Synchronal Algorithm For a Countable Family of Strict Pseudocontractions in q-uniformly Smooth Banach Spaces

Transcription

1 Int. Journal of Math. Analysis, Vol. 8, 2014, no. 15, HIKARI Ltd, Synchronal Algorithm For a Countable Family of Strict Pseudocontractions in q-uniformly Smooth Banach Spaces Abba Auwalu Government Day Secondary School, P.M.B. 1008, Gumel Ministry of Education, Science and Technology, Dutse Jigawa State, Nigeria Copyright c 2014 Abba Auwalu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let E be a real q-uniformly smooth Banach space whose duality map is weakly sequentially continuous and C be a nonempty, closed and convex subset of E. Let {T i } i=1 : C E be a family of k-strict pseudocontractions for k (0, 1) such that i=1 F (T i), f be a contraction with coefficient β (0, 1) and {λ i } i=1 be a real sequence in (0, 1) such that i=1 λ i = 1. Let G : C E be an η-strongly accretive and L-Lipschitzian operator with L > 0, η > 0. Let {α n } and {β n } be sequences in (0, 1) satisfying some conditions. For some positive real numbers γ, µ appropriately chosen, let {x n } be a sequence defined by x 0 C arbitrarily chosen, T βn = β n I + (1 β n ) i=1 λ it i, x n+1 = α n γf(x n ) + (I α n µg)t βn x n, n 0. Then, we prove that {x n } converges strongly to a common fixed point x of the countable family {T i } i=1, which solves the variational inequality: (γf µg)x, j q (x x ) 0, x F (T i ). Mathematics Subject Classification: 47H06, 47H09, 47H10, 47J05, 47J20, 47J25 Keywords: q-uniformly smooth Banach space; strict pseudocontractions; variational inequality; Synchronal algorithm; common fixed point i=1

2 728 Abba Auwalu 1 Introduction Let E be a real Banach space and E be the dual of E. For some real number q (1 < q < ), the generalized duality mapping J q : E 2 E is defined by J q (x) = { x E : x, x = x q, x = x q 1}, x E, (1) where.,. denotes the duality pairing between elements of E and those of E. In particular, J = J 2 is called the normalized duality mapping and J q (x) = x q 2 J 2 (x) for x 0. If E is a real Hilbert space, then J = I, where I is the identity mapping. It is well known that if E is smooth, then J q is single-valued, which is denoted by j q (see [16]). The duality mapping J q from a smooth Banach space E into E is said to be weakly sequentially continuous generalized duality mapping if for all {x n } E with x n x implies J q (x n ) J q (x). Let C be a nonempty closed convex subset of E, and G : E E be a nonlinear map. Then, a variational inequality problem with respect to C and G is to find a point x C such that Gx, j q (x x ) 0, x C and j q (x x ) J q (x x ). (2) We denotes by V I(G, C) the set of solutions of this variational inequality problem. If E = H, a real Hilbert space, the variational inequality problem reduces to the following: Find a point x C such that Gx, x x 0, x C. (3) A mapping T : E E is said to be L-Lipschitzian if there exists L > 0 such that T x T y L x y, x, y E. (4) If L = 1, then T is called Nonexpansive and if 0 L < 1, T is called Contraction. A point x E is called a fixed point of the map T if T x = x. We denote by F (T ) the set of all fixed points of the mapping T, that is F (T ) = {x C : T x = x}. We assume that F (T ) in the sequel. It is well known that F (T ) above, is closed and convex (see e.g. Goebel and Kirk [7]). An Operator F : E E is said to be Accretive if x, y E, there exists j q (x y) J q (x y) such that F x F y, j q (x y) 0. (5) For some positive real numbers η, λ, the mapping F is said to be η-strongly accretive if for any x, y E, there exists j q (x y) J q (x y) such that F x F y, j q (x y) η x y q, (6)

3 Synchronal algorithm for a strict pseudocontractions 729 and it is called λ-strictly pseudocontractive if F x F y, j q (x y) x y q λ x y (F x F y) q. (7) It is clear that (7) is equivalent to the following (I F )x (I F )y, j q (x y) λ x y (F x F y) q, (8) where I denotes the identity operator. In Hilbert spaces, accretive operators are called monotone where inequality (5) holds with j q replace by identity map of H. The modulus of smoothness of E, with dim E 2, is a function ρ E : [0, ) [0, ) defined by { } x + y + x y ρ E (τ) = sup 1: x = 1, y τ. 2 A Banach space E is said to be uniformly smooth if lim ρ E(t) t 0 + = 0. For q > 1, a t Banach space E is said to be q-uniformly smooth, if there exists a fixed constant c > 0 such that ρ E (t) ct q, t > 0. It is well known (see [5]) that Hilbert spaces and L p (p > 1) spaces are uniformly smooth. More precisely, L p (or l p ) spaces are { 2 uniformly smooth, if 2 p <, p uniformly smooth, if 1 < p 2. Also, Every l p space, (1 < p < ) has a weakly sequentially continuous duality map. Let K be a nonempty closed convex and bounded subset of a Banach space E and let the diameter of K be defined by d(k) := sup{ x y : x, y K}. For each x K, let r(x, K) := sup{ x y : y K} and let r(k) := inf{r(x, K) : x K} denote the Chebyshev radius of K relative to itself. The normal structure coefficient N(E) of E (see, e.g., [3]) is defined by N(E) := inf { d(k) : d(k) > 0}. A space E such that N(E) > 1 r(k) is said to have uniform normal structure. It is known that all uniformly convex and uniformly smooth Banach spaces have uniform normal structure (see, e.g., [6, 9]). Let µ be a continuous linear functional on l and (a 0, a 1,...) l. We write µ n (a n ) instead of µ((a 0, a 1,...)). We call µ a Banach limit if µ satisfies µ = µ n (1) = 1 and µ n (a n+1 ) = µ n (a n ) for all (a 0, a 1,...) l. If µ is a Banach limit, then lim inf a n µ n (a n ) lim sup a n for all (a 0, a 1,...) l. (see, e.g., [5, 6]). The Variational inequality problem was initially introduced and studied by Stampacchia [14] in In the recent years, variational inequality problems have been extended to study a large variety of problems arising in structural analysis, economics and optimization. Thus, the problem of solving a variational inequality of the form (2) has

4 730 Abba Auwalu been intensively studied by numerous authors (see for example, [10, 17, 18, 20] and the references therein). Let H be a real Hilbert space. In 2001, Yamada [20] proposed a hybrid steepest descent method for solving variational inequality as follows; Let x 0 H be chosen arbitrary and define a sequence {x n } by x n+1 = T x n µα n F (T x n ), n 0, (9) where T is a nonexpansive mapping on H, F is L-Lipschitzian and η-strongly monotone with L > 0, η > 0, 0 < µ < 2η/L 2. If {α n } is a sequence in (0, 1) satisfying the following conditions: (C1) lim α n = 0; (C2) n=0 α n = ; (C3) n=1 α n+1 α n <, then he prove that the sequence {x n } converges strongly to the unique solution of the variational inequality: F x, x x 0, x F (T ). (10) In 2006, Marino and Xu [10] considered the following general iterative method: starting with an arbitrary initial point x 0 H, define a sequence {x n } by x n+1 = α n γf(x n ) + (I α n A)T x n, n 0, (11) where T is a nonexpansive mapping of H, f is a contraction, A is a linear bounded strongly positive operator, and {α n } is a sequence in (0, 1) satisfying the conditions (C1) (C3). They proved that the sequence {x n } converges strongly to a fixed point x of T which solves the variational inequality: (γf A) x, x x 0, x F (T ). (12) In 2010, Tian [17] combined the iterative method (11) with that of Yamada s (9) and considered the following general iterative method x n+1 = α n γf(x n ) + (I µα n F )T x n, n 0, (13) where T is a nonexpansive mapping on H, f is a contraction, F is k-lipschitzian and η-strongly monotone with k > 0, η > 0, 0 < µ < 2η/k 2. He proved that if the sequence {α n } of parameters satisfies conditions (C1) (C3), then the sequence {x n } generated by (13) converges strongly to a fixed point x of T which solves the variational inequality: (γf µf ) x, x x 0, x F (T ). (14) Very recently, in 2011, Tian and Di [18] studied an algorithm, based on Tian [17] general Iterative algorithm, and proved the following theorem:

5 Synchronal algorithm for a strict pseudocontractions 731 Theorem 1.1 (Synchronal Algorithm) Let H be a real Hilbert space and Let T i : H H be a k i -strictly pseudocontractions for some k i (0, 1) such that N i=1 F (T i), and f be a contraction with coefficient β (0, 1) and λ i be a positive constants such that N i=1 λ i = 1. Let G : H H be an η-strongly monotone and L-Lipschitzian operator with L > 0, η > 0. Assume that 0 < µ < 2η/L 2, 0 < γ < µ(η µl2 )/β = τ/β. Let x 2 0 H be chosen arbitrarily and let {α n }, {β n } be sequences in (0, 1), satisfying the following conditions: (T 1) lim α n = 0, n=0 α n = ; (T 2) n=1 α n+1 α n <, n=1 β n+1 β n < ; (T 3) 0 < max k i β n < a < 1, n 0. Let {x n } be a sequences defined by the composite process { T β n = β n I + (1 β n ) N i=1 λ it i, x n+1 = α n γf(x n ) + (I α n µg)t βn x n, n 0. Then {x n } converges strongly to a common fixed point x of {T i } N i=1 which solves the variational inequality: (15) (γf µg)x, x x 0, N x F (T i ). (16) i=1 The following questions naturally arise in connection with above results: Question 1. Can Theorem of Tian and Di [18] be extend from a real Hilbert space to a general Banach space? such as q-uniformly smooth Banach space. Question 2. Can we extend the iterative method of scheme (15) to a general iterative scheme define over the set of fixed points of a countable infinite family of strict pseudocontractions. The purpose of this paper is to give the affirmative answers to these questions mentioned above. Throughout this paper, we will use the following notations: 1. for weak convergence and for strong convergence. 2. ω ω (x n ) = {x : x nj x} denotes the weak ω-limit set of {x n }. 2 Preliminaries In the sequel, we shall make use of the following lemmas. Lemma 2.1 (Boonchari and Saejung, [1, 2]) Let C be a nonempty, closed and convex subset of a smooth Banach space E. Suppose that {T i } i=1 : C E is a family of λ-strictly pseudocontractive mappings with i=1 F (T i) and {µ i } i=1 is a real sequence in (0, 1) such that i=1 µ i = 1. Then the following conclusions hold: (i) A mapping G : C E defined by G := i=1 µ it i is a λ-strictly pseudocontractive

6 732 Abba Auwalu mapping. (ii) F (G) = i=1 F (T i). Lemma 2.2 (Lim and Xu, [9]) Suppose E is a Banach space with uniform normal structure, K is a nonempty bounded subset of E, and T : K K is uniformly k- Lipschitzian mapping with k < N(E) 1 2. Suppose also there exists nonempty bounded closed convex subset C of K with the following property (P ) : x C implies ω ω (x) C, where ω ω (x) is the ω-limit set of T at x, i.e., the set Then T has a fixed point in C. {y E : y = weak lim j T n j x for some n j }. Lemma 2.3 (Sunthrayuth and Kumam, [15]) Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space E which admits a weakly sequentially continuous generalized duality mapping j q from E into E. Let T : C C be a nonexpansive mapping with F (T ). Then, for all {x n } C, if x n x and x n T x n 0, then x = T x. Lemma 2.4 (Petryshyn, [12]) Let E be a real q-uniformly smooth Banach space and let J q : E 2 E be the generalized duality mapping. Then for any x, y E and j q (x + y) J q (x + y), x + y q x q + q y, j q (x + y). Lemma 2.5 (Sunthrayuth and Kumam, [15]) Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space E. Let F : C E be a η-strongly ( ) 1 q 1 qη accretive and L-Lipschitzian operator with η > 0, L > 0. Assume that 0 < µ < d ql ( ) q and τ = µ η dqµq 1 L q. Then for t ( 0, min { }) 1, 1 q τ, the mapping T := (I tµf ) : C E is a contraction with coefficient (1 tτ). Lemma 2.6 (Zhang and Guo, [21]) Let E be a real q-uniformly smooth Banach space and C be a nonempty closed convex subset of E. Suppose T : C E are λ-strict pseudocontractions such that F (T ). For any α (0, 1), we define T α : C E by T α x = αx+(1 α)t x, for each x C. Then, as α [µ, 1), µ [ max { } ) 0, 1 ( λq d q ) 1 q 1, 1, T α is a nonexpansive mapping such that F (T α ) = F (T ). Lemma 2.7 (Xu, [19]) Let {a n } be a sequence of nonnegative real numbers such that a n+1 (1 γ n )a n + δ n, n 0, where {γ n } is a sequence in (0, 1) and {δ n } is a sequence in R such that: (i) lim γ n = 0 and n=0 γ n = ; δ (ii) lim sup n γn 0 or n=1 δ n <. Then lim a n = 0.

7 Synchronal algorithm for a strict pseudocontractions 733 Lemma 2.8 (Chang et al., [4]) Let E be a real q-uniformly smooth Banach space, then the generalized duality mapping J q : E 2 E is single-valued and uniformly continuous on each bounded subset of E from the norm topology of E to the norm topology of E. Lemma 2.9 (Shioji and Takahashi, [13]) Let a be a real number and a sequence {a n } l such that µ n (a n ) 0 for all Banach limit µ and lim sup (a n+1 a n ) 0. Then, lim sup a n 0. Lemma 2.10 (Mitrinović, [11]) Suppose that q > 1. Then, for any arbitrary positive real numbers x, y, the following inequality holds: xy 1 ( ) q 1 q xq + y q q 1. q Lemma 2.11 Let E be a real q-uniformly smooth Banach space. Let f : E E be a contraction mapping with coefficient α (0, 1). Let T : E E be a nonexpansive mapping such that F (T ) and G : E E be an η-strongly accretive mapping which is also L- Lipschitzian. Assume that 0 < µ < ( qη d ql ) 1 q q 1 Then for each t ( 0, min{1, 1}), the sequence {x τ t} define by x t = tγf(x t ) + (I tµg)t x t and 0 < γ < τ, where τ := µ( η µq 1 d ql q α q converges strongly as t 0 to a fixed point x of T which solves the variational inequality: (µg γf)x, j q (x x) 0, x F (T ). (17) Proof. The definition of {x t } is well definition. Now, for each t ( 0, min{1, 1 τ }), define a mapping T t on C by Then, by Lemma 2.5, we have T t x = tγf(x) + (I tµg)t x, x C. T t x T t y = [tγf(x) + (I tµg)t x] [tγf(y) + (I tµg)t y] tγ[f(x) f(y)] + (I tµf )T x (I tµf )T y tγ f(x) f(y) + (1 tτ) T x T y tγα x y + (1 tτ) x y = [1 t(τ γα)] x y, which implies that T t is a contraction. Hence, T t has a unique fixed point, denoted by x t, which uniquely solve the fixed point equation: x t = tγf(x t ) + (I tµg)t x t. (18) ).

8 734 Abba Auwalu We observe that {x t } is bounded. Indeed, from (18) and Lemma 2.5, we have It follows that x t x = tγf(x t ) + (I tµg)t x t x = t[γf(x t ) µg x] + (I tµg)t x t (I tµg) x (I tµg)t x t (I tµg) x + t γf(x t ) µg x (1 tτ) x t x + tγ f(x t ) f( x) + t γf( x) µg x (1 tτ) x t x + tγα x t x + t γf( x) µg x = [1 t(τ γα)] x t x + t γf( x) µg x. x t x γf( x) µg x. τ γα Hence, {x t } is bounded. Furthermore {f(x t )} and {G(T x t )} are also bounded. Also, from (18), we have x t T x t = t γf(x t ) µg(t x t ) 0 as t 0. (19) Take t, t 0 (0, 1 ). From (18) and Lemma 2.5, we have τ x t x t0 = [tγf(x t ) + (I tµg)t x t ] [t 0 γf(x t0 ) + (I t 0 µg)t x t0 ] = (t t 0 )γf(x t ) + t 0 γ[f(x t ) f(x t0 )] + (t 0 t)µg(t x t ) It follows that +(I t 0 µg)t x t (I t 0 µg)t x t0 (γ f(x t ) + µ G(T x t ) ) t t 0 + t 0 γ f(x t ) f(x t0 ) + (I t 0 µg)t x t (I t 0 µg)t x t0 (γ f(x t ) + µ G(T x t ) ) t t 0 + t 0 γα x t x t0 + (1 t 0 τ) T x t T x t0 (γ f(x t ) + µ G(T x t ) ) t t 0 + [1 t 0 (τ γα)] x t x t0. x t x t0 γ f(x t) + µ G(T x t ) t t 0. t 0 (τ γα) This shows that {x t } is locally Lipschitzian and hence continuous. We next show the uniqueness of a solution of the variational inequality (17). Suppose both x F (T ) and ỹ F (T ) are solutions to (17). From (17), we know that (µg γf) x, j q ( x ỹ) 0. (20) and (µg γf)ỹ, j q (ỹ x) 0. (21) Adding up (20) and (21), we have (µg γf) x (µg γf)ỹ, j q ( x ỹ) 0.

9 Synchronal algorithm for a strict pseudocontractions 735 Observe that d q µ q 1 L q q > 0 η d qµ q 1 L q < η q µ (η d ) qµ q 1 L q < µη q τ < µη. It follows that We notice that 0 < γα < τ < µη. (µg γf) x (µg γf)ỹ, j q ( x ỹ) = (µ(g x Gỹ) γ(f( x) f(ỹ)), j q ( x ỹ) = µ G x Gỹ, j q ( x ỹ) γ f( x) f(ỹ), j q ( x ỹ) µη x ỹ q γ f( x) f(ỹ) x ỹ q 1 µη x ỹ q γα x ỹ q = (µη γα) x ỹ q. Therefore, x = ỹ and the uniqueness is proved. Below, we use x F (T ) to denote the unique solution of the variational inequality (17). Next, we prove that x t x as t 0. Define a map φ : E R by φ(x) = µ n x n x q, x E, where µ n is a Banach limit for each n. Then φ is continuous, convex, and φ(x) as x. Since E is reflexive, there exists y E such that φ(y ) = min φ(u). Hence u E the set K min := {x E : φ(x) = min φ(u)}. u E Therefore, applying Lemma 2.2, we have K min F (T ). Without loss of generality, assume x = y K min F (T ). Let t (0, 1). Then, it follows that φ(x ) φ(x +t(γf µg)x ) and using Lemma 2.4, we obtain that x n x t(γf µg)x q x n x q qt (γf µg)x, j q (x n x t(γf µg)x ). Thus, taking Banach limit over n 1 gives µ n x n x t(γf µg)x q µ n x n x q qtµ n (γf µg)x, j q (x n x t(γf µg)x ). This implies, qtµ n (γf µg)x, j q (x n x t(γf µg)x ) φ(x ) φ(x + t(γf µg)x ) 0.

10 736 Abba Auwalu Therefore µ n (γf µg)x, j q (x n x t(γf µg)x ) 0, n 1. Moreover, µ n (γf µg)x, j q (x n x ) = µ n (γf µg)x, j q (x n x ) j q (x n x t(γf µg)x ) + µ n (γf µg)x, j q (x n x t(γf µg)x ) µ n (γf µg)x, j q (x n x ) j q (x n x t(γf µg)x ). By Lemma 2.8, the duality mapping J q is norm-to-norm uniformly continuous on bounded subset of E, we have that Now, using (18) and Lemma 2.5, we have µ n (γf µg)x, j q (x n x ) 0. (22) x n x q = t n γf(x n ) µgx, j q (x n x ) + (I t n µg)(t x n x ), j q (x n x ) So, = t n γf(x n ) µgx, j q (x n x ) + (I t n µg)t x n (I t n µg)x, j q (x n x ) [1 t n (τ γα)] x n x q + t n (γf µg)x, j q (x n x ). x n x q Again, taking Banach limit, we obtain µ n x n x q 1 τ γα (γf µg)x, j q (x n x ). 1 τ γα µ n (γf µg)x, j q (x n x ) 0, which implies that µ n x n x q = 0. Hence, there exists a subsequence of {x n } which will still be denoted by {x n } such that lim x n = x. We next prove that x solves the variational inequality (17). Since we can derive that x t = tγf(x t ) + (I tµg)t x t, (µg γf)x t = 1 t (I T )x t + µ(gx t GT x t ) Notice that (I T )x t (I T )z, j q (x t z) x t z q T x t T z x t z q 1 x t z q x t z q = 0.

11 Synchronal algorithm for a strict pseudocontractions 737 It follows that, for all z F (T ), (µg γf)x t, j q (x t z) = 1 t (I T )x t (I T )z, j q (x t z) +µ Gx t G(T x t ), j q (x t z) µ Gx t G(T x t ), j q (x t z) µl x t T x t x t z q 1 x t T x t M, (23) where ( M{ is}) an appropriate constant such that M = sup{µl x t z q }, where t 0, min 1, 1 τ. Now replacing t in (23) with tn and letting n, noticing that (I T )x tn (I T )x = 0 for x F (T ), we obtain (µg γf)x, j q (x z) 0. That is, x F (T ) is the solution of (17). Hence, x = x by uniqueness. We have shown that each cluster point of {x t } (at t 0) equals x. Therefore, x t x as t 0. This completes the proof. 3 Main Results Theorem 3.1 (Synchronal Algorithm) Let E be a real q-uniformly smooth Banach space whose duality map is weakly sequentially continuous and C be a nonempty, closed and convex subset of E. Let {T i } i=1 : C E be a family of k-strict pseudocontractions for k (0, 1) such that i=1 F (T i), f be a contraction with coefficient β (0, 1) and {λ i } i=1 be a real sequence in (0, 1) such that i=1 λ i = 1. Let G : C E be an η-strongly accretive and L-Lipschitzian operator with L > 0, η > 0. Assume that 0 < µ < (qη/d q L q ) 1/q 1, 0 < γ < µ(η d q µ q 1 L q /q)/β = τ/β. Let {α n } and {β n } be sequences in (0, 1) satisfying the following conditions: (K1) lim α n = 0, n=0 α n = ; (K2) n=1 α n+1 α n <, n=1 β n+1 β n < ; (K3) 0 < β n < a < 1; β n [µ, 1), µ [ max { 0, 1 ( kq d q ) 1 q 1 }, 1 ). Let {x n } be a sequence defined by the iterative algorithm x 0 C arbitrarily chosen, T βn = β n I + (1 β n ) i=1 λ it i, x n+1 = α n γf(x n ) + (I α n µg)t βn x n, n 0, then {x n } converges strongly to a common fixed point x of {T i } i=1 which solves the variational inequality: (24) (γf µg)x, j q (x x ) 0, x F (T i ). (25) i=1

12 738 Abba Auwalu Proof. Put T := i=1 λ it i, then by Lemma 2.1, we conclude that T is a k-strict pseudocontraction and F (T ) = i=1 F (T i). We can then rewrite the algorithm (24) as x 0 E arbitrarily chosen, T βn = β n I + (1 β n )T, x n+1 = α n γf(x n ) + (I α n µg)t βn x n, n 0. Furthermore, by using Lemma 2.6, we conclude that T βn is a nonexpansive mapping and F (T βn ) = F (T ). From the condition (K1), we may assume, without loss of generality, that α n ( 0, min { }) 1, 1 τ. We shall carry out the proof in six steps as follows: Step 1. We show that {x n } is bounded. Take p i=1 F (T i), then the sequence {x n } satisfies { x n p max x 0 p, } γf(p) µgp, n 0. τ γβ We prove this by Mathematical induction as follows; Obviously, it is true for n = 0. Assume it is true for n = k for some k N. From (24) and Lemma 2.5, we have x k+1 p = α k γf(x k ) + (I α k µg)t β k x k p = α k [γf(x k ) µgp] + (I α k µg)(t β k x k p) (1 α k τ) x k p + α k γ[f(x k ) f(p)] + γf(p) µgp (1 α k τ) x k p + α k γβ x k p + α k γf(p) µgp γf(p) µgp = [1 α k (τ γβ)] x k p + α k (τ γβ) τ γβ max { x k p, γf(p) µgp τ γβ }. Hence the proved. Thus, the sequence {x n } is bounded and so are {T x n }, {GT βn x n } and {f(x n )}. Step 2. We show that lim x n+1 x n = 0.

13 Synchronal algorithm for a strict pseudocontractions 739 Observe that so that x n+2 x n+1 = [α n+1 γf(x n+1 ) + (I α n+1 µg)t β n+1 x n+1 ] [α n γf(x n ) + (I α n µg)t βn x n ] = [α n+1 γf(x n+1 ) α n+1 γf(x n )] + [α n+1 γf(x n ) α n γf(x n )] +[(I α n+1 µg)t β n+1 x n+1 (I α n+1 µg)t βn x n ] +[α n µgt βn x n α n+1 µgt βn x n ] = α n+1 γ[f(x n+1 ) f(x n )] + [(I α n+1 µg)t β n+1 x n+1 (I α n+1 µg)t βn x n ] + (α n+1 α n )γf(x n ) +(α n α n+1 )µgt βn x n, x n+2 x n+1 α n+1 γβ x n+1 x n + (1 α n+1 τ) T β n+1 x n+1 T βn x n + α n+1 α n (γ f(x n ) + µ GT βn x n ) α n+1 γβ x n+1 x n + (1 α n+1 τ) T β n+1 x n+1 T βn x n + α n+1 α n M 1, (26) where M 1 is an appropriate constant such that M 1 sup n 1 {γ f(x n ) + µ GT βn x n }. On the other hand, we note that T β n+1 x n+1 T βn x n T β n+1 x n+1 T β n+1 x n + T β n+1 x n T βn x n x n+1 x n + [β n+1 x n + (1 β n+1 )T x n ] [β n x n + (1 β n )T x n ] = x n+1 x n + β n+1 (x n T x n ) β n (x n T x n ) x n+1 x n + β n+1 β n M 2, (27) where M 2 is an appropriate constant such that M 2 sup n 1 { x n T x n }. Now, substituting (27) into (26) yields x n+2 x n+1 α n+1 γβ x n+1 x n + (1 α n+1 τ) x n+1 x n + α n+1 α n M 1 + β n+1 β n M 2 [1 α n+1 (τ γβ)] x n+1 x n + ( α n+1 α n + β n+1 β n )M 3, where M 3 is an appropriate constant such that M 3 max{m 1, M 2 }. By Lemma 2.7 and the conditions (K1), (K2), we have lim x n+1 x n = 0. (28) Step 3. We show that lim x n T x n = 0.

14 740 Abba Auwalu From (24) and condition (K1), we have x n+1 T βn x n = α n γf(x n ) + (I α n µg)t βn x n T βn x n α n γf(x n ) + µgt βn x n 0 as n. (29) On the other hand, x n+1 T βn x n = x n+1 [β n x n + (1 β n )T x n ] = x n+1 x n + (1 β n )(x n T x n ) (1 β n ) x n T x n x n+1 x n, which implies, by condition (K3), that x n T x n 1 ( x n+1 x n + x n+1 T βn x n ) 1 β n 1 1 a ( x n+1 x n + x n+1 T βn x n ). Hence, from (28) and (29), we have Step 4. lim x n T x n = 0. (30) We show that ω ω (x n ) F (T ). From the boundedness of {x n }, without loss of generality, we may assume that x n y. Hence, by Lemma 2.3 and (30), we obtain T y = y. So, we have Step 5. ω ω (x n ) F (T ). (31) We show that lim sup (γf µg)x, j q (x n x ) 0, where x is obtained in Lemma Put a n := (γf µg)x, j q (x n x ). Then, by (22), we have µ n (a n ) 0 for any Banach limit µ. Furthermore, by (28), x n+1 x n 0 as n, we therefore conclude that lim sup ( (a n+1 a n ) = lim sup (γf µg)x, j q (x n+1 x ) (γf µg)x, j q (x n x ) ) = lim sup (γf µg)x, j q (x n+1 x ) j q (x n x ) = 0. Hence, by Lemma 2.9, we have lim sup a n 0, that is, lim sup (γf µg)x, j q (x n x ) 0. (32)

15 Synchronal algorithm for a strict pseudocontractions 741 Step 6. We show that Using (24), Lemmas 2.5 and 2.10, we have x n+1 x q = x n+1 x, j q (x n+1 x ) lim x n x = 0. = α n [γf(x n ) µgx ] + (I α n µg)(t βn x n x ), j q (x n+1 x ) = α n γf(x n ) γf(x ), j q (x n+1 x ) + α n γf(x ) µgx, j q (x n+1 x ) + (I α n µg)(t βn x n x ), j q (x n+1 x ) α n γ f(x n ) f(x ) x n+1 x q 1 + α n (γf µg)x, j q (x n+1 x ) This implies that + (I α n µg)(t βn x n x ) x n+1 x q 1 (1 α n τ) x n x x n+1 x q 1 + α n γβ x n x x n+1 x q 1 +α n (γf µg)x, j q (x n+1 x ) [1 α n (τ γβ)] [ 1 q x n x q + ( q 1) xn+1 x q] q +α n (γf µg)x, j q (x n+1 x ). x n+1 x q 1 α n (τ γβ) 1 + α n (q 1)(τ γβ) x n x q qα n α n (q 1)(τ γβ) (γf µg)x, j q (x n+1 x ) [1 α n (τ γβ)] x n x q qα n α n (q 1)(τ γβ) (γf µg)x, j q (x n+1 x ) (1 γ n ) x n x q + δ n, where γ n := α n (τ γβ) and δ n := lim γ n = 0, γ n =. Now δn γ n = n=0 δ lim sup n γ n 0. Hence, by Lemma 2.7, we conclude that This completes the proof. qα n 1+α n(q 1)(τ γβ) (γf µg)x, j q (x n+1 x ). From (K1), q [1+α n(q 1)(τ γβ)](τ γβ) (γf µg)x, j q (x n+1 x ). So, lim x n x = 0. 4 Conclusion The following Corollaries are consequences of Theorem 3.1.

16 742 Abba Auwalu Corollary 4.1 Let E be a real q-uniformly smooth Banach space whose duality map is weakly sequentially continuous and C be a nonempty, closed and convex subset of E. Let {T i } N i=1 : C E be a family of k i -strict pseudocontractions for k i (0, 1), (i = 1, 2,..., N), such that N i=1 F (T i) and k = min{k i : 1 i N}, f be a contraction with coefficient β (0, 1) and {λ i } N i=1 be a real sequence in (0, 1) such that N i=1 λ i = 1. Let G : C E be an η-strongly accretive and L-Lipschitzian operator with L > 0, η > 0. Assume that 0 < µ < (qη/d q L q ) 1/q 1, 0 < γ < µ(η d q µ q 1 L q /q)/β = τ/β. Let {α n } and {β n } be sequences in (0, 1) satisfying the following conditions (K1) (K3). Let {x n } be a sequence defined by the iterative algorithm x 0 C arbitrarily chosen, T βn = β n I + (1 β n ) N i=1 λ it i, x n+1 = α n γf(x n ) + (I α n µg)t βn x n, n 0, then {x n } converges strongly to a common fixed point x of {T i } N i=1 which solves the variational inequality: (33) (γf µg)x, j q (x x ) 0, N x F (T i ). (34) i=1 Corollary 4.2 Let E be a real 2-uniformly smooth Banach space whose duality map is weakly sequentially continuous and C be a nonempty, closed and convex subset of E. Let {T i } i=1 : C E be a family of k-strict pseudocontractions for k (0, 1), such that i=1 F (T i), f be a contraction map with coefficient β (0, 1) and {λ i } i=1 be a real sequence such that i=1 λ i = 1. Let G : C E be an η-strongly accretive and L-Lipschitzian operator with L > 0, η > 0. Assume that 0 < µ < 2η/d 2 L 2, 0 < γ < µ(η d 2 µl 2 /2)/β = τ/β. Let {α n } and {β n } be sequences in (0, 1) satisfying the conditions (K1) (K3). Let {x n } be a sequence defined by the algorithm (24), then {x n } converges strongly to a common fixed point x of {T i } i=1 which solves the variational inequality: (γf µg)x, j(x x ) 0, x F (T i ). (35) i=1 Corollary 4.3 Let E = L p (or l p ) space, (1 < p < ). Let {T i } i=1 : E E be k i -strict pseudocontractions for k i (0, 1) such that i=1 F (T i), f be a contraction map with coefficient β (0, 1) and λ i be positive constants such that i=1 λ i = 1. Let G : E E be an η-strongly accretive and L-Lipschitzian operator with L > 0, η > 0. Assume that 0 < µ < 2η/d 2 L 2, 0 < γ < µ(η d 2 µl 2 /2)/β = τ/β. Let {α n } and {β n } be sequences in (0, 1) satisfying the conditions (K1) (K3). Let {x n } be a sequence defined by the composite process (24), then {x n } converges strongly to a common fixed point x of {T i } i=1 which solves the variational inequality (35).

17 Synchronal algorithm for a strict pseudocontractions 743 Corollary 4.4 (Tian and Di, [18]) Let E = H be a real Hilbert space. Let {x n } be a sequence generated by (15). Assume that {α n } are {β n } are sequences in (0, 1) satisfying the conditions (K1) (K3), then {x n } converges strongly to a common fixed point x of {T i } N i=1 which solves the variational inequality (16). Corollary 4.5 (Tian, [17]) Let E = H be a real Hilbert space. Let {x n } be a sequence generated by (33). Assume that {α n } is a sequence in (0, 1) satisfying (K1) and (K2), then {x n } converges strongly to a common fixed point of T which solves the variational inequality (14). Corollary 4.6 (Marino and Xu, [10]) Let E = H be a real Hilbert space. Let {x n } be a sequence generated by (11). Assume that {α n } is a sequence in (0, 1) satisfying (K1) and (K2), then {x n } converges strongly to a common fixed point of T which solves the variational inequality (12). Corollary 4.7 (Yamada, [20]) Let E = H be a real Hilbert space. Let {x n } be a sequence generated by (9). Assume that {α n } is a sequence in (0, 1) satisfying (K1) and (K2), then {x n } converges strongly to a common fixed point of T which solves the variational inequality (10). References [1] D. Boonchari and S. Saejung, Weak and strong convergence theorems of an implicit iteration for a countable family of continuous pseudocontractive mappings, J. Comput. Appl. Math., 233 (2009), [2] D. Boonchari and S. Saejung, Construction of common fixed points of a countable family of λ-demicontractive mappings in arbitrary Banach spaces, Appl. Math. Comput., 216 (2010), [3] W.L. Bynum, Normal structure coefficients for Banach spaces, Pacific J. of Maths., 86 (1980), [4] S.-S. Chang, Iterative Methods for Nonlinear Operator Equations in Banach spaces, Nova Science Publishers, New York, [5] C.E. Chidume, Geometric Properties of Banach Spaces And Nonlinear Iterations, Springer Verlag Series: Lecture Notes in Mathematics, 1965 (2009), XVII, 326p, ISBN , [6] C.E. Chidume, J. Li and A. Udomene, Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings, Proc. of the Amer. Math. Soc., 133 (2005),

18 744 Abba Auwalu [7] K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, New York, [8] D. Kinderlehrer and G. Stampacchia, An Introduction To Variational Inequalities and Their Appl., Vol. 88 of Pure and Applied Maths., Academic Press, New York, USA, [9] T.C. Lim and H.-K. Xu, Fixed Point Theorems for Asymptotically Nonexpansive Mappings, Nonlinear Anal., 22 (1994), [10] G. Marino and H.-K. Xu, A General Iterative Method for Nonexpansive Mappings in Hilbert Spaces, J. Math. Anal. Appl., 318 (2006), [11] D.S. Mitrinović, Analytical inequalities, Springer Verlag, New York, [12] W. V. Petryshyn, A Characterization of strict convexity of Banach spaces and other use of duality mappings, J. Funct. Anal., 6 (1970), [13] N. Shioji and W. Takahashi, Strong Convergence of Approximated Sequences for Nonexpansive Mappings in Banach spaces, Proc. Am. Math. Soc., 125 (1997), [14] G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, Comptes Rendus de l Academie des Sciences, 258 (1964), [15] P. Sunthrayuth and P. Kumam, Iterative methods for variational inequality problems and fixed point problems of a countable family of strict pseudo-contractions in a q-uniformly smooth Banach space, Fixed Point Theory and Appl., (2012), doi: / , 31 pages. [16] W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokahama, Japan, [17] M. Tian, A General Iterative Method for Nonexpansive Mappings in Hilbert Spaces, Nonlinear Anal., 73 (2010), [18] M. Tian and L.Y. Di, Synchronal Algorithm and Cyclic Algorithm for Fixed Point Problems and Variational Inequality Problems in Hilbert spaces, Fixed Point Theory and Appl., (2011). doi: / , 27 pages. [19] H.-K. Xu, Iterative Algorithm for Nonlinear Operators, J. Lond. Math. Soc., 66 (2002), [20] I. Yamada, The Hybrid Steepest-descent Method for Variational Inequality Problems Over the intersection of the Fixed Point Sets of Nonexpansive Mappings, In: Butnarin, D, Censor, Y, Reich S (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Application. North-Holland, Amsterdam, 8 (2001),

19 Synchronal algorithm for a strict pseudocontractions 745 [21] Y. Zhang and Y. Guo, Weak Convergence Theorems of Three Iterative Methods for Strictly Pseudocontractive Mappings of Browder-Petryshyn Type, Fixed Point Theory and Applications, (2008), Article ID , 13 pages. Received: October 14, 2012