Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems

and Applied Analysis 3 for all x, y ∈ C. A mapping A : C → H is said to be α-inverse strongly g-monotone if and only if 〈 Ax −Ay, g x − gy ≥ α∥Ax −Ay∥2, 2.2 for some α > 0 and for all x, y ∈ C. A mapping g : C → C is said to be strongly monotone if there exists a constant γ > 0 such that 〈 g x − gy, x − y ≥ γ∥x − y∥2, 2.3 for all x, y ∈ C. Let B be a mapping ofH into 2 . The effective domain of B is denoted by dom B , that is, dom B {x ∈ H : Bx/ ∅}. A multivalued mapping B is said to be a monotone operator on H if and only if 〈 x − y, u − v ≥ 0, 2.4 for all x, y ∈ dom B , u ∈ Bx, and v ∈ By. Amonotone operator B onH is said to bemaximal if and only if its graph is not strictly contained in the graph of any other monotone operator on H. Let B be a maximal monotone operator on H and let B−10 {x ∈ H : 0 ∈ Bx}. It is well known that, for any u ∈ H, there exists a unique u0 ∈ C such that ‖u − u0‖ inf{‖u − x‖ : x ∈ C}. 2.5 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. It is easy to see that the following is true: u ∈ GVIC,A, g ⇐⇒ g u PC ( g u − λA u , ∀λ > 0. 2.6 We use the following notation: i xn ⇀ x stands for the weak convergence of xn to x; ii xn → x stands for the strong convergence of xn to x. We need the following lemmas for the next section. Lemma 2.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let G : C → C be a nonlinear mapping and let the mapping A : C → H be α-inverse strongly g-monotone. Then, for any λ > 0, one has ∥PC [ g x − λAx − PC [ g ( y ) − λAy∥2 ≤ ∥g x − gy∥2 λ λ − 2α ∥Ax −Ay∥2, x, y ∈ C. 2.7 4 Abstract and Applied Analysis Proof. Consider the following: ∥ ∥PC [ g x − λAx − PC [ g ( y ) − λAy∥2 ≤ ∥g x − gy − λAx −Ay∥2 ∥ ∥g x − gy∥2 − 2λAx −Ay, g x − gy λ2∥Ax −Ay∥2 ≤ ∥g x − gy∥2 − 2λα∥Ax −Ay∥2 λ2∥Ax −Ay∥2 ≤ ∥g x − gy∥2 λ λ − 2α ∥Ax −Ay∥2. 2.8 If λ ∈ 0, 2α , we have ∥PC [ g x − λAx − PC [ g ( y ) − λAy∥ ≤ ∥g x − gy − λAx −Ay∥ ≤ ∥g x − gy∥. 2.9 Lemma 2.2 see 36 . Let C be a closed convex subset of a Hilbert space H. Let S : C → C be a nonexpansive mapping. Then Fix S is a closed convex subset ofC and the mapping I−S is demiclosed at 0, that is, whenever {xn} ⊂ C is such that xn ⇀ x and I − S xn → 0, then I − S x 0. Lemma 2.3 see 37 . Let {xn} and {yn} be bounded sequences in a Banach spaceX and let {βn} be a sequence in 0, 1 with 0 < lim infn→∞βn ≤ lim supn→∞βn < 1. Suppose xn 1 1−βn yn βnxn for all n ≥ 0 and lim supn→∞ ‖yn 1 − yn‖ − ‖xn 1 − xn‖ ≤ 0. Then, limn→∞‖yn − xn‖ 0. Lemma 2.4 see 38 . Assume {an} is a sequence of nonnegative real numbers such that an 1 ≤ ( 1 − γn ) an δnγn, 2.10 where {γn} is a sequence in 0, 1 and {δn} is a sequence such that 1 ∑∞ n 1γn ∞; 2 lim supn→∞ δn ≤ 0 or ∑∞ n 1|δnγn| < ∞. Then limn→∞an 0. 3. Main Results In this section, we will prove our main results. Theorem 3.1. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F : C → H be an L-Lipschitz continuous mapping, g : C → C be a weakly continuous and γ-strongly monotone mapping such that R g C. Let A : C → H be an α-inverse strongly g-monotone mapping and let S : C → C be a nonexpansive mapping. Suppose that Ω/ ∅. Let β ∈ 0, 1 and γ ∈ L, 2α . For given x0 ∈ C, let {xn} ⊂ C be a sequence generated by g xn 1 βg xn ( 1 − βSPC [ αnF xn 1 − αn ( g xn − λAxn )] , n ≥ 0, 3.1 Abstract and Applied Analysis 5and Applied Analysis 5 where {αn} ⊂ 0, 1 satisfies C1 : limn→∞αn 0 and C2 : ∑ nαn ∞. Then the sequence {xn} generated by 3.1 converges strongly to x∗ ∈ Ω which is the unique solution of the following variational inequality: 〈 F x∗ − g x∗ , g x − g x∗ 〉 ≤ 0, ∀x ∈ Ω. 3.2 Proof. First, we show the solution set of variational inequality 3.2 is singleton. Assume x̃ ∈ Ω also solves 3.2 . Then, we have 〈 F x∗ − g x∗ , g x̃ − g x∗ 〉 ≤ 0, F x̃ − g x̃ , g x∗ − g x̃ 〉 ≤ 0. 3.3 It follows that 〈 F x̃ − g x̃ − F x∗ g x∗ , g x∗ − g x̃ 〉 ≤ 0 ⇒ ∥g x∗ − g x̃ ∥2 ≤ F x∗ − F x̃ , g x∗ − g x̃ 〉 ⇒ ∥g x∗ − g x̃ ∥2 ≤ F x∗ − F x̃ , g x∗ − g x̃ 〉 ≤ ‖F x∗ − F x̃ ‖∥g x∗ − g x̃ ∥ ⇒ ∥g x∗ − g x̃ ∥ ≤ ‖F x∗ − F x̃ ‖. 3.4 Since g is γ-strongly monotone, we have γ ∥x − y∥2 ≤ g x − gy, x − y ≤ ∥g x − gy∥∥x − y∥, ∀x, y ∈ C. 3.5 Hence, γ ∥x − y∥ ≤ ∥g x − gy∥, ∀x, y ∈ C. 3.6 In particular, γ‖x∗ − x̃‖ ≤ ‖g x∗ − g x̃ ‖. By 3.4 , we deduce γ‖x∗ − x̃‖ ≤ ∥g x∗ − g x̃ ∥ ≤ ‖F x∗ − F x̃ ‖ ≤ L‖x∗ − x̃‖, 3.7 which implies that x̃ x∗ because of L < γ by the assumption. Therefore, the solution of variational inequality 3.2 is unique. Pick up any u ∈ Ω. It is obvious that u ∈ GVI C,A, g and g u ∈ Fix S . Set un PC αnF xn 1−αn g xn −λAxn , n ≥ 0. From 2.6 , we know g u PC g u −μAu for any μ > 0. Hence, we have g u PC [ g u − 1 − αn λAu ] PC [ αng u 1 − αn ( g u − λAu, ∀n ≥ 0. 3.8 6 Abstract and Applied Analysis From 3.6 , 3.8 , and Lemma 2.1, we get ∥ ∥un − g u ∥ ∥ ∥ ∥PC [ αnF xn 1 − αn ( g xn − λAxn )] −PC [ αng u 1 − αn ( g u − λAu∥ ≤ αn ∥ ∥F xn − g u ∥ ∥ 1 − αn ∥ ∥g xn − λAxn ) − g u − λAu∥ ≤ αn‖F xn − F u ‖ αn ∥ ∥F u − g u ∥ 1 − αn ∥ ∥g xn − g u ∥ ∥ ≤ αnL‖xn − u‖ αn ∥ ∥F u − g u ∥ 1 − αn ∥ ∥g xn − g u ∥ ∥ ≤ αnL γ ∥ ∥g xn − g u ∥ ∥ αn ∥ ∥F u − g u ∥ 1 − αn ∥ ∥g xn − g u ∥ ∥ [ 1 − ( 1 − L γ ) αn ]∥ ∥g xn − g u ∥ ∥ αn ∥ ∥F u − g u ∥. 3.9 It follows from 3.1 that ∥g xn 1 − g u ∥ ≤ β∥g xn − g u ∥ ( 1 − β∥Sun − Sg u ∥ ≤ β∥g xn − g u ∥ ( 1 − β∥un − g u ∥ ≤ β∥g xn − g u ∥∥ ( 1 − β [ 1 − ( 1 − L γ ) αn ]∥∥g xn − g u ∥∥ ( 1 − βαn ∥F u − g u ∥∥ [ 1 − ( 1 − L γ ) ( 1 − βαn ]∥∥g xn − g u ∥ ( 1 − L γ ) ( 1 − βαn ∥F u − g u ∥ 1 − L/γ . 3.10 This indicates by induction that ∥g xn 1 − g u ∥ ≤ max { ∥g xn − g u ∥, ∥F u − g u ∥ 1 − L/γ } .