DSpace at VNU: Parallel Hybrid Iterative Methods for Variational Inequalities, Equilibrium Problems, and Common Fixed Po...
Vietnam J Math DOI 10.1007/s10013-015-0129-z Parallel Hybrid Iterative Methods for Variational Inequalities, Equilibrium Problems, and Common Fixed Point Problems Pham Ky Anh · Dang Van Hieu Received: 15 October 2014 / Accepted: 16 November 2014 © Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015 Abstract In this paper, we propose two strongly convergent parallel hybrid iterative methods for finding a common element of the set of fixed points of a family of asymptotically quasi φ-nonexpansive mappings, the set of solutions of variational inequalities and the set of solutions of equilibrium problems in uniformly smooth and 2-uniformly convex Banach spaces A numerical experiment is given to verify the efficiency of the proposed parallel algorithms Keywords Asymptotically quasi φ-nonexpansive mapping · Variational inequality · Equilibrium problem · Hybrid method · Parallel computation Mathematics Subject Classification (2010) 47H05 · 47H09 · 47H10 · 47J25 · 65J15 · 65Y05 Introduction Let C be a nonempty closed convex subset of a Banach space E The variational inequality for a possibly nonlinear mapping A : C → E ∗ , consists of finding p ∗ ∈ C such that Ap ∗ , p − p ∗ ≥ ∀p ∈ C (1) The set of solutions of (1) is denoted by V I (A, C) Takahashi and Toyoda [19] proposed a weakly convergent method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an α-inverse strongly monotone mapping in a Hilbert space Dedicated to Professor Nguyen Khoa Son’s 65th birthday P K Anh ( ) · D V Hieu Hanoi University of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam e-mail: anhpk@vnu.edu.vn D V Hieu e-mail: dv.hieu83@gmail.com P K Anh, D V Hieu Theorem [19] Let K be a closed convex subset of a real Hilbert space H Let α > Let A be an α-inverse strongly monotone mapping of K into H, and let S be a nonexpansive mapping of K into itself such that F (S) V I (K, A) = ∅ Let {xn } be a sequence generated by x0 ∈ K, xn+1 = αn xn + (1 − αn )SPK (xn − λn Axn ) for every n = 0, 1, 2, , where λn ∈ [a, b] for some a, b ∈ (0, 2α) and αn ∈ [c, d] for some c, d ∈ (0, 1) Then, {xn } converges weakly to z ∈ F (S) V I (K, A), where z = limn→∞ PF (S) V I (K,A) xn In 2008, Iiduka and Takahashi [8] considered problem (1) in a 2-uniformly convex, uniformly smooth Banach space under the following assumptions: (V1) (V2) (V3) A is α-inverse-strongly monotone V I (A, C) = ∅ Ay ≤ Ay − Au for all y ∈ C and u ∈ V I (A, C) Theorem [8] Let E be a 2-uniformly convex, uniformly smooth Banach space whose duality mapping J is weakly sequentially continuous, and let C be a nonempty, closed convex subset of E Assume that A is a mapping of C into E ∗ satisfying conditions (V1)–(V3) Suppose that x1 = x ∈ C and {xn } is given by xn+1 = CJ −1 (J xn − λn Axn ) for every n = 1, 2, , where {λn } is a sequence of positive numbers If λn is chosen so that λn ∈ [a, b] for some a, b with < a < b < c 2α , then the sequence {xn } converges weakly to some element z in V I (C, A) Here, 1/c is the 2-uniform convexity constant of E, and z = limn→∞ V I (A,C) xn In 2009, Zegeye and Shahzad [22] studied the following hybrid iterative algorithm in a 2uniformly convex and uniformly smooth Banach space for finding a common element of the set of fixed points of a weakly relatively nonexpansive mapping T and the set of solutions of a variational inequality involving an α-inverse strongly monotone mapping A: ⎧ −1 ⎪ ⎪ yn = C J (J xn − λn Axn ) , ⎪ ⎪ zn = T yn , ⎪ ⎪ ⎪ ⎪ ⎨ H0 = {v ∈ C : φ(v, z0 ) ≤ φ(v, y0 ) ≤ φ(v, x0 )} , Hn = v ∈ Hn−1 Wn−1 : φ(v, zn ) ≤ φ(v, yn ) ≤ φ(v, xn ) , ⎪ ⎪ W ⎪ = C, ⎪ ⎪ ⎪ = v ∈ Hn−1 Wn−1 : xn − v, J x0 − J xn ≥ , W ⎪ ⎪ ⎩ n xn+1 = PHn Wn x0 , n ≥ 1, where J is the normalized duality mapping on E The strong convergence of {xn } to F (T ) V I (A,C) x0 has been established Kang et al [9] extended this algorithm to a weakly relatively nonexpansive mapping, a variational inequality and an equilibrium problem Recently, Saewan and Kumam [14] have constructed a sequential hybrid block iterative algorithm for an infinite family of closed and uniformly asymptotically quasi φ-nonexpansive mappings, a variational inequality for an α-inverse-strongly monotone mapping, and a system of equilibrium problems Parallel Hybrid Iterative Methods for VIs, EPs, and FPPs Qin et al [12] considered the following sequential hybrid method for a pair of inverse strongly monotone and a quasi φ-nonexpansive mappings in a 2-uniformly convex and uniformly smooth Banach space: ⎧ x0 = E, C1 = C, x1 = C1 x0 , ⎪ ⎪ ⎪ −1 (J x − η Bx ) , ⎪ u ⎪ n = C J n n n ⎪ ⎨ −1 zn = C J (J un − λn Aun ) , yn = T zn , ⎪ ⎪ ⎪ ⎪ C = {v ∈ Cn : φ(v, yn ) ≤ φ(v, zn ) ≤ φ(v, un ) ≤ φ(v, xn )} , ⎪ ⎪ ⎩ n+1 xn+1 = Cn+1 x0 , n ≥ They proved the strong convergence of the sequence {xn } to F x0 , where F = F (T ) V I (A, C) V I (B, C) Let f be a bifunction from C × C to a set of real numbers R The equilibrium problem for f consists of finding an element x ∈ C, such that f (x, y) ≥ ∀y ∈ C (2) The set of solutions of the equilibrium problem (2) is denoted by EP (f ) Equilibrium problems include several problems such as: variational inequalities, optimization problems, fixed point problems, etc In recent years, equilibrium problems have been studied widely and several solution methods have been proposed (see [3, 9, 14, 15, 18]) On the other hand, for finding a common element in F (T ) EP (f ), Takahashi and Zembayashi [20] introduced the following algorithm in a uniformly smooth and uniformly convex Banach space: ⎧ x0 ∈ C, ⎪ ⎪ ⎪ ⎪ yn = J −1 (αn J xn + (1 − αn )J T yn ), ⎪ ⎪ ⎨ un ∈ C, s.t., f (un , y) + r1n y − un , J un − J yn ≥ ∀y ∈ C, ⎪ Hn = {v ∈ C : φ(v, un ) ≤ φ(v, xn )} , ⎪ ⎪ ⎪ ⎪ W = {v ∈ C : xn − v, J x0 − J xn ≥ 0} , ⎪ ⎩ n xn+1 = PHn Wn x0 , n ≥ The strong convergence of the sequences {xn } and {un } to F (T ) EP (f ) x0 has been established Recently, the above-mentioned algorithms have been generalized and modified for finding a common point of the set of solutions of variational inequalities, the set of fixed points of (asymptotically) quasi φ-nonexpansive mappings, and the set of solutions of equilibrium problems by several authors, such as Takahashi and Zembayashi [20], Wang et al [21], and others Very recently, Anh and Chung [4] have considered the following parallel hybrid method for a finite family of relatively nonexpansive mappings {Ti }N i=1 : ⎧ ⎪ ⎪ x0i ∈ C,−1 ⎪ ⎪ y = J (αn J xn + (1 − αn )J Ti xn ), i = 1, , N, ⎪ ⎪ ⎨ n in = arg max1≤i≤N yni − xn , y¯n := ynin , ⎪ Cn = {v ∈ C : φ(v, y¯n ) ≤ φ(v, xn )} , ⎪ ⎪ ⎪ ⎪ Q = {v ∈ C : J x0 − J xn , xn − v ≥ 0} , ⎪ ⎩ n xn+1 = Cn Qn x0 , n ≥ This algorithm was extended, modified and generalized by Anh and Hieu [5] for a finite family of asymptotically quasi φ-nonexpansive mappings in Banach spaces Note that the proposed parallel hybrid methods in [4, 5] can be used for solving simultaneous systems P K Anh, D V Hieu of maximal monotone mappings Other parallel methods for solving accretive operator equations can be found in [3] In this paper, motivated and inspired by the above-mentioned results, we propose two novel parallel iterative methods for finding a common element of the set of fixed points of a family of asymptotically quasi φ-nonexpansive mappings {F (Sj )}N j =1 , the set of solutions M of variational inequalities {V I (Ai , C)}i=1 , and the set of solutions of equilibrium problems {EP (fk )}K k=1 in uniformly smooth and 2-uniformly convex Banach spaces, namely: Method A ⎧ x0 ∈ C is chosen arbitrarily, ⎪ ⎪ i −1 (J x − λ A x ) , ⎪ ⎪ i = 1, 2, M, y n n i n ⎪ n = C J ⎪ ⎪ i i ⎪ ⎪ in = arg max yn − xn : i = 1, , M , y¯n = ynn , ⎪ ⎪ ⎪ j −1 α J x + (1 − α )J S n y¯ ⎪ j = 1, , N, ⎪ n n n j n , ⎨ zn = J j j jn = arg max zn − xn : j = 1, , N , z¯ n = znn , ⎪ ⎪ ⎪ ⎪ ukn = Trkn z¯ n , k = 1, , K, ⎪ ⎪ ⎪ kn ⎪ k ⎪ ⎪ kn = arg max un − xn : k = 1, 2, K , u¯ n = un , ⎪ ⎪ ⎪ ⎪ ⎩ Cn+1 = {z ∈ Cn : φ(z, u¯ n ) ≤ φ(z, z¯ n ) ≤ φ(z, xn ) + n } , xn+1 = Cn+1 x0 , n ≥ 0, (3) where, Tr x := z is a unique solution to a regularized equilibrium problem y − z, J z − J x ≥ ∀y ∈ C r Further, the control parameter sequences {λn }, {αn }, and {rn } satisfy the conditions f (z, y) + ≤ αn ≤ 1, lim sup αn < 1, n→∞ λn ∈ [a, b], rn ≥ d, (4) for some a, b ∈ (0, αc2 /2), d > 0, where 1/c is the 2-uniform convexity constant of E Concerning the sequence { n }, we consider two cases If the mappings {Si } are asymptotically quasi φ-nonexpansive, we assume that the solution set F is bounded, i.e., there exists a positive number ω, such that F ⊂ := {u ∈ C : u ≤ ω} and put n := (kn − 1)(ω + xn ) If the mappings {Si } are quasi φ-nonexapansive, then kn = 1, and we put n = Method B ⎧ x ∈ C is chosen arbitrarily, ⎪ ⎪ 0i −1 (J x − λ A x ) , i = 1, , M, ⎪ ⎪ y n n i n ⎪ n = C J ⎪ ⎪ ⎪ in = arg max yni − xn : i = 1, , M , y¯n = ynin , ⎪ ⎪ ⎪ ⎨ n zn = J −1 αn,0 J xn + N j =1 αn,j J Sj y¯n , ⎪ ⎪ ukn = Trkn zn , k = 1, , K, ⎪ ⎪ ⎪ ⎪ kn = arg max ukn − xn : k = 1, , K , u¯ n = uinn , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Cn+1 = {z ∈ Cn : φ(z, u¯ n ) ≤ φ(z, xn ) + n } , xn+1 = Cn+1 x0 , n ≥ 0, (5) where the control parameter sequences {λn }, {αn,j }, and {rn } satisfy the conditions N ≤ αn,j ≤ 1, αn,j = 1, j =0 lim inf αn,0 αn,j > 0, n→∞ λn ∈ [a, b], rn ≥ d (6) Parallel Hybrid Iterative Methods for VIs, EPs, and FPPs In Method A (3), knowing xn we find the intermediate approximations yni , i = 1, , M j in parallel Using the farthest element among yni from xn , we compute zn , j = 1, , N in j parallel Further, among zn , we choose the farthest element from xn and determine solutions of regularized equilibrium problems ukn , k = 1, , K in parallel Then the farthest from xn element among ukn , denoted by u¯ n is chosen Based on u¯ n , a closed convex subset Cn+1 is constructed Finally, the next approximation xn+1 is defined as the generalized projection of x0 onto Cn+1 A similar idea of parallelism is employed in Method B (5) However, the subset Cn+1 in Method B is simpler than that in Method A The results obtained in this paper extend and modify the corresponding results of Zegeye and Shahzad [22], Takahashi and Zembayashi [20], Anh and Chung [4], Anh and Hieu [5], and others The paper is organized as follows: In Section 2, we collect some definitions and results needed for further investigation Section deals with the convergence analysis of the methods (3) and (5) In Section 4, a novel parallel hybrid iterative method for variational inequalities and closed, quasi φ-nonexpansive mappings is studied Finally, a numerical experiment is considered in Section to verify the efficiency of the proposed parallel hybrid methods Preliminaries In this section, we recall some definitions and results which will be used later The reader is referred to [2] for more details Definition A Banach space E is called 1) strictly convex if the unit sphere S1 (0) = {x ∈ X : x = 1} is strictly convex, i.e., the inequality x + y < holds for all x, y ∈ S1 (0), x = y; 2) uniformly convex if for any given > there exists δ = δ( ) > such that for all x, y ∈ E with x ≤ 1, y ≤ 1, x − y = the inequality x + y ≤ 2(1 − δ) holds; 3) smooth if the limit x + ty − x lim (7) t→0 t exists for all x, y ∈ S1 (0); 4) uniformly smooth if the limit (7) exists uniformly for all x, y ∈ S1 (0) The modulus of convexity of E is the function δE : [0, 2] → [0, 1] defined by δE ( ) = inf − x+y : x = y = 1, x − y = for all ∈ [0, 2] Note that E is uniformly convex if and only if δE ( ) > for all < ≤ and δE (0) = Let p > 1, E is said to be p-uniformly convex if there exists some constant c > such that δE ( ) ≥ c p It is well-known that the spaces Lp , l p , and p Wm are p-uniformly convex if p > and 2-uniformly convex if < p ≤ and a Hilbert space H is uniformly smooth and 2-uniformly convex Let E be a real Banach space with its dual E ∗ The dual product of f ∈ E ∗ and x ∈ E is denoted by x, f or f, x For the sake of simplicity, the norms of E and E ∗ are denoted P K Anh, D V Hieu ∗ · The normalized duality mapping J : E → 2E is defined by the same symbol by J (x) = f ∈ E ∗ : f, x = x = f The following properties can be found in [7]: i) If E is a smooth, strictly convex, and reflexive Banach space, then the normalized ∗ duality mapping J : E → 2E is single-valued, one-to-one, and onto; ii) If E is a reflexive and strictly convex Banach space, then J −1 is norm to weak ∗ continuous; iii) If E is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E; iv) A Banach space E is uniformly smooth if and only if E ∗ is uniformly convex; v) Each uniformly convex Banach space E has the Kadec–Klee property, i.e., for any sequence {xn } ⊂ E, if xn x ∈ E and xn → x , then xn → x Lemma [22] If E is a 2-uniformly convex Banach space, then x − y ≤ J x − Jy ∀x, y ∈ E, c where J is the normalized duality mapping on E and < c ≤ The best constant 1c is called the 2-uniform convexity constant of E Next, we assume that E is a smooth, strictly convex, and reflexive Banach space In the sequel, we always use φ : E × E → [0, ∞) to denote the Lyapunov functional defined by φ(x, y) = x − x, J y + y ∀x, y ∈ E From the definition of φ, we have + y )2 (8) φ(x, y) = φ(x, z) + φ(z, y) + z − x, J y − J z (9) ( x − y )2 ≤ φ(x, y) ≤ ( x Moreover, the Lyapunov functional satisfies the identity for all x, y, z ∈ E The generalized projection C : E → C is defined by C (x) = arg φ(x, y) y∈C In what follows, we need the following properties of the functional φ and the generalized projection C Lemma [1] Let E be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset of E Then the following conclusions hold: i) ii) iii) φ(x, C (y)) + φ( C (y), y) ≤ φ(x, y) ∀x ∈ C, y ∈ E; if x ∈ E, z ∈ C, then z = C (x) if and only if z − y, J x − J z ≥ ∀y ∈ C; φ(x, y) = if and only if x = y Lemma [10] Let C be a nonempty closed convex subset of a smooth Banach E, x, y, z ∈ E and λ ∈ [0, 1] For a given real number a, the set D := {v ∈ C : φ(v, z) ≤ λφ(v, x) + (1 − λ)φ(v, y) + a} is closed and convex Parallel Hybrid Iterative Methods for VIs, EPs, and FPPs Lemma [1] Let {xn } and {yn } be two sequences in a uniformly convex and uniformly smooth real Banach space E If φ(xn , yn ) → and either {xn } or {yn } is bounded, then xn − yn → as n → ∞ Lemma [6] Let E be a uniformly convex Banach space, r be a positive number, and Br (0) ⊂ E be the closed ball with center at origin and radius r Then, for any given subset {x1 , x2 , , xN } ⊂ Br (0) and for any positive numbers λ1 , λ2 , , λN with N i=1 λi = 1, there exists a continuous, strictly increasing, and convex function g : [0, 2r) → [0, ∞) with g(0) = such that, for any i, j ∈ {1, 2, , N } with i < j , N λk xk k=1 N ≤ λk xk − λi λj g( xi − xj ) k=1 Definition A mapping A : E → E ∗ is called 1) monotone, if A(x) − A(y), x − y ≥ ∀x, y ∈ E; 2) uniformly monotone, if there exists a strictly increasing function ψ [0, ∞), ψ(0) = 0, such that : [0, ∞) → A(x) − A(y), x − y ≥ ψ( x − y ) ∀x, y ∈ E; (10) 3) η-strongly monotone, if there exists a positive constant η, such that in (10), ψ(t) = ηt ; 4) α-inverse strongly monotone, if there exists a positive constant α, such that A(x) − A(y), x − y ≥ α A(x) − A(y) ∀x, y ∈ E; 5) L-Lipschitz continuous if there exists a positive constant L, such that A(x) − A(y) ≤ L x − y ∀x, y ∈ E If A is α-inverse strongly monotone then it is α1 -Lipschitz continuous If A is η-strongly monotone and L-Lipschitz continuous then it is Lη2 -inverse strongly monotone Lemma [17] Let C be a nonempty, closed convex subset of a Banach space E and A be a monotone, hemicontinuous mapping of C into E ∗ Then V I (C, A) = {u ∈ C : v − u, A(v) ≥ ∀v ∈ C} Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, T : C → C be a mapping The set F (T ) = {x ∈ C : T x = x} is called the set of fixed points of T A point p ∈ C is said to be an asymptotic fixed point p and xn − T xn → as of T if there exists a sequence {xn } ⊂ C such that xn n → + ∞ The set of all asymptotic fixed points of T will be denoted by F˜ (T ) Definition A mapping T : C → C is called i) relatively nonexpansive mapping if F (T ) = ∅, F˜ (T ) = F (T ), and φ(p, T x) ≤ φ(p, x) ∀p ∈ F (T ), ∀x ∈ C; P K Anh, D V Hieu ii) iii) closed if for any sequence {xn } ⊂ C, xn → x and T xn → y, then T x = y; quasi φ-nonexpansive mapping (or hemi-relatively nonexpansive mapping) if F (T ) = ∅ and φ(p, T x) ≤ φ(p, x) iv) ∀p ∈ F (T ), ∀x ∈ C; asymptotically quasi φ-nonexpansive if F (T ) = ∅ and there exists a sequence {kn } ⊂ [1, +∞) with kn → as n → + ∞ such that φ(p, T n x) ≤ kn φ(p, x) ∀n ≥ 1, ∀p ∈ F (T ), ∀x ∈ C; v) uniformly L-Lipschitz continuous, if there exists a constant L > such that T nx − T ny ≤ L x − y ∀n ≥ 1, ∀x, y ∈ C The reader is referred to [6, 16] for examples of closed and asymptotically quasi φ-nonexpansive mappings It has been shown that the class of asymptotically quasi φnonexpansive mappings contains properly the class of quasi φ-nonexpansive mappings, and the class of quasi φ-nonexpansive mappings contains the class of relatively nonexpansive mappings as a proper subset Lemma [6] Let E be a real uniformly smooth and strictly convex Banach space with Kadec–Klee property, and C be a nonempty closed convex subset of E Let T : C → C be a closed and asymptotically quasi φ-nonexpansive mapping with a sequence {kn } ⊂ [1, +∞), kn → Then F (T ) is a closed convex subset of C Next, for solving the equilibrium problem (2), we assume that the bifunction f satisfies the following conditions: (A1) (A2) (A3) f (x, x) = for all x ∈ C; f is monotone, i.e., f (x, y) + f (y, x) ≤ for all x, y ∈ C; For all x, y, z ∈ C, lim sup f (tz + (1 − t)x, y) ≤ f (x, y); t→0+ (A4) For all x ∈ C, f (x, ·) is convex and lower semicontinuous The following results show that in a smooth (uniformly smooth), strictly convex and reflexive Banach space, the regularized equilibrium problem has a solution (unique solution, respectively) Lemma [20] Let C be a closed and convex subset of a smooth, strictly convex and reflexive Banach space E, f be a bifunction from C × C to R satisfying conditions (A1)–(A4) and let r > 0, x ∈ E Then, there exists z ∈ C such that f (z, y) + y − z, J z − J x ≥ ∀y ∈ C r Lemma [20] Let C be a closed and convex subset of a uniformly smooth, strictly convex, and reflexive Banach spaces E, f be a bifunction from C × C to R satisfying conditions (A1)–(A4) For all r > and x ∈ E, define the mapping Tr x = {z ∈ C : f (z, y) + Then the following hold: y − z, J z − J x ≥ ∀y ∈ C} r Parallel Hybrid Iterative Methods for VIs, EPs, and FPPs (B1) (B2) Tr is single-valued; Tr is a firmly nonexpansive-type mapping, i.e., for all x, y ∈ E, Tr x − Tr y, J Tr x − J Tr y ≤ Tr x − Tr y, J x − J y ; (B3) (B4) F (Tr ) = F˜ (T ) = EP (f ); EP (f ) is closed and convex and Tr is a relatively nonexpansive mapping Lemma 10 [20] Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E Let f be a bifunction from C × C to R satisfying (A1)–(A4) and let r > Then, for x ∈ E and q ∈ F (Tr ), φ(q, Tr x) + φ(Tr x, x) ≤ φ(q, x) Let E be a real Banach space Alber [1] studied the function V : E ×E ∗ → R defined by V (x, x ∗ ) = x − x, x ∗ + x ∗ Clearly, V (x, x ∗ ) = φ(x, J −1 x ∗ ) Lemma 11 [1] Let E be a reflexive, strictly convex and smooth Banach space with E ∗ as its dual Then V (x, x ∗ ) + J −1 x − x ∗ , y ∗ ≤ V (x, x ∗ + y ∗ ) ∀x ∈ E, ∀x ∗ , y ∗ ∈ E ∗ Consider the normal cone NC to a set C at a point x ∈ C defined by x ∗ ∈ E ∗ : x − y, x ∗ ≥ NC (x) = ∀y ∈ C We have the following result Lemma 12 [13] Let C be a nonempty closed convex subset of a Banach space E and let A be a monotone and hemi-continuous mapping of C into E ∗ with D(A) = C Let Q be a mapping defined by: Ax + NC (x) if x ∈ C, Q(x) = ∅ if x ∈ / C Then Q is a maximal monotone and Q−1 = V I (A, C) Convergence Analysis Throughout this section, we assume that C is a nonempty closed convex subset of a real uniformly smooth and 2-uniformly convex Banach space E Denote ⎞ ⎛ M F = V I (Ai , C) i=1 ⎝ N j =1 F (Sj )⎠ K EP (fk ) k=1 and assume that the set F is nonempty We prove convergence theorems for methods (3) and (5) with the control parameter sequences satisfying conditions (4) and (6), respectively We also propose similar parallel hybrid methods for quasi φ-nonexpansive mappings, variational inequalities and equilibrium problems P K Anh, D V Hieu ∗ Theorem Let {Ai }M i=1 be a finite family of mappings from C to E satisfying condiK tions (V1)–(V3) Let {fk }k=1 : C × C → R be a finite family of bifunctions satisfying conditions (A1)–(A4) Let {Sj }N j =1 : C → C be a finite family of uniform LLipschitz continuous and asymptotically quasi-φ-nonexpansive mappings with the same sequence {kn } ⊂ [1, +∞), kn → Assume that there exists a positive number ω such that F ⊂ := {u ∈ C : u ≤ ω} If the control parameter sequences {αn }, {λn }, and {rn } satisfy condition (4), then the sequence {xn } generated by (3) converges strongly to F x0 Proof We divide the proof of Theorem into seven steps Step Claim that F, Cn are closed convex subsets of C Indeed, since each mapping Si is uniformly L-Lipschitz continuous, it is closed By Lemmas 6, 7, and 9, F (Si ), V I (Aj , C) N M and EP (fk ) are closed convex sets, therefore, j =1 (F (Sj )), i=1 V I (Ai , C) and K k=1 EP (fk ) are also closed and convex Hence, F is a closed and convex subset of C It is obvious that Cn is closed for all n ≥ We prove the convexity of Cn by induction Clearly, C0 := C is closed convex Assume that Cn is closed convex for some n ≥ From the construction of Cn+1 , we find Cn+1 = Cn {z ∈ E : φ(z, u¯ n ) ≤ φ(z, z¯ n ) ≤ φ(z, xn ) + n } Lemma ensures that Cn+1 is convex Thus, Cn is closed convex for all n ≥ Hence, F x0 and xn+1 := Cn+1 x0 are well-defined Step Claim that F ⊂ Cn for all n ≥ By Lemma 10 and the relative nonexpansiveness of Trn , we obtain φ(u, u¯ n ) = φ(u, Trn z¯ n ) ≤ φ(u, z¯ n ) From the convexity of · for all u ∈ F and the asymptotical quasi φ-nonexpansiveness of Sj , we find φ(u, z¯ n ) = φ u, J −1 αn J xn + (1 − αn )J Sjnn y¯n = u − 2αn u, xn − 2(1 − αn ) u, J Sjnn y¯n + αn J xn + (1 − αn )J Sjnn y¯n ≤ u − 2αn u, xn − 2(1 − αn ) u, J Sjnn y¯n + αn xn 2 + (1 − αn ) Sjnn y¯n = αn φ(u, xn ) + (1 − αn )φ(u, Sjnn y¯n ) ≤ αn φ(u, xn ) + (1 − αn )kn φ(u, y¯n ) (11) for all u ∈ F By the hypotheses of Theorem 3, Lemmas 1, 2, and 11, and u ∈ F , we have φ(u, y¯n ) = φ u, C J −1 (J xn − λn Ain xn ) ≤ φ(u, J −1 (J xn − λn Ain xn )) = V (u, J xn − λn Ain xn ) ≤ V (u, J xn − λn Ain xn + λn Ain xn ) − J −1 J xn − λn Ain xn − u, λn Ain xn = φ(u, xn ) − 2λn J −1 J xn − λn Ain xn − J −1 (J xn ) , Ain xn −2λn xn − u, Ain xn − Ain (u) − 2λn xn − u, Ain u Parallel Hybrid Iterative Methods for VIs, EPs, and FPPs 4λn J xn − λn Ain xn − J xn Ain xn − 2λn α Ain xn − Ain u c2 4λ2 ≤ φ(u, xn ) + 2n Ain xn − 2λn α Ain xn − Ain u c 2b Ain xn − Ain u ≤ φ(u, xn ) − 2a α − c ≤ φ(u, xn ) (12) ≤ φ(u, xn ) + From (11), (12) and the estimate (8), we obtain φ(u, z¯ n ) ≤ αn φ(u, xn ) + (1 − αn )kn φ(u, xn ) − 2a(1 − αn ) α − ≤ φ(u, xn ) + (kn − 1)φ(u, xn ) − 2a(1 − αn ) α − 2b c2 ≤ φ(u, xn ) + (kn − 1) (ω + xn )2 − 2a(1 − αn ) α − ≤ φ(u, xn ) + 2b c2 Ain xn − Ain u Ain xn − Ain u 2b c2 2 Ain xn − Ain u (13) n Therefore F ⊂ Cn for all n ≥ j Step Claim that the sequence {xn }, {yni }, {zn } and {ukn } converge strongly to p ∈ C By Lemma and xn = Cn x0 , we have φ(xn , x0 ) ≤ φ(u, x0 ) − φ(u, xn ) ≤ φ(u, x0 ) for all u ∈ F Hence, {φ(xn , x0 )} is bounded By (8), {xn } is bounded, and so are the sequences {y¯n }, {u¯ n }, and {¯zn } By the construction of Cn , xn+1 = Cn+1 x0 ∈ Cn+1 ⊂ Cn From Lemma and xn = Cn x0 , we get φ(xn , x0 ) ≤ φ(xn+1 , x0 ) − φ(xn+1 , xn ) ≤ φ(xn+1 , x0 ) Therefore, the sequence {φ(xn , x0 )} is nondecreasing, hence, it has a finite limit Note that for all m ≥ n, xm ∈ Cm ⊂ Cn , and by Lemma we obtain φ(xm , xn ) ≤ φ(xm , x0 ) − φ(xn , x0 ) → (14) as m, n → ∞ From (14) and Lemma 4, we have xn − xm → This shows that {xn } ⊂ C is a Cauchy sequence Since E is complete and C is a closed convex subset of E, {xn } converges strongly to p ∈ C From (14), φ(xn+1 , xn ) → as n → ∞ Taking into account that xn+1 ∈ Cn+1 , we find φ(xn+1 , u¯ n ) ≤ φ(xn+1 , z¯ n ) ≤ φ(xn+1 , xn ) + Since {xn } is bounded, we can put M = sup{ xn n n (15) : n = 0, 1, 2, }, hence := (kn − 1)(ω + xn ) ≤ (kn − 1)(ω + M)2 → (16) By (15), (16) and φ(xn+1 , xn ) → 0, we find that lim φ(xn+1 , u¯ n ) = lim φ(xn+1 , z¯ n ) = lim φ(xn+1 , xn ) = n→∞ n→∞ n→∞ Therefore, from Lemma 4, lim n→∞ xn+1 − u¯ n = lim n→∞ xn+1 − z¯ n = lim n→∞ xn+1 − xn = (17) P K Anh, D V Hieu This together with xn+1 − xn → implies that xn − u¯ n = lim lim n→∞ n→∞ xn − z¯ n = By the definitions of jn and kn , we obtain j xn − ukn = lim lim n→∞ x n − zn = n→∞ (18) for all ≤ k ≤ K and ≤ j ≤ N Hence, j lim xn = lim ukn = lim zn = p n→∞ n→∞ (19) n→∞ for all ≤ k ≤ K and ≤ j ≤ N By the hypotheses of Theorem 3, Lemmas 1, 2, and 11, we also have φ(xn , y¯n ) = φ xn , C J −1 (J xn − λn Ain xn ) ≤ φ(xn , J −1 (J xn − λn Ain xn )) = V (xn , J xn − λn Ain xn ) ≤ V (xn , J xn − λn Ain xn + λn Ain xn ) − J −1 J xn −λn Ain xn −xn , λn Ain xn = −2λn J −1 J xn − λn Ain xn − J −1 J xn , Ain xn 4λn J xn − λn Ain xn − J xn c2 4λ2 ≤ 2n Ain xn c 4b2 ≤ Ain xn − Ain u c ≤ for all u ∈ M i=1 V I (Ai , C) 2(1 − αn )a α − Ain xn (20) From (13), we obtain 2b c2 Ain xn − Ain u = u, J z¯ n − J xn + ( xn 2 ≤ (φ(u, xn ) − φ(u, z¯ n )) + − z¯ n ) + n n ≤ u J z¯ n − J xn + xn − z¯ n ( xn + z¯ n ) + n (21) Using the fact that xn − z¯ n → and J is uniformly continuous on each bounded set, we can conclude that J z¯ n − J xn → as n → ∞ This, together with (21), and the relations lim supn→∞ αn < and n → imply that lim n→∞ Ain xn − Ain u = (22) From (20) and (22), we obtain lim φ(xn , y¯n ) = n→∞ Therefore, limn→∞ xn −y¯n Hence, = By the definition of in , we get limn→∞ xn − yni lim y i n→∞ n for all ≤ i ≤ M =p = (23) Parallel Hybrid Iterative Methods for VIs, EPs, and FPPs j n −1 (α J x n n j =1 F (Sj ) The relation zn = J n αn J xn + (1 − αn )J Sj y¯n Therefore, Step Claim that p ∈ implies that j J zn = + (1 − αn )J Sjn y¯n ) j J xn − J zn = (1 − αn ) J xn − J Sjn y¯n Since xn − j J xn − J zn implies that (24) j zn → and J is uniformly continuous on each bounded subset of E, → as n → ∞ This, together with (24) and lim supn→∞ αn < lim n→∞ J xn − J Sjn y¯n = Therefore lim n→∞ Since limn→∞ xn − y¯n xn − Sjn y¯n = 0, limn→∞ y¯n − lim S n y¯n n→∞ j = Sjn y¯n (25) = 0, hence = p (26) Further, Sjn+1 y¯n − Sjn y¯n ≤ Sjn+1 y¯n − Sjn+1 y¯n+1 + Sjn+1 y¯n+1 − y¯n+1 + y¯n+1 − y¯n + y¯n − Sjn y¯n ≤ (L + 1) y¯n+1 − y¯n + Sjn+1 y¯n+1 − y¯n+1 + y¯n − Sjn y¯n → 0, therefore lim S n+1 y¯n n→∞ j = lim Sj Sjn y¯n = p n→∞ (27) From (26), (27) and the closedness of Sj , we obtain p ∈ F (Sj ) for all ≤ j ≤ N N Hence, p ∈ j =1 F (Sj ) Step Claim that p ∈ M i=1 V I (Ai , C) Qi (x) = Lemma 12 ensures that the mapping Ai x + NC (x) if x ∈ C, ∅ if x ∈ /C is maximal monotone, where NC (x) is the normal cone to C at x ∈ C For all (x, y) in the graph of Qi , i.e., (x, y) ∈ G(Qi ), we have y − Ai (x) ∈ NC (x) By the definition of NC (x), we find that x − z, y − Ai (x) ≥ for all z ∈ C Since yni ∈ C, x − yni , y − Ai (x) ≥ Therefore Taking into account yni = x − yni , y ≥ x − yni , Ai (x) −1 (J x − λ A x )) and Lemma 2, we get C (J n n i n x − yni , J yni − J xn + λn Ai xn ≥ (28) (29) Therefore, from (28), (29) and the monotonicity of Ai , we find that x − yni , y ≥ x − yni , Ai (x) = x − yni , Ai (x) − Ai (yni ) + x − yni , Ai (yni ) − Ai (xn ) + x − yni , Ai (xn ) ≥ x − yni , Ai (yni ) − Ai (xn ) + x − yni , J xn − J yni λn (30) P K Anh, D V Hieu Since xn −yni → and J is uniform continuous on each bounded set, J xn − J yni By λn ≥ a > 0, we obtain J xn − J yni lim = n→∞ λn → (31) Since Ai is α-inverse strongly monotone, Ai is α1 -Lipschitz continuous This, together with xn − yni → implies that lim n→∞ Ai (yni ) − Ai (xn ) = (32) From (30), (31), (32), and yni → p, we obtain x − p, y ≥ for all (x, y) ∈ G(Qi ) M Therefore, p ∈ Q−1 i=1 V I (Ai , C) i = V I (Ai , C) for all ≤ i ≤ M Hence, p ∈ K k Step Claim that p ∈ k=1 EP (fk ) Since limn→∞ un − z¯ n continuous on every bounded subset of E, we have lim n→∞ = and J is uniformly J ukn − J z¯ n = This together with rn ≥ d > implies that lim n→∞ J ukn − J z¯ n = rn (33) We have ukn = Trkn z¯ n , and fk (ukn , y) + y − ukn , J ukn − J z¯ n ≥ ∀y ∈ C rn (34) From (34) and condition (A2), we get y − ukn , J ukn − J z¯ n ≥ −fk (ukn , y) ≥ fk (y, ukn ) rn ∀y ∈ C (35) Letting n → ∞, by (33), (35) and (A4), we obtain fk (y, p) ≤ ∀y ∈ C (36) Putting yt = ty + (1 − t)p, where < t ≤ and y ∈ C, we get yt ∈ C Hence, for sufficiently small t, from (A3) and (36), we have fk (yt , p) = fk (ty + (1 − t)p, p) ≤ By the properties (A1) and (A4), we find = fk (yt , yt ) = fk (yt , ty + (1 − t)p) ≤ tfk (yt , y) + (1 − t)fk (yt , p) ≤ tfk (yt , y) Dividing both sides of the last inequality by t > 0, we obtain fk (yt , y) ≥ for all y ∈ C, i.e., fk (ty + (1 − t)p, y) ≥ ∀y ∈ C Passing t → 0+ , from (A3) we get fk (p, y) ≥ ∀y ∈ C and ≤ k ≤ K, i.e., K p ∈ k=1 EP (fk ) Step Claim that the sequence {xn } converges strongly to F x0 Indeed, since x † := F (x0 ) ∈ F ⊂ Cn , xn = Cn (x0 ) from Lemma 2, we have φ(xn , x0 ) ≤ φ(x † , x0 ) − φ(x † , xn ) ≤ φ(x † , x0 ) (37) Parallel Hybrid Iterative Methods for VIs, EPs, and FPPs Therefore φ(x † , x0 ) ≥ lim φ(xn , x0 ) = lim n→∞ = p xn n→∞ − p, J x0 + x0 − x n , J x0 + x0 2 = φ(p, x0 ) From the definition of x † , it follows that p = x † The proof of Theorem is complete Remark Assume that {Ai }M i=1 is a finite family of η-strongly monotone and LLipschitz continuous mappings Then each Ai is Lη -inverse strongly monotone and V I (Ai , C) = A−1 i Hence, Ai x ≤ Ai x − Ai u for all x ∈ C and u ∈ V I (Ai , C) Thus, all the conditions (V1)–(V3) for the variational inequalities V I (Ai , C) hold ∗ Theorem Let {Ai }M i=1 be a finite family of mappings from C to E satisfying conditions (V1)–(V3) Let {fk }K k=1 : C × C → R be a finite family of bifunctions satisfying conditions (A1)–(A4) Let {Sj }N j =1 : C → C be a finite family of uniform L-Lipschitz continuous and asymptotically quasi-φ-nonexpansive mappings with the same sequence {kn } ⊂ [1, +∞), kn → Assume that F is a subset of , and suppose that the control parameter sequences {αn }, {λn }, {rn } satisfy condition (6) Then, the sequence {xn } generated by method (5) converges strongly to F x0 Proof Arguing similarly as in Step of the proof of Theorem 3, we conclude that F, Cn are closed convex for all n ≥ Now, we show that F ⊂ Cn for all n ≥ For all u ∈ F , by Lemma and the convexity of · we obtain N φ(u, zn ) = φ u, J −1 αn,0 J xn + αn,l J Sln y¯n l=1 N = u − 2αn,0 u, xn − 2 N αn,l u, Sln y¯n + αn,0 J xn + l=1 αn,l J Sln y¯n l=1 N ≤ u αn,l u, Sln y¯n + αn,0 xn − 2αn,0 u, xn − 2 l=1 N + αn,l Sln y¯n − αn,0 αn,j g J xn − J Sjn y¯n l=1 N ≤ αn,0 φ(u, xn ) + αn,l φ(u, Sln y¯n ) − αn,0 αn,j g J xn − J Sjn y¯n αn,l kn φ(u, y¯n ) − αn,0 αn,j g J xn − J Sjn y¯n l=1 N ≤ αn,0 φ(u, xn ) + l=1 (38) From (12), we get φ(u, y¯n ) ≤ φ(u, xn ) − 2a α − 2b c2 Ain xn − Ain u (39) P K Anh, D V Hieu Using (38), (39), and the estimate (8), we find φ(u, u¯ n ) = φ(u, Trknn zn ) ≤ φ(u, zn ) N ≤ φ(u, xn ) + αn,l (kn − 1)φ(u, xn ) − αn,0 αn,j g J xn − J Sjn y¯n l=1 N 2b c2 αn,l a α − −2 l=1 Ain xn − Ain u ≤ φ(u, xn ) + (kn − 1) (ω + xn )2 − αn,0 αn,j g N 2b c2 αn,l a α − −2 l=1 ≤ φ(u, xn ) + Ain xn − Ain u J xn − J Sjn y¯n (40) n Therefore φ(u, u¯ n ) ≤ φ(u, xn ) + n for all u ∈ F This implies that F ⊂ Cn for all n ≥ Using (40) and arguing similarly as in Steps 3, 5, and of Theorem 3, we obtain lim u¯ n = lim ukn = lim y¯n = lim yni = lim xn = p ∈ C, n→∞ n→∞ K k=1 EP (fk )) and p ∈ ( Next, we show that p ∈ αn,0 αn,j g ( n→∞ n→∞ n→∞ M i=1 V I (Ai , C)) N j =1 F (Sj ) Indeed, from (40), we have J xn − J Sjn y¯n ≤ (φ(u, xn ) − φ(u, u¯ n )) + n (41) Since xn − u¯ n → 0, |φ(u, xn ) − φ(u, u¯ n )| → as n → ∞ This, together with (41) and the facts that n → and lim infn→∞ αn,0 αn,j > imply that lim g n→∞ J xn − J Sjn y¯n = By Lemma 5, we get lim n→∞ J xn − J Sjn y¯n = Since J is uniformly continuous on each bounded subset of E, we conclude that lim n→∞ xn − Sjn y¯n = Using the last equality and a similar argument for proving relations (25), (26), (27), N and acting as in Step of the proof of Theorem 3, we obtain p ∈ j =1 F (Sj ) and p = x † = F x0 The proof of Theorem is complete Next, we consider two parallel hybrid methods for solving variational inequalities, equilibrium problems and quasi φ-nonexpansive mappings, when the boundedness of the solution set F and the uniform Lipschitz continuity of Si are not required K Theorem Assume that {Ai }M i=1 , {fk }k=1 , {αn }, {rn }, and {λn } satisfy all conditions of N Theorem and {Sj }j =1 is a finite family of closed and quasi φ-nonexpansive mappings In Parallel Hybrid Iterative Methods for VIs, EPs, and FPPs addition, suppose that the solution set F is nonempty For an initial point x0 ∈ C, define the sequence {xn } as follows: ⎧ yni = C J −1 (J xn − λn Ai xn ) , i = 1, 2, M, ⎪ ⎪ ⎪ ⎪ ⎪ in = arg max yni − xn : i = 1, 2, , M , y¯n = ynin , ⎪ ⎪ ⎪ j ⎪ ⎪ zn = J −1 αn J xn + (1 − αn )J Sj y¯n , j = 1, 2, , N, ⎪ ⎪ ⎨ j j jn = arg max zn − xn : j = 1, 2, , N , z¯ n = znn , (42) ⎪ ⎪ ukn = Trkn z¯ n , k = 1, 2, , K, ⎪ ⎪ ⎪ ⎪ kn = arg max ukn − xn : k = 1, 2, , K , u¯ n = uknn , ⎪ ⎪ ⎪ ⎪ ⎪ Cn+1 = {z ∈ Cn : φ(z, u¯ n ) ≤ φ(z, z¯ n ) ≤ φ(z, xn )} , ⎪ ⎩ xn+1 = Cn+1 x0 , n ≥ Then the sequence {xn } converges strongly to F x0 Proof Since Si is a closed and quasi φ-nonexpansive mapping, it is closed and asymptotically quasi φ-nonexpansive mapping with kn = for all n ≥ Hence, n = by definition Arguing similarly as in the proof of Theorem 3, we come to the desired conclusion K Theorem Assume that {Ai }M i=1 , {fk }k=1 , {rn }, {αn,j }, and {λn } satisfy all conditions of Theorem and {Sj }N j =1 is a finite family of closed and quasi φ-nonexpansive mappings In addition, suppose that the solution set F is nonempty For an initial approximation x0 ∈ C, let the sequence {xn } be defined by ⎧ i yn = C J −1 (J xn − λn Ai xn ) , i = 1, 2, , M, ⎪ ⎪ ⎪ ⎪ in = arg max yni − xn : i = 1, 2, , M , y¯n = ynin , ⎪ ⎪ ⎪ ⎪ N ⎪ ⎨ zn = J −1 αn,0 J xn + j =1 αn,j J Sj y¯n , (43) ukn = Trkn zn , k = 1, 2, , K, ⎪ ⎪ ⎪ k = arg max uk − x : k = 1, 2, , K , u¯ = ukn , ⎪ n n n ⎪ n n ⎪ ⎪ ⎪ {z = ∈ C : φ(z, u ¯ ) ≤ φ(z, x )} , C ⎪ n n n n+1 ⎩ xn+1 = Cn+1 x0 , n ≥ Then the sequence {xn } converges strongly to F x0 Proof The proof is similar to that of Theorem for Si being a closed and quasi φasymptotically nonexpansive mapping with kn = for all n ≥ A Parallel Iterative Method for Quasi φ-Nonexpansive Mappings and Variational Inequalities In 2004, using Mann’s iteration, Matsushita and Takahashi [11] proposed the following scheme for finding a fixed point of a relatively nonexpansive mapping T: xn+1 = CJ −1 (αn J xn + (1 − αn )J T xn ) , n = 0, 1, 2, , (44) where x0 ∈ C is given They proved that if the interior of F (T ) is nonempty then the sequence {xn } generated by (44) converges strongly to some point in F (T ) Recently, using Halpern’s and Ishikawa’s iterative processes, Zhang et al [23] have proposed modified iterative algorithms of (44) for a relatively nonexpansive mapping P K Anh, D V Hieu In this section, employing the ideas of Matsushita and Takahashi [11] and Anh and Chung [4], we propose a parallel hybrid iterative algorithm for finite families of closed and quasi M φ- nonexpansive mappings {Sj }N j =1 and variational inequalities {V I (Ai , C)}i=1 : ⎧ x0 ∈ C is chosen arbitrarily, ⎪ ⎪ ⎪ yi = −1 (J x − λ A x ) , i = 1, 2, , M, ⎪ C J n n i n ⎪ n ⎪ ⎪ ⎨ in = arg max yni − xn : i = 1, 2, , M , y¯n = ynin , j zn = J −1 αn J xn + (1 − αn )J Sj y¯n , j = 1, 2, , N, ⎪ ⎪ ⎪ j j ⎪ ⎪ jn = arg max zn − xn : j = 1, 2, , N , z¯ n = znn , ⎪ ⎪ ⎩ xn+1 = C z¯ n , n ≥ 0, (45) where {αn } ⊂ [0, 1] such that limn→∞ αn = Remark One can employ method (45) for a finite family of relatively nonexpansive mappings without assuming their closedness Remark Method (45) modifies the corresponding method (44) in the following aspects: – – A relatively nonexpansive mapping T is replaced with a finite family of quasi φ-nonexpansive mappings, where the restriction F (Sj ) = F˜ (Sj ) is not required A parallel hybrid method for finite families of closed and quasi φ-nonexpansive mappings and variational inequalities is considered instead of an iterative method for a relatively nonexpansive mapping Theorem Let E be a real uniformly smooth and 2-uniformly convex Banach space with dual space E ∗ and C be a nonempty closed convex subset of E Assume that {Ai }M i=1 is a finite family of mappings satisfying conditions (V1)–(V3), {Sj }N is a finite family of j =1 closed and quasi φ-nonexpansive mappings, and {αn } ⊂ [0, 1] satisfies limn→∞ αn = 0, λn ∈ [a, b] for some a, b ∈ (0, αc2 /2) In addition, suppose that the interior of N F = ( M i=1 V I (Ai , C)) ( j =1 F (Sj )) is nonempty Then, the sequence {xn } generated by (45) converges strongly to some point u ∈ F Moreover, u = limn→∞ F xn Proof By Lemma 7, the subset F is closed and convex, hence the generalized projections F , C are well-defined We now show that the sequence {xn } is bounded Indeed, for every u ∈ F , from Lemma and the convexity of · , we have φ(u, xn+1 ) = φ(u, C z¯ n ) ≤ φ(u, z¯ n ) = u − u, J z¯ n + z¯ n = u − 2αn u, J xn − 2(1 − αn ) u, J Sjn y¯n + αn J xn +(1 − αn )J Sjn y¯n ≤ u − 2αn u, J xn − 2(1 − αn ) u, J Sjn y¯n +αn y¯n + (1 − αn ) Sjn y¯n 2 = αn φ(u, xn ) + (1 − αn )φ(u, Sjn y¯n ) ≤ αn φ(u, xn ) + (1 − αn )φ(u, y¯n ) Arguing similarly to (12) and (13), we obtain φ(u, xn+1 ) ≤ φ(u, xn ) − 2a(1 − αn ) α − 2b c2 Ain xn − Ain u ≤ φ(u, xn ) (46) Parallel Hybrid Iterative Methods for VIs, EPs, and FPPs Therefore, the sequence {φ(u, xn )} is decreasing Hence, there exists a finite limit of {φ(u, xn )} This, together with (8) and (46) imply that the sequences {xn } is bounded and lim n→∞ Ain xn − Ain u = (47) Next, we show that {xn } converges strongly to some element u in C Since the interior of F is nonempty, there exist p ∈ F and r > such that p + rh ∈ F for all h ∈ E and h we have ≤ Since the sequence {φ(u, xn )} is decreasing for all u ∈ F , φ(p + rh, xn+1 ) ≤ φ(p + rh, xn ) (48) From (9), we find that φ(u, xn ) = φ(u, xn+1 ) + φ(xn+1 , xn ) + xn+1 − u, J xn − J xn+1 for all u ∈ F Therefore φ(p + rh, xn ) = φ(p + rh, xn+1 ) + φ(xn+1 , xn ) +2 xn+1 − (p + rh), J xn − J xn+1 (49) From (48), (49), we obtain φ(xn+1 , xn ) + xn+1 − (p + rh), J xn − J xn+1 ≥ This inequality is equivalent to {φ(xn+1 , xn ) + xn+1 − p, J xn − J xn+1 } 2r (50) φ(p, xn ) = φ(p, xn+1 ) + φ(xn+1 , xn ) + xn+1 − p, J xn − J xn+1 (51) h, J xn − J xn+1 ≤ From (9), we also have From (50), (51), we obtain h, J xn − J xn+1 for all h {φ(p, xn ) − φ(p, xn+1 )} 2r ≤ ≤ Hence, sup h, J xn − J xn+1 h ≤1 ≤ {φ(p, xn ) − φ(p, xn+1 )} 2r The last relation is equivalent to {φ(p, xn ) − φ(p, xn+1 )} 2r Therefore, for all n, m ∈ N and n > m, we have J xn − J xn+1 ≤ J xn − J xm = J xn − J xn−1 + J xn−1 − J xn−2 + · · · + J xm+1 − J xm n−1 J xi+1 − J xi ≤ i=m ≤ = 2r n−1 {φ(p, xi ) − φ(p, xi+1 )} i=m (φ(p, xm ) − φ(p, xn )) 2r P K Anh, D V Hieu Letting m, n → ∞, we obtain J xn − J xm = lim m,n→∞ Since E is a uniformly convex and uniformly smooth Banach space, J −1 is uniformly continuous on every bounded subset of E From the last relation we have lim m,n→∞ xn − xm = Therefore, {xn } is a Cauchy sequence Since E is complete and C is closed and convex, {xn } converges strongly to some element u in C By arguing similarly to (20), we obtain 4b2 Ain xn − Ain u c2 This relation together with (47) implies that φ(xn , y¯n ) → Therefore, xn − y¯n → By the definition of in , we conclude that xn − yni → for all ≤ i ≤ M Hence, φ(xn , y¯n ) ≤ lim y i n→∞ n for all ≤ i ≤ M From xn+1 = C z¯ n =u∈C and Lemma 2, we have φ(Sjn y¯n , xn+1 ) + φ(xn+1 , z¯ n ) = φ(Sjn y¯n , Using the convexity of · (52) C z¯ n ) + φ( C z¯ n , z¯ n ) ≤ φ(Sjn y¯n , z¯ n ) (53) we have φ(Sjn y¯n , z¯ n ) = Sjn y¯n − Sjn y¯n , J z¯ n + z¯ n = Sjn y¯n − 2αn Sjn y¯n , J xn − 2(1 − αn ) Sjn y¯n , J Sjn y¯n + αn J xn + (1 − αn )J Sjn y¯n ≤ Sjn y¯n +αn xn 2 − 2αn Sjn y¯n , J xn − 2(1 − αn ) Sjn y¯n , J Sjn y¯n + (1 − αn ) Sjn y¯n = αn φ(Sjn y¯n , xn ) + (1 − αn )φ(Sjn y¯n , Sjn y¯n ) = αn φ(Sjn y¯n , xn ) The last inequality together with (53) implies that φ(xn+1 , z¯ n ) ≤ αn φ(Sjn y¯n , xn ) Therefore, from the boundedness of {φ(Sjn y¯n , xn )} and limn→∞ αn = 0, we get lim φ(xn+1 , z¯ n ) = n→∞ Hence, xn+1 − z¯ n → Since xn − xn+1 → 0, we find xn − z¯ n → 0, and by j the definition of jn , we obtain xn − zn → for all ≤ j ≤ N Thus, j lim zn = p (54) n→∞ Arguing similarly to Steps 4, in the proof of Theorem 3, we obtain ⎛ ⎞ M p ∈ F = V I (Ai , C) i=1 The proof of Theorem is complete ⎝ N j =1 F (Sj )⎠ Parallel Hybrid Iterative Methods for VIs, EPs, and FPPs A Numerical Example Let E = R1 be a Hilbert space with the standard inner product x, y := xy and the norm x := |x| for all x, y ∈ E Let C := [0, 1] ⊂ E The normalized dual mapping J = I and the Lyapunov functional φ(x, y) = |x − y|2 It is well known that, the modulus of convexity of Hilbert space E is δE ( ) = − − /4 ≥ 14 Therefore, E is 2-uniformly convex Moreover, the best constant 1c satisfying relations |x − y| ≤ c22 |J x − J y| = c22 |x − y| with < c ≤ is This implies that c = Define the mappings Ai (x) := x − the variational inequalities Ai (p ∗ ), p − p ∗ x i+1 i+1 , x ≥ ∈ C, i = 1, , M, and consider ∀p ∈ C, for i = 1, , M Clearly, V I (Ai , C) = {0}, i = 1, , M Since each mapping i+1 Ui (x) := xi+1 is nonexpansive, the mapping Ai = I − Ui , i = 1, , M, is 12 inverse strongly monotone Besides, |Ai (y)| = |Ai (y) − Ai (0)|, hence all the assumptions (V1)–(V3) for the variational inequalities are satisfied N Further, let {ti }N i=1 and {si }i=1 be two sequences of positive numbers, such that < t1 < · · · < tN < and si ∈ (1, 1−t ]; i = 1, , N Define the mappings i Si : C → C, i = 1, , N , by putting Si (x) = for x ∈ [0, ti ], and Si (x) = si (x−ti ) if x ∈ [ti , 1] It is easy to verify that F (Si ) = {0}, φ(Si (x), 0) = |Si (x)|2 ≤ |x|2 = φ(x, 0) for every x ∈ C, and |Si (1) − Si (ti )| = si (1 − ti ) > |1 − ti | Hence, the mappings Si are quasi φ-nonexpansive but not nonexpansive Finally, let < ξ1 < · · · < ξK < and ηk ∈ (0, ξk ), k = 1, , K be two given sequences Consider K bifunctions fk (x, y) := Bk (x)(y − x), k = 1, , K, where Bk (x) = ηξkk x if ≤ x ≤ ξk , and Bk (x) = ηk if ξk ≤ x ≤ It is easy to verify that all the assumptions (A1)–(A4) for the bifunctions fk (x, y) are fulfilled Besides, EP (fk ) = {0} Thus, the solution set ⎞ ⎛ M F := ⎝ V I (Ai , C) N F (Sj )⎠ j =1 i=1 K EP (fk ) = {0} k=1 According to Theorem 6, the iteration sequence {xn } generated by yni = C xn − λn (xn − (xn )i+1 ) , i+1 i = 1, 2, , M, in = arg max |yni − xn | : i = 1, 2, M , j zn = αn xn + (1 − αn )Sj y¯n , j = 1, 2, , N, j jn = arg max |zn − xn | : j = 1, 2, , N , ukn = Trkn z¯ n , y¯n = ynin , j z¯ n = znn , k = 1, 2, , K, kn = arg max |ukn − xn | : k = 1, 2, , K , u¯ n = uknn , Cn+1 = {z ∈ Cn : φ(z, u¯ n ) ≤ φ(z, z¯ n ) ≤ φ(z, xn )} , xn+1 = Cn+1 x0 , n≥0 P K Anh, D V Hieu Table Experiment with αn = log(log(n+10)) TOL Tp Ts 10−7 21.84 42.42 10−9 26.46 51.24 strongly converges to x † := A straightforward calculation yields i+1 n) , i = 1, 2, M Further, the element u := ukn = Trkn z¯ n yni = (1 − λn )xn − λn (xi+1 is a solution of the following inequality (y − u)[Bk (u) + u − z¯ n ] ≥ ∀y ∈ [0, 1] (55) From (55), we find that u = if and only if z¯ n = Therefore, if z¯ n = 0, then the algorithm stops and x † = If z¯ n = then inequality (55) is equivalent to the system of inequalities −u(Bk (u) + u − z¯ n ) ≥ 0, (1 − u)(Bk (u) + u − z¯ n ) ≥ The last system yields Bk (u) + u = z¯ n Hence, if < z¯ n ≤ ηk + ξk then k ukn := u = ξk ξ+η z¯ n Otherwise, if ξk + ηk < z¯ n ≤ 1, then ukn := u = z¯ n − ηk k Using the fact that F = {0} ⊂ Cn+1 we can conclude that ≤ u¯ n ≤ z¯ n ≤ xn ≤ From the definition of Cn+1 we find Cn+1 = Cn 0, z¯ n + u¯ n According to Step of the proof of Theorem 3, Cn is a closed convex subset, hence zn [0, xn ] ⊂ Cn because 0, xn ∈ Cn Further, since u¯ n +¯ ≤ xn , it implies that u¯ n +¯zn u¯ n +¯zn [0, ] ⊂ [0, xn ] ⊂ Cn Thus, Cn+1 = [0, ] For the sake of comparison between the computing times in the parallel and sequential modes, we choose sufficiently large numbers N , K, and M and a slowly convergent to zero sequence {αn } The numerical experiment is performed on a LINUX cluster 1350 with eight computing nodes Each node contains two Intel Xeon dual core 3.2 GHz and GB Ram All the programs are written in C For given tolerances we compare the execution time of algorithm (55) in both parallel and sequential modes We denote by TOL-the tolerance xk − x ∗ ; Tp — the execution time in parallel mode using two CPUs (in seconds), and Ts —the execution time in sequential mode (in seconds) The computing times in both modes are given in Tables and According to Tables and 2, in the most favorable cases, the speed-up and the efficiency of the parallel algorithm are Sp = Ts /Tp ≈ 2; Ep = Sp /2 ≈ 1, respectively Table Experiment with αn = logn n TOL Tp Ts 10−7 6.09 11.97 10−9 8.10 14.28 Parallel Hybrid Iterative Methods for VIs, EPs, and FPPs Conclusions In this paper, we proposed two strongly convergent parallel hybrid iterative methods for finding a common element of the set of fixed points of quasi φ-asymptotically nonexpansive mappings, the set of solutions of variational inequalities, and the set of solutions of equilibrium problems in uniformly smooth and 2-uniformly convex Banach spaces A numerical example was given to demonstrate the efficiency of the proposed parallel algorithms Acknowledgments The 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Iterative Methods for VIs, EPs, and FPPs Conclusions In this paper, we proposed two strongly convergent parallel hybrid iterative methods for finding a common element of the set of fixed points... definitions and results needed for further investigation Section deals with the convergence analysis of the methods (3) and (5) In Section 4, a novel parallel hybrid iterative method for variational