3. Main Result In this section we will state and prove our main result. Theorem 3.1. Let C be a closed convex subset of a real Hilbert space H. Let A : C → H be an αinverse-strongly monotone mapping. Suppose that VI C,A / ∅. For given x0 ∈ C arbitrarily, define a sequence {xn} iteratively by yn PC 1 − αn xn − λAxn , xn 1 PC ( yn − λAyn ) , n ≥ 0, 3.1 where {αn} is a sequence in 0, 1 and λ ∈ a, b ⊂ 0, 2α is a constant. Assume the following conditions are satisfied: C1 : limn→∞αn 0; C2 : ∑∞ n 1 αn ∞; C3 : limn→∞ αn 1/αn 1. Then the sequence {xn} generated by 3.1 converges strongly to PVI C,A 0 which is the minimumnorm element in VI C,A . 4 Abstract and Applied Analysis We will divide our detailed proofs into several conclusions. Proof. Take x∗ ∈ VI C,A . First we need to use the following facts: 1 x∗ PC x∗ − λAx∗ for all λ > 0; in particular, x∗ PC x∗ − λ 1 − αn Ax∗ PC αnx∗ 1 − αn x∗ − λAx∗ , ∀n ≥ 0; 3.2 2 I − λA is nonexpansive and for all x, y ∈ C ∥∥ I − λA x − I − λA y∥2 ≤ ∥x − y∥2 λ λ − 2α ∥Ax −Ay∥2. 3.3 From 3.1 , we have ∥yn − x∗ ∥∥ ‖PC 1 − αn xn − λAxn − PC αnx∗ 1 − αn x∗ − λAx∗ ‖ ≤ ‖αn −x∗ 1 − αn xn − λAxn − x∗ − λAx∗ ‖ ≤ αn‖x∗‖ 1 − αn ‖ I − λA xn − I − λA x∗‖ ≤ αn‖x∗‖ 1 − αn ‖xn − x∗‖. 3.4 Thus, ‖xn 1 − x∗‖ ∥PC ( yn − λAyn ) − PC x∗ − λAx∗ ∥∥ ≤ ∥yn − λAyn ) − x∗ − λAx∗ ∥∥ ≤ ∥yn − x∗ ∥∥ ≤ αn‖x∗‖ 1 − αn ‖xn − x∗‖ ≤ max{‖x∗‖, ‖x0 − x∗‖}. 3.5 Therefore, {xn} is bounded and so are {yn}, {Axn}, and {Ayn}. From 3.1 , we have ‖xn 1 − xn‖ ∥PC ( yn − λAyn ) − PC ( yn−1 − λAyn−1 )∥∥ ≤ ∥yn − λAyn ) − yn−1 − λAyn−1 )∥∥ ≤ ∥yn − yn−1 ∥∥ ‖PC 1 − αn xn − λAxn − PC 1 − αn−1 xn−1 − λAxn−1 ‖ ≤ ‖ 1 − αn I − λA xn − I − λA xn−1 − αn − αn−1 I − λA xn−1‖ ≤ 1 − αn ‖ I − λA xn − I − λA xn−1‖ |αn − αn−1|‖ I − λA xn−1‖ ≤ 1 − αn ‖xn − xn−1‖ |αn − αn−1|M, 3.6 Abstract and Applied Analysis 5 where M > 0 is a constant such that supn{‖ I − λA xn‖, ‖ I − λA xn‖ ‖ I − λA xn‖ 2‖xn − x∗‖ } ≤ M. Hence, by Lemma 2.1, we obtainand Applied Analysis 5 where M > 0 is a constant such that supn{‖ I − λA xn‖, ‖ I − λA xn‖ ‖ I − λA xn‖ 2‖xn − x∗‖ } ≤ M. Hence, by Lemma 2.1, we obtain lim n→∞ ‖xn 1 − xn‖ 0. 3.7 From 3.4 , 3.5 and the convexity of the norm, we deduce ‖xn 1 − x∗‖2 ≤ ‖αn −x∗ 1 − αn xn − λAxn − x∗ − λAx∗ ‖ ≤ αn‖x∗‖ 1 − αn ‖ I − λA xn − I − λA x∗‖2 ≤ αn‖x∗‖ 1 − αn [ ‖xn − x∗‖2 λ λ − 2α ‖Axn −Ax∗‖2 ] ≤ αn‖x∗‖ ‖xn − x∗‖2 1 − αn a b − 2α ‖Axn −Ax∗‖2. 3.8