Modified Noor’s Extragradient Method for Solving Generalized Variational Inequalities in Banach Spaces

and Applied Analysis 3 2. Preliminaries Let C be a nonempty closed convex subset of a real Banach space E. Recall that a mapping A of C into E is said to be accretive if there exists j x − y ∈ J x − y such that 〈 Ax −Ay, j(x − y)〉 ≥ 0, 2.1 for all x, y ∈ C. A mapping A of C into E is said to be α-strongly accretive if, for α > 0, 〈 Ax −Ay, j(x − y)〉 ≥ α∥∥x − y∥∥2, 2.2 for all x, y ∈ C. A mapping A of C into E is said to be α-inverse-strongly accretive if, for α > 0, 〈 Ax −Ay, j(x − y)〉 ≥ α∥∥Ax −Ay∥∥2, 2.3 for all x, y ∈ C. Let U {x ∈ E : ‖x‖ 1}. A Banach space E is said to uniformly convex if, for each ∈ 0, 2 , there exists δ > 0 such that for any x, y ∈ U, ∥∥x − y∥∥ ≥ implies ∥∥∥ x y 2 ∥∥∥ ≤ 1 − δ. 2.4 It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space E is said to be smooth if the limit lim t→ 0 ∥∥x ty∥∥ − ‖x‖ t 2.5 exists for all x, y ∈ U. It is also said to be uniformly smooth if the limit 2.5 is attained uniformly for x, y ∈ U. The norm of E is said to be Fréchet differentiable if, for each x ∈ U, the limit 2.5 is attained uniformly for y ∈ U. And we define a function ρ : 0,∞ → 0,∞ called the modulus of smoothness of E as follows: ρ τ sup { 1 2 (∥∥x y∥∥ ∥∥x − y∥∥) − 1 : x, y ∈ X, ‖x‖ 1, ∥∥y∥∥ τ } . 2.6 It is known that E is uniformly smooth if and only if limτ → 0 ρ τ /τ 0. Let q be a fixed real number with 1 < q ≤ 2. Then a Banach space E is said to be q-uniformly smooth if there exists a constant c > 0 such that ρ τ ≤ cτ for all τ > 0. We need the following lemmas for proof of our main results. Lemma 2.1 see 28 . Let q be a given real number with 1 < q ≤ 2 and let E be a q-uniformly smooth Banach space. Then ∥∥x y∥∥q ≤ ‖x‖ q〈y, Jq x 〉 2 ∥∥Ky∥∥q 2.7 4 Abstract and Applied Analysis for all x, y ∈ E, whereK is the q-uniformly smoothness constant of E and Jq is the generalized duality mapping from E into 2 ∗ defined by Jq x { f ∈ E∗ : 〈x, f〉 ‖x‖, ∥∥f∥∥ ‖x‖q−1 } , ∀x ∈ E. 2.8 LetD be a subset of C and letQ be a mapping of C intoD. ThenQ is said to be sunny if Q Qx t x −Qx Qx, 2.9 whenever Qx t x − Qx ∈ C for x ∈ C and t ≥ 0. A mapping Q of C into itself is called a retraction if Q2 Q. If a mapping Q of C into itself is a retraction, then Qz z for every z ∈ R Q , where R Q is the range ofQ. A subsetD of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C ontoD. One knows the following lemma concerning sunny nonexpansive retraction. Lemma 2.2 see 29 . Let C be a closed convex subset of a smooth Banach space E, let D be a nonempty subset of C, and letQ be a retraction from C ontoD. ThenQ is sunny and nonexpansive if and only if 〈 u −Qu, j(y −Qu)〉 ≤ 0 2.10 for all u ∈ C and y ∈ D. Remark 2.3. 1 It is well known that if E is a Hilbert space, then a sunny nonexpansive retraction QC is coincident with the metric projection from E onto C. 2 Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and let T be a nonexpansive mapping of C into itself with F T / ∅. Then the set F T is a sunny nonexpansive retract of C. The following lemma characterized the set of solution of 1.1 by using sunny nonexpansive retractions. Lemma 2.4 see 25 . Let C be a nonempty closed convex subset of a smooth Banach space X. Let QC be a sunny nonexpansive retraction from X onto C and letA be an accretive operator of C into X. Then for all λ > 0, S C,A F QC I − λA , 2.11 where S C,A {x∗ ∈ C : 〈Ax∗, J x − x∗ 〉 ≥ 0, ∀x ∈ C}. Lemma 2.5 see 30 . Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E and let T be nonexpansive mapping of C into itself. If {xn} is a sequence of C such that xn → x weakly and xn − Txn → 0 strongly, then x is a fixed point of T . Lemma 2.6 see 31 . Assume that {an} is a sequence of nonnegative real numbers such that an 1 ≤ ( 1 − γn ) an δn, n ≥ 0, 2.12 Abstract and Applied Analysis 5 where {γn} is a sequence in 0, 1 and {δn} is a sequence in R such that a ∑∞ n 0 γn ∞; b lim supn→∞ δn/γn ≤ 0 or ∑∞ n 0 |δn| < ∞. Then limn→∞an 0.and Applied Analysis 5 where {γn} is a sequence in 0, 1 and {δn} is a sequence in R such that a ∑∞ n 0 γn ∞; b lim supn→∞ δn/γn ≤ 0 or ∑∞ n 0 |δn| < ∞. Then limn→∞an 0. 3. Main Results In this section, we will state and prove our main result. Theorem 3.1. Let E be a uniformly convex and 2-uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let QC be a sunny nonexpansive retraction from E onto C and letA : C → E be an α-strongly accretive and L-Lipschitz continuous mapping with S C,A / ∅. For given x0 ∈ C, let the sequence {xn} be generated iteratively by yn QC xn − λAxn , xn 1 QC [ 1 − αn ( yn − λAyn )] , n ≥ 0, 3.1 where {αn} and {βn} are two sequences in 0, 1 and λ is a constant in a, b for some a, b with 0 < a < b < α/K2L2. Assume that the following conditions hold: a limn→∞αn 0 and ∑∞ n 1 αn ∞; b limn→∞ αn 1/αn 1. Then {xn} defined by 3.1 converges strongly to Q′ 0 , where Q′ is a sunny nonexpansive retraction of E onto S C,A . Proof. First, we note that A must be α/L2-inverse-strongly accretive mapping. Take p ∈ S C,A . By using Lemmas 2.1 and 2.4, we easily obtain the following facts. 1 p QC p − λAp for all λ > 0; in particular, p QC [ p − λ 1 − αn Ap ] QC [ αnp 1 − αn ( p − λAp)], n ≥ 0. 3.2 2 If λ ∈ 0, α/K2L2 , then I − λA is nonexpansive and for all x, y ∈ C ∥∥ I − λA x − I − λA y∥∥2 ≤ ∥∥x − y∥∥2 2λ ( K2λ − α L2 ∥∥Ax −Ay∥∥2. 3.3 Indeed, from Lemma 2.1, we have ∥∥ I − λA x − I − λA y∥∥2 ∥∥(x − y) − λ(Ax −Ay)∥∥2 ≤ ∥∥x − y∥∥2 − 2λ〈Ax −Ay, j(x − y)〉 2K2λ2∥∥Ax −Ay∥∥2 ≤ ∥∥x − y∥∥2 − 2λ α L2 ∥∥Ax −Ay∥∥2 2K2λ2∥∥Ax −Ay∥∥2 ∥∥x − y∥∥2 2λ ( K2λ − α L2 ∥∥Ax −Ay∥∥2. 3.4 6 Abstract and Applied Analysis From 3.1 , we have ∥yn − p ∥∥ ∥QC xn − λAxn −QC ( p − λAp)∥∥ ≤ ∥∥ xn − λAxn − ( p − λAp)∥∥ ≤ ∥xn − p ∥∥. 3.5 By 3.1 and 3.5 , we have ∥xn 1 − p ∥∥ ∥QC [ 1 − αn ( yn − λAyn )] −QC [ αnp 1 − αn ( p − λAp)]∥∥ ≤ ∥∥[ 1 − αn ( yn − λAyn )] − αnp 1 − αn ( p − λAp)]∥∥ ≤ αn ∥∥p∥∥ 1 − αn ∥yn − λAyn ) − (p − λAp)∥∥ ≤ αn ∥∥p∥∥ 1 − αn ∥yn − p ∥∥ ≤ αn ∥∥p∥∥ 1 − αn ∥xn − p ∥∥ ≤ max∥p∥,∥x0 − p ∥∥}. 3.6 Therefore, {xn} is bounded. We observe that ∥yn − yn−1 ∥∥ ‖QC xn − λAxn −QC xn−1 − λAxn−1 ‖ ≤ ‖ xn − λAxn − xn−1 − λAxn−1 ‖ ≤ ‖xn − xn−1‖, 3.7 and hence ‖xn 1 − xn‖ ∥QC [ 1 − αn ( yn − λAyn )] −QC [ 1 − αn−1 ( yn−1 − λAyn−1 )]∥∥ ≤ ∥∥[ 1 − αn ( yn − λAyn )] − [ 1 − αn−1 ( yn−1 − λAyn−1 )]∥∥ ∥∥ 1 − αn [( yn − λAyn ) − yn−1 − λAyn−1 )] αn−1 − αn ( yn−1 − λAyn−1 )∥∥ ≤ 1 − αn ∥yn − λAyn ) − yn−1 − λAyn−1 )∥∥ |αn − αn−1| ∥yn−1 − λAyn−1 ∥∥ ≤ 1 − αn ∥yn − yn−1 ∥∥ |αn − αn−1| ∥yn−1 − λAyn−1 ∥∥ ≤ 1 − αn ‖xn − xn−1‖ |αn − αn−1| ∥yn−1 − λAyn−1 ∥∥. 3.8 By Lemma 2.6, we obtain lim n→∞ ‖xn 1 − xn‖ 0. 3.9 Abstract and Applied Analysis 7 From 3.1 , we also have ∥yn − p ∥∥2 ∥QC xn − λAxn −QC [ p − λAp]∥∥2 ≤ ∥∥ xn − λAxn − ( p − λAp)∥∥2 ≤ ∥xn − p ∥∥2 2λ ( K2λ − α L2 ∥∥Axn −Ap ∥∥2. 3.10and Applied Analysis 7 From 3.1 , we also have ∥yn − p ∥∥2 ∥QC xn − λAxn −QC [ p − λAp]∥∥2 ≤ ∥∥ xn − λAxn − ( p − λAp)∥∥2 ≤ ∥xn − p ∥∥2 2λ ( K2λ − α L2 ∥∥Axn −Ap ∥∥2. 3.10 By 3.1 and 3.10 , we obtain ∥xn 1 − p ∥∥2 ≤ ∥αn (−p) 1 − αn [( yn − λAyn ) − (p − λAp)]∥∥2 ≤ αn ∥∥p∥∥2 1 − αn ∥yn − λAyn ) − (p − λAp)∥∥2 ≤ αn ∥∥p∥∥2 1 − αn ∥∥yn − p ∥∥2 2λ ( K2λ − α L2 ∥∥Ayn −Ap ∥∥2 ] ≤ αn ∥∥p∥∥2 1 − αn ∥∥xn − p ∥∥2 2λ ( K2λ − α L2 ∥∥Axn −Ap ∥∥2 ] 2 1 − αn λ ( K2λ − α L2 ∥∥Ayn −Ap ∥∥2 ≤ αn ∥∥p∥∥2 ∥xn − p ∥∥2 2 1 − αn λ ( K2λ − α L2 ∥∥Axn −Ap ∥∥2 2 1 − αn λ ( K2λ − α L2 ∥∥Ayn −Ap ∥∥2. 3.11