Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces

Mathematical and Computer Modelling 55 (2012) 538–546

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Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces Gang Cai ∗ , Shangquan Bu Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, China

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Article history: Received 2 June 2011 Received in revised form 18 August 2011 Accepted 22 August 2011

In this paper, we study a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of variational inequalities for two inverse strongly accretive mappings and the set of common fixed points of an infinite family of nonexpansive mappings in a real 2-uniformly smooth and uniformly convex Banach space. We prove some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the corresponding results announced by many others. © 2011 Elsevier Ltd. All rights reserved.

Keywords: Strong convergence Variational inequality Fixed point Nonexpansive mappings Banach space

1. Introduction Throughout this paper, we denote by X and X ∗ a real Banach space and the dual space of X , respectively. Let C be a subset of X and T be a self-mapping of C . We use F (T ) to denote the  fixed points of T .  ∗ The duality mapping J : X → 2X is defined by J (x) = x∗ ∈ X ∗ : ⟨x, x∗ ⟩ = ‖x‖2 , ‖x∗ ‖ = ‖x‖ , ∀x ∈ X . If X is a Hilbert space, then J = I, where I is the identity mapping. It is well-known that if X is smooth, then J is single-valued,which is denoted by j. Recall that a mapping f : C → C is a contraction on C , if there exists a constant α ∈ (0, 1) such that

‖f (x) − f (y)‖ ≤ α‖x − y‖, ∏

∀x, y ∈ C .

We use C to denote the collection of all contractions on C . A mapping T : C → C is said to be nonexpansive, if

‖Tx − Ty‖ ≤ ‖x − y‖,

∀x, y ∈ C .

(1.1)

A mapping A : C → X is said to be accretive if there exists j(x − y) ∈ J (x − y) such that

⟨Ax − Ay, j(x − y)⟩ ≥ 0,

∀x, y ∈ C .

(1.2)

A mapping A : C → X is said to be α -inverse strongly accretive if there exists j(x − y) ∈ J (x − y) and α > 0 such that

⟨Ax − Ay, j(x − y)⟩ ≥ α‖Ax − Ay‖2 ,

∀x, y ∈ C .

(1.3)

The variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral, free, moving, equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework.

∗

Corresponding author. E-mail addresses: [email protected] (G. Cai), [email protected] (S. Bu).

0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.08.031

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Several numerical methods have been developed for solving variational inequalities and related optimization problems; see [1–6] and the references therein. Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that the classical variational inequality is to find an x∗ such that



Ax∗ , x − x∗ ≥ 0,

∀x ∈ C ,



(1.4)

where A : C → H is a nonlinear mapping. The set of solutions of (1.4) is denoted by VI (A, C ). Let A, B : C → H be two mappings. Ceng et al. [1] considered the following problem of finding (x∗ , y∗ ) ∈ C × C such that

  λAy∗ + x∗ − y∗ , x − x∗ ≥ 0, ∀x ∈ C ,   µBx∗ + y∗ − x∗ , x − y∗ ≥ 0, ∀x ∈ C ,

(1.5)

which is called a general system of variational inequalities, where λ > 0 and µ > 0 are two constants. Very recently, Yao et al. [5] introduced an iterative scheme 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. Let C be a nonempty closed convex subset of a real Banach space X . For given two operators A, B : C → X , we consider the problem of finding (x∗ , y∗ ) ∈ C × C such that

  λAy∗ + x∗ − y∗ , j(x − x∗ ) ≥ 0, ∀x ∈ C ,   µBx∗ + y∗ − x∗ , j(x − y∗ ) ≥ 0, ∀x ∈ C ,

(1.6)

which is called the system of general variational inequalities in a real Banach space. In this paper, motivated by the above facts, we study a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of variational inequalities for two inverse strongly accretive mappings and the set of common fixed points of an infinite family of nonexpansive mappings in a real 2-uniformly smooth and uniformly convex Banach space. We prove some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the corresponding results announced by many others. 2. Preliminaries A Banach space X is said to be strictly convex, if whenever x and y are not collinear, then: ‖x + y‖ < ‖x‖ + ‖y‖. Then the modulus of convexity of X is defined by

  1 δX (ϵ) = inf 1 − ‖x + y‖ : ‖x‖, ‖y‖ ≤ 1, ‖x − y‖ ≥ ϵ , 2

for all ϵ ∈ [0, 2]. X is said to be uniformly convex if δX (0) = 0, and δX (ϵ) > 0 for all 0 < ϵ ≤ 2. Hilbert space H is 2-uniformly convex, while Lp is max {p, 2}-uniformly convex for every p > 1. Let ρX : [0, ∞) −→ [0, ∞) be the modulus of smoothness of X defined by

ρX (t ) = sup



1 2



(‖x + y‖ + ‖x − y‖) − 1 : x ∈ S (X ), ‖y‖ ≤ t . ρ (t )

A Banach space X is said to be uniformly smooth if Xt → 0 as t → 0. A Banach space X is said to be q-uniformly smooth, if there exists a fixed constant c > 0 such that ρX (t ) ≤ ct q . A typical example of uniformly smooth Banach spaces is Lp , where p > 1. More precisely, Lp is min {p, 2}-uniformly smooth for every p > 1. Recall that, if C and D are nonempty subsets of a Banach space X such that C is nonempty closed convex and D ⊂ C , then a mapping P : C → D is sunny [7] provided P (x + t (x − P (x))) = P (x) for all x ∈ C and t ≥ 0, whenever x + t (x − P (x)) ∈ C . A mapping P : C → D is called a retraction if Px = x for all x ∈ D. Furthermore, P is a sunny nonexpansive retraction from C onto D if P is a retraction from C onto D which is also sunny and nonexpansive. A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. The following propositions concern the sunny nonexpansive retraction. Proposition 2.1 ([7]). Let C be a closed convex subset of a smooth Banach space X . Let D be a nonempty subset of C . Let P : C → D be a retraction and let J be the normalized duality mapping on X . Then the following are equivalent: (a) P is sunny and nonexpansive. (b) ‖Px − Py‖2 ≤ ⟨x − y, J (Px − Py)⟩ , ∀x, y ∈ C . (c) ⟨x − Px, J (y − Px)⟩ ≤ 0, ∀x ∈ C , y ∈ D. Proposition 2.2 ([8]). If X is strictly convex and uniformly smooth and if T : C → C is a nonexpansive mapping having a nonempty fixed point set F (T ), then the set F (T ) is a sunny nonexpansive retraction of C .

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Lemma 2.3 ([9]). Let {xn } and {zn } be bounded sequences in a Banach space X and let {βn } be a sequence in [0, 1] which satisfies the following condition: 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1. Suppose xn+1 = βn xn + (1 − βn )zn , n ≥ 0 and lim supn→∞ (‖zn+1 − zn ‖ − ‖xn+1 − xn ‖) ≤ 0. Then limn→∞ ‖zn − xn ‖ = 0. Lemma 2.4 ([10]). Assume {an } is a sequence of nonnegative real numbers such that an+1 ≤ (1 − αn )an + δn , n ≥ 0, where ∑ δn {αn } is a sequence in (0, 1) and {δn } is a sequence in R such that ∞ n=0 αn = ∞ and lim supn→∞ α ≤ 0. Then limn→∞ an = 0. n

Lemma 2.5 ([11]). Let C be a nonempty closed convex subset of a Banach space X . Let S1 , S2 , . . . be a sequence of mappings ∑∞ of C into itself. Suppose that n=1 sup {‖Sn+1 x − Sn x‖ : x ∈ C } < ∞. Then for each y ∈ C , {Sn y} converges strongly to some point of C . Moreover, let S be a mapping of C into itself defined by Sy = limn→∞ Sn y for all y ∈ C . Then limn→∞ sup {‖Sx − Sn x‖ : x ∈ C } = 0. Lemma 2.6 ([12]). Let X be a real 2-uniformly smooth Banach space, then there exists a best smooth constant K > 0 such that ‖x + y‖2 ≤ ‖x‖2 + 2 ⟨y, jx⟩ + 2‖Ky‖2 , ∀x, y ∈ X . Lemma 2.7 ([13]). Let X be a real smooth and uniformly convex Banach space and let r > 0. Then there exists a strictly increasing, continuous and convex function g : [0, 2r ] → R such that g (0) = 0 and g (‖x − y‖) ≤ ‖x‖2 − 2 ⟨x, jy⟩ + ‖y‖2 , for all x, y ∈ Br . Lemma 2.8. Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X . Let the mapping A : C → X be α -inverse-strongly accretive. Then, we have

‖(I − λA)x − (I − λA)y‖2 ≤ ‖x − y‖2 + 2λ(λK 2 − α)‖Ax − Ay‖2 , where λ > 0. In particular, if 0 < λ ≤ Kα2 , then I − λA is nonexpansive. Proof. Indeed, for all x, y ∈ C , it follows from Lemma 2.6 that

‖(I − λA)x − (I − λA)y‖2 = ≤ ≤ =

‖x − y − λ(Ax − Ay)‖2 ‖x − y‖2 − 2λ ⟨Ax − Ay, j(x − y)⟩ + 2λ2 K 2 ‖Ax − Ay‖2 ‖x − y‖2 − 2λα‖Ax − Ay‖2 + 2λ2 K 2 ‖Ax − Ay‖2 ‖x − y‖2 + 2λ(λK 2 − α)‖Ax − Ay‖2 .

It is clear that if 0 < λ ≤ Kα2 , then I − λA is nonexpansive. This completes the proof.



Lemma 2.9. Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X . Let QC be the sunny nonexpansive retraction from X onto C . Let the mappings A, B : C → X be α -inverse-strongly accretive and β -inverse-strongly accretive, respectively. Let G : C → C be a mapping defined by G(x) = QC [QC (x − µBx) − λAQC (x − µBx)] , If 0 < λ ≤

α

K2

and 0 < µ ≤

β K2

∀x ∈ C .

, then G : C → C is nonexpansive.

Proof. For all x, y ∈ C , by Lemma 2.8, we have

‖G(x) − G(y)‖ = ≤ ≤ ≤ ≤

‖QC [QC (x − µBx) − λAQC (x − µBx)] − QC [QC (y − µBy) − λAQC (y − µBy)] ‖ ‖(I − λA)QC (I − µB)x − (I − λA)QC (I − µB)y‖ ‖QC (I − µB)x − QC (I − µB)y‖ ‖(I − µB)x − (I − µB)y‖ ‖x − y‖,

which implies that G is nonexpansive. This completes the proof.



Lemma 2.10. Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X . Let QC be the sunny nonexpansive retraction from X onto C . Let A, B : C → X be two nonlinear mappings. For given x∗ , y∗ ∈ C , (x∗ , y∗ ) is a solution of problem (1.6) if and only if x∗ = QC (y∗ −λAy∗ ), where y∗ = QC (x∗ −µBx∗ ), that is, x∗ = Gx∗ , where G is defined by Lemma 2.9. Proof. We rewrite (1.6) as

 ∗  ∗ ∗ ∗ (y∗ − λAy ∗) − x ∗, j(x − x ∗)  ≤ 0, ∀x ∈ C , (x − µBx ) − y , j(x − y ) ≤ 0, ∀x ∈ C . From Proposition 2.1, we deduce that (2.1) is equivalent to



x∗ = QC (y∗ − λAy∗ ), y∗ = QC (x∗ − µBx∗ ).

That is, x∗ = Gx∗ . This completes the proof.



(2.1)

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Lemma 2.11 ([10]). Let X be a uniformly smooth Banach space, C be a closed convex subset of X , T : C → C be a nonexpansive mapping with F (T ) ̸= ∅ and let f ∈ ΠC . Then the sequence {xt } define by xt = tf (xt ) + (1 − t )Txt converges strongly to a point in F (T ). If we define a mapping Q : ΠC → F (T ) by Q (f ) := limt →0 xt , ∀f ∈ ΠC . Then Q (f ) solves the following variational inequality:

⟨(I − f )Q (f ), j(Q (f ) − p)⟩ ≤ 0,

∀f ∈ ΠC , p ∈ F (T ).

Lemma 2.12 ([14]). Let C be a closed convex subset of a strictly convex Banach space X . Let {Tn : n ∈ N} be ∑ a sequence of ∞ nonexpansive mappings on C . Suppose ∩∞ F (Tn ) is nonempty. Let {λn } be a sequence of positive numbers with n=1 λn = 1. n = 1 ∑∞ ∞ Then a mapping S on C defined by Sx = n=1 λn Tn x for x ∈ C is well defined, non-expansive and F (S ) = ∩n=1 F (Tn ) holds. Lemma 2.13 ([15]). In a Banach space X , the following inequality holds:

‖x + y‖2 ≤ ‖x‖2 + 2 ⟨y, j(x + y)⟩ ,

∀x, y ∈ X ,

where j(x + y) ∈ J (x + y). 3. Main results Theorem 3.1. Let C be a nonempty closed subset of a uniformly convex and 2-uniformly smooth Banach space X . Let QC be the sunny nonexpansive retraction from X onto C . Let the mappings A, B : C → X be α -inverse-strongly accretive and β -inversestrongly accretive, respectively. Let f be a contraction of C into itself with coefficient δ ∈ (0, 1). Let {Sn }∞ n=1 be a family of nonexpansive mappings of C into itself such that F = ∩∞ F ( S ) ∩ F ( G ) = ̸ ∅ , where G is defined by Lemma 2.9. For arbitrarily i i=1 given x1 ∈ C , let {xn } be the sequence generated by

 x = βn xn + (1 − βn )Sn yn ,   n+1 yn = αn f (xn ) + (1 − αn )zn ,  zn = QC (un − λAun ), un = QC (xn − µBxn ),

(3.1)

β where 0 < λ < Kα2 , 0 < µ < K 2 . Suppose {αn } and {βn } be two sequences satisfy the following conditions:

(i) 0 < lim infn→∞ β∑ n ≤ lim supn→∞ βn < 1; ∞ (ii) limn→∞ αn = 0, n=1 αn = ∞.

Assume that n=1 supx∈D ‖Sn+1 x − Sn x‖ < ∞ for any bounded subset D of C and let S be a mapping of C into X defined by Sx = limn→∞ Sn x for all x ∈ C and suppose that F (S ) = ∩∞ n=1 F (Sn ). Then {xn } converges strongly to q ∈ F , which solves the following variational inequality:

∑∞

⟨q − f (q), j(q − p)⟩ ≤ 0,

∀p ∈ F .

Proof. Let x ∈ F , from Lemma 2.10, we see x∗ = QC (QC (x∗ − µBx∗ ) − λAQC (x∗ − µBx∗ )). Put y∗ = QC (x∗ − µBx∗ ), then x∗ = QC (y∗ − λAy∗ ). From Lemma 2.9, we have ∗

‖zn − x∗ ‖ = ‖Gxn − Gx∗ ‖ ≤ ‖xn − x∗ ‖.

(3.2)

It follows from (3.2) that

‖yn − x∗ ‖ = ‖αn (f (xn ) − x∗ ) + (1 − αn )(zn − x∗ )‖ ≤ αn ‖f (xn ) − f (x∗ )‖ + αn ‖f (x∗ ) − x∗ ‖ + (1 − αn )‖zn − x∗ ‖ ≤ αn δ‖xn − x∗ ‖ + αn ‖f (x∗ ) − x∗ ‖ + (1 − αn )‖xn − x∗ ‖ = [1 − αn (1 − δ)]‖xn − x∗ ‖ + αn ‖f (x∗ ) − x∗ ‖. From (3.3), we obtain

‖βn (xn − x∗ ) + (1 − βn )(Sn yn − x∗ )‖ βn ‖xn − x∗ ‖ + (1 − βn )‖Sn yn − x∗ ‖ βn ‖xn − x∗ ‖ + (1 − βn )‖yn − x∗ ‖ βn ‖xn − x∗ ‖ + (1 − βn )[(1 − αn (1 − δ))‖xn − x∗ ‖ + αn ‖f (x∗ ) − x∗ ‖] ‖f (x∗ ) − x∗ ‖ = [1 − (1 − βn )(1 − δ)αn ]‖xn − x∗ ‖ + (1 − βn )(1 − δ)αn 1−δ  ∗ ∗  ‖ f ( x ) − x ‖ ≤ max ‖x1 − x∗ ‖, , 1−δ

‖ x n +1 − x ∗ ‖ = ≤ ≤ ≤

which implies that {xn } is bounded. From (3.2) and (3.3), we know that {zn } and {yn } are also bounded.

(3.3)

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Next we show that ‖xn − Sn yn ‖ → 0 as n → ∞. We observe

‖zn+1 − zn ‖ = ‖Gxn+1 − Gxn ‖ ≤ ‖xn+1 − xn ‖.

(3.4)

It follows that

‖yn+1 − yn ‖ = ‖αn+1 f (xn+1 ) + (1 − αn+1 )zn+1 − αn f (xn ) − (1 − αn )zn ‖ ≤ αn+1 ‖f (xn+1 ) − zn+1 ‖ + αn ‖f (xn ) − zn ‖ + ‖zn+1 − zn ‖ ≤ αn+1 ‖f (xn+1 ) − zn+1 ‖ + αn ‖f (xn ) − zn ‖ + ‖xn+1 − xn ‖. Hence we have

‖Sn+1 yn+1 − Sn yn ‖ ≤ ‖Sn+1 yn+1 − Sn+1 yn ‖ + ‖Sn+1 yn − Sn yn ‖ ≤ ‖yn+1 − yn ‖ + ‖Sn+1 yn − Sn yn ‖ ≤ αn+1 ‖f (xn+1 ) − zn+1 ‖ + αn ‖f (xn ) − zn ‖ + ‖xn+1 − xn ‖ + ‖Sn+1 yn − Sn yn ‖, which implies

‖Sn+1 yn+1 − Sn yn ‖ − ‖xn+1 − xn ‖ ≤ αn+1 ‖f (xn+1 ) − zn+1 ‖ + αn ‖f (xn ) − zn ‖ + ‖Sn+1 yn − Sn yn ‖. From condition (ii) and the assumption on {Sn }, we have lim sup(‖Sn+1 yn+1 − Sn yn ‖ − ‖xn+1 − xn ‖) ≤ 0. n→∞

It follows from Lemma 2.3 that lim ‖Sn yn − xn ‖ = 0.

(3.5)

n→∞

Hence we obtain lim ‖xn+1 − xn ‖ = lim (1 − βn )‖Sn yn − xn ‖ = 0.

n→∞

n→∞

(3.6)

Next we show that ‖xn − zn ‖ → 0 as n → ∞. From Lemma 2.8, we have

‖un − y∗ ‖2 = ‖QC (xn − µBxn ) − QC (x∗ − µBx∗ )‖2 ≤ ‖xn − x∗ − µ(Bxn − Bx∗ )‖2 ≤ ‖xn − x∗ ‖2 − 2µ(β − K 2 µ)‖Bxn − Bx∗ ‖2 ,

(3.7)

and

‖zn − x∗ ‖2 = ‖QC (un − λAun ) − QC (y∗ − λAy∗ )‖2 ≤ ‖un − y∗ − λ(Aun − Ay∗ )‖2 ≤ ‖un − y∗ ‖2 − 2λ(α − K 2 λ)‖Aun − Ay∗ ‖2 .

(3.8)

Substituting (3.7) into (3.8), we obtain

‖zn − x∗ ‖2 ≤ ‖xn − x∗ ‖2 − 2µ(β − K 2 µ)‖Bxn − Bx∗ ‖2 − 2λ(α − K 2 λ)‖Aun − Ay∗ ‖2 .

(3.9)

2

By the convexity of ‖.‖ and (3.1), we have

‖xn+1 − x∗ ‖2 = ≤ ≤ = ≤ ≤

‖βn (xn − x∗ ) + (1 − βn )(Sn yn − x∗ )‖2 βn ‖xn − x∗ ‖2 + (1 − βn )‖Sn yn − x∗ ‖2 βn ‖xn − x∗ ‖2 + (1 − βn )‖yn − x∗ ‖2 βn ‖xn − x∗ ‖2 + (1 − βn )‖αn (f (xn ) − x∗ ) + (1 − αn )(zn − x∗ )‖2 βn ‖xn − x∗ ‖2 + (1 − βn )αn ‖f (xn ) − x∗ ‖2 + (1 − βn )(1 − αn )‖zn − x∗ ‖2 αn ‖f (xn ) − x∗ ‖2 + βn ‖xn − x∗ ‖2 + (1 − βn )‖zn − x∗ ‖2

≤ αn M1 + βn ‖xn − x∗ ‖2 + (1 − βn )‖zn − x∗ ‖2 , where M1 = supn≥1 ‖f (xn ) − x ‖ . Substituting (3.9) into (3.10), we have ∗ 2

‖xn+1 − x∗ ‖2 ≤ αn M1 + βn ‖xn − x∗ ‖2 + (1 − βn )[‖xn − x∗ ‖2 − 2µ(β − K 2 µ)‖Bxn − Bx∗ ‖2 − 2λ(α − K 2 λ)‖Aun − Ay∗ ‖2 ] = αn M1 + ‖xn − x∗ ‖2 − 2(1 − βn )µ(β − K 2 µ)‖Bxn − Bx∗ ‖2 − 2(1 − βn )λ(α − K 2 λ)‖Aun − Ay∗ ‖2 ,

(3.10)

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which implies 2(1 − βn )µ(β − K 2 µ)‖Bxn − Bx∗ ‖2 + 2(1 − βn )λ(α − K 2 λ)‖Aun − Ay∗ ‖2

≤ αn M1 + ‖xn − x∗ ‖2 − ‖xn+1 − x∗ ‖2 ≤ αn M1 + ‖xn − xn+1 ‖(‖xn − x∗ ‖ + ‖xn+1 − x∗ ‖). β Since 0 < λ < Kα2 , 0 < µ < K 2 , conditions (i), (ii) and (3.6), we obtain

lim ‖Bxn − Bx∗ ‖ = 0,

n→∞

lim ‖Aun − Ay∗ ‖ = 0.

n→∞

(3.11)

From Proposition 2.1 and Lemma 2.7, we have

‖un − y∗ ‖2 = ‖QC (xn − µBxn ) − QC (x∗ − µBx∗ )‖2   ≤ xn − µBxn − (x∗ − µBx∗ ), j(un − y∗ )     = xn − x∗ , j(un − y∗ ) + µ Bx∗ − Bxn , j(un − y∗ ) ≤

1 2

[‖xn − x∗ ‖2 + ‖un − y∗ ‖2 − g1 (‖xn − un − (x∗ − y∗ )‖)] + µ‖Bx∗ − Bxn ‖‖un − y∗ ‖,

which implies

‖un − y∗ ‖2 ≤ ‖xn − x∗ ‖2 − g1 (‖xn − un − (x∗ − y∗ )‖) + 2µ‖Bx∗ − Bxn ‖ ‖un − y∗ ‖.

(3.12)

In the same way, we obtain

‖zn − x∗ ‖2 = ‖QC (un − λAun ) − QC (y∗ − λAy∗ )‖2   ≤ un − λAun − (y∗ − λAy∗ ), j(zn − x∗ )     = un − y∗ , j(zn − x∗ ) + λ Ay∗ − Aun , j(zn − x∗ ) ≤

1 2

[‖un − y∗ ‖2 + ‖zn − x∗ ‖2 − g2 (‖un − zn + (x∗ − y∗ )‖)] + λ‖Ay∗ − Aun ‖ ‖zn − x∗ ‖,

which implies

‖zn − x∗ ‖2 ≤ ‖un − y∗ ‖2 − g2 (‖un − zn + (x∗ − y∗ )‖) + 2λ‖Ay∗ − Aun ‖ ‖zn − x∗ ‖.

(3.13)

Substituting (3.12) into (3.13), we get

‖zn − x∗ ‖2 ≤ ‖xn − x∗ ‖2 − g1 (‖xn − un − (x∗ − y∗ )‖) − g2 (‖un − zn + (x∗ − y∗ )‖) + 2µ‖Bx∗ − Bxn ‖ ‖un − y∗ ‖ + 2λ‖Ay∗ − Aun ‖ ‖zn − x∗ ‖.

(3.14)

From (3.10) and (3.14), we have

‖xn+1 − x∗ ‖2 ≤ αn M1 + βn ‖xn − x∗ ‖2 + (1 − βn )[‖xn − x∗ ‖2 − g1 (‖xn − un − (x∗ − y∗ )‖) − g2 (‖un − zn + (x∗ − y∗ )‖) + 2µ‖Bx∗ − Bxn ‖ ‖un − y∗ ‖ + 2λ‖Ay∗ − Aun ‖ ‖zn − x∗ ‖] ≤ αn M1 + ‖xn − x∗ ‖2 − (1 − βn )g1 (‖xn − un − (x∗ − y∗ )‖) − (1 − βn )g2 (‖un − zn + (x∗ − y∗ )‖) + 2µ‖Bx∗ − Bxn ‖ ‖un − y∗ ‖ + 2λ‖Ay∗ − Aun ‖ ‖zn − x∗ ‖, which implies

(1 − βn )g1 (‖xn − un − (x∗ − y∗ )‖) + (1 − βn )g2 (‖un − zn + (x∗ − y∗ )‖) ≤ αn M1 + ‖xn − x∗ ‖2 − ‖xn+1 − x∗ ‖2 + 2µ‖Bx∗ − Bxn ‖ ‖un − y∗ ‖ + 2λ‖Ay∗ − Aun ‖ ‖zn − x∗ ‖ ≤ αn M1 + ‖xn − xn+1 ‖(‖xn − x∗ ‖ + ‖xn+1 − x∗ ‖) + 2µ‖Bx∗ − Bxn ‖ ‖un − y∗ ‖ + 2λ‖Ay∗ − Aun ‖ ‖zn − x∗ ‖. Since conditions (i), (ii), (3.6) and (3.11), we have lim g1 (‖xn − un − (x∗ − y∗ )‖) = 0,

n→∞

lim g2 (‖un − zn + (x∗ − y∗ )‖) = 0.

n→∞

(3.15)

It follows from the properties of g1 and g2 , we have lim ‖xn − un − (x∗ − y∗ )‖ = 0,

n→∞

lim ‖un − zn + (x∗ − y∗ )‖ = 0.

n→∞

(3.16)

From (3.16), we obtain

‖xn − zn ‖ ≤ ‖xn − un − (x∗ − y∗ )‖ + ‖un − zn + (x∗ − y∗ )‖ → 0 as n → ∞.

(3.17)

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G. Cai, S. Bu / Mathematical and Computer Modelling 55 (2012) 538–546

On the other hand, we observe yn − zn = αn (f (xn ) − zn ). Since limn→∞ αn = 0, we have lim ‖yn − zn ‖ = 0.

(3.18)

n→∞

We note that

‖Sn zn − zn ‖ ≤ ‖Sn zn − Sn yn ‖ + ‖Sn yn − xn ‖ + ‖xn − zn ‖ ≤ ‖zn − yn ‖ + ‖Sn yn − xn ‖ + ‖xn − zn ‖. From (3.5), (3.17) and (3.18), we obtain lim ‖Sn zn − zn ‖ = 0.

(3.19)

n→∞

By (3.19) and Lemma 2.5, we have

‖Szn − zn ‖ ≤ ‖Szn − Sn zn ‖ + ‖Sn zn − zn ‖ → 0 as n → ∞.

(3.20)

In view of (3.17) and (3.20), we have

‖xn − Sxn ‖ ≤ ‖xn − zn ‖ + ‖zn − Szn ‖ + ‖Szn − Sxn ‖ ≤ 2‖xn − zn ‖ + ‖zn − Szn ‖ → 0 as n → ∞.

(3.21)

Define a mapping Wx = (1 − θ )Sx + θ Gx, where G is defined by Lemma 2.9, θ ∈ (0, 1) is a constant. Then by Lemma 2.12, we have that F (W ) = F (S ) ∩ F (G) = F . We observe that

‖xn − Wxn ‖ = ‖(1 − θ )(xn − Sxn ) + θ (xn − Gxn )‖ ≤ (1 − θ )‖xn − Sxn ‖ + θ ‖xn − Gxn ‖ = (1 − θ )‖xn − Sxn ‖ + θ ‖xn − zn ‖. By (3.17) and (3.21), we obtain lim ‖xn − Wxn ‖ = 0.

(3.22)

n→∞

Then, we claim that lim sup ⟨f (q) − q, j(xn − q)⟩ ≤ 0,

(3.23)

n→∞

where q = limt →0 xt with xt being the fixed point of the contraction x → tf (x) + (1 − t )Wx. Then xt solves the fixed point equation xt = tf (xt ) + (1 − t )Wxt . Thus we have

‖xt − xn ‖ = ‖(1 − t )(Wxt − xn ) + t (f (xt ) − xn )‖. It follows from Lemma 2.13 that

‖x t − x n ‖2 = ≤ ≤ ≤ =

‖(1 − t )(Wxt − xn ) + t (f (xt ) − xn )‖2 (1 − t )2 ‖Wxt − xn ‖2 + 2t ⟨f (xt ) − xn , j(xt − xn )⟩ (1 − t )2 (‖Wxt − Wxn ‖ + ‖Wxn − xn ‖)2 + 2t ⟨f (xt ) − xn , j(xt − xn )⟩ (1 − t )2 (‖xt − xn ‖ + ‖Wxn − xn ‖)2 + 2t ⟨f (xt ) − xn , j(xt − xn )⟩   (1 − t )2 ‖xt − xn ‖2 + 2‖xt − xn ‖ ‖Wxn − xn ‖ + ‖Wxn − xn ‖2 + 2t ⟨f (xt ) − xt , j(xt − xn )⟩ + 2t ⟨xt − xn , j(xt − xn )⟩

= (1 − 2t + t 2 )‖xt − xn ‖2 + fn (t ) + 2t ⟨f (xt ) − xt , j(xt − xn )⟩ + 2t ‖xt − xn ‖2 ,

(3.24)

where fn (t ) = (1 − t )2 (2‖xt − xn ‖ + ‖xn − Wxn ‖)‖xn − Wxn ‖ → 0,

as n → ∞.

(3.25)

G. Cai, S. Bu / Mathematical and Computer Modelling 55 (2012) 538–546

545

It follows from (3.24) that

⟨xt − f (xt ), j(xt − xn )⟩ ≤

t 2

‖xt − xn ‖2 +

1 2t

fn (t ).

(3.26)

Let n → ∞ in (3.26) and note that (3.25) yields t

lim sup ⟨xt − f (xt ), j(xt − xn )⟩ ≤

2

n→∞

M2 ,

(3.27)

where M2 > 0 is a constant such that M2 ≥ ‖xt − xn ‖2 for all t ∈ (0, 1) and n ≥ 1. Taking t → 0 from (3.27), we have lim sup lim sup ⟨xt − f (xt ), j(xt − xn )⟩ ≤ 0. t →0

(3.28)

n→∞

On the other hand, we have

⟨f (q) − q, j(xn − q)⟩ = ⟨f (q) − q, j(xn − q)⟩ − ⟨f (q) − q, j(xn − xt )⟩ + ⟨f (q) − q, j(xn − xt )⟩ − ⟨f (q) − xt , j(xn − xt )⟩ + ⟨f (q) − xt , j(xn − xt )⟩ − ⟨f (xt ) − xt , j(xn − xt )⟩ + ⟨f (xt ) − xt , j(xn − xt )⟩ = ⟨f (q) − q, j(xn − q) − j(xn − xt )⟩ + ⟨xt − q, j(xn − xt )⟩ + ⟨f (q) − f (xt ), j(xn − xt )⟩ + ⟨f (xt ) − xt , j(xn − xt )⟩ . It follows that lim sup ⟨f (q) − q, j(xn − q)⟩ ≤ lim sup ⟨f (q) − q, j(xn − q) − j(xn − xt )⟩ n→∞

n→∞

+ ‖xt − q‖ lim sup ‖xn − xt ‖ + α‖q − xt ‖ lim sup ‖xn − xt ‖ n→∞

n→∞

+ lim sup ⟨f (xt ) − xt , j(xn − xt )⟩ . n→∞

Noticing that j is norm-to-norm uniformly continuous on bounded subsets of C , it follows from (3.28), we have lim sup ⟨f (q) − q, j(xn − q)⟩ = lim sup lim sup ⟨f (q) − q, j(xn − q)⟩ n→∞

t →0

n→∞

≤ 0. Therefore, (3.23) holds. From (3.17) and (3.18), we have yn − q → xn − q. Noticing that j is norm-to-norm uniformly continuous on bounded subsets of C , it follows from (3.23) that lim sup ⟨f (q) − q, j(yn − q)⟩ = lim sup ⟨f (q) − q, j(xn − q)⟩ n→∞

n→∞

≤ 0.

(3.29)

Finally we prove that xn → q as n → ∞. We observe

αn ⟨f (xn ) − q, j(yn − q)⟩ + (1 − αn ) ⟨zn − q, j(yn − q)⟩ αn ⟨f (xn ) − f (q), j(yn − q)⟩ + αn ⟨f (q) − q, j(yn − q)⟩ + (1 − αn ) ⟨zn − q, j(yn − q)⟩ αn δ‖xn − q‖ ‖yn − q‖ + (1 − αn )‖zn − q‖ ‖yn − q‖ + αn ⟨f (q) − q, j(yn − q)⟩ αn δ‖xn − q‖ ‖yn − q‖ + (1 − αn )‖xn − q‖ ‖yn − q‖ + αn ⟨f (q) − q, j(yn − q)⟩ [1 − αn (1 − δ)]‖xn − q‖ ‖yn − q‖ + αn ⟨f (q) − q, j(yn − q)⟩ 1 − αn (1 − δ) ≤ (‖xn − q‖2 + ‖yn − q‖2 ) + αn ⟨f (q) − q, j(yn − q)⟩

‖yn − q‖2 = = ≤ ≤ =

≤

2 1 − αn (1 − δ) 2

1

‖xn − q‖2 + ‖yn − q‖2 + αn ⟨f (q) − q, j(yn − q)⟩ , 2

which implies

‖yn − q‖2 ≤ [1 − αn (1 − δ)]‖xn − q‖2 + αn (1 − δ) By the convexity of ‖.‖2 and (3.1), we get

‖xn+1 − q‖2 ≤ βn ‖xn − q‖2 + (1 − βn )‖yn − q‖2 .

2 ⟨f (q) − q, j(yn − q)⟩ 1−δ

.

(3.30)

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G. Cai, S. Bu / Mathematical and Computer Modelling 55 (2012) 538–546

It follows from (3.30) that

‖xn+1 − q‖2 ≤ [1 − αn (1 − βn )(1 − δ)]‖xn − q‖2 + αn (1 − βn )(1 − δ)

2 ⟨f (q) − q, j(yn − q)⟩

Apply Lemma 2.4 to (3.31), we obtain xn → q as n → ∞. This completes the proof.

1−δ

.

(3.31)



Corollary 3.2. Let C be a nonempty closed subset of a uniformly convex and 2-uniformly smooth Banach space X . Let QC be the sunny nonexpansive retraction from X onto C . Let the mappings A, B : C → X be α -inverse-strongly accretive and β -inversestrongly accretive, respectively. Let f be a contraction of C into itself with coefficient δ ∈ (0, 1). Let S be a nonexpansive mapping of C into itself such that F = F (S ) ∩ F (G) ̸= ∅, where G is defined by Lemma 2.9. For arbitrarily given x1 ∈ C , let {xn } be the sequence generated by

 x = βn xn + (1 − βn )Syn ,   n+1 yn = αn f (xn ) + (1 − αn )zn ,  zn = QC (un − λAun ), un = QC (xn − µBxn ),

(3.32)

β where 0 < λ < Kα2 , 0 < µ < K 2 . Suppose {αn } and {βn } be two sequences satisfy the following conditions:

(i) 0 < lim infn→∞ β∑ n ≤ lim supn→∞ βn < 1; ∞ (ii) limn→∞ αn = 0, n=1 αn = ∞. Then {xn } converges strongly to q ∈ F , which solves the following variational inequality:

⟨q − f (q), j(q − p)⟩ ≤ 0,

∀p ∈ F .

Acknowledgment This work was supported by the NSF of China (10731020) and the Specialized Research Fund for the Doctoral Program of Higher Education (200800030059). References [1] L.C. Ceng, C. Wang, J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res. 67 (2008) 375–390. [2] Y. Yao, M.A. Noor, On viscosity iterative methods for variational inequalities, J. Math. Anal. Appl. 325 (2007) 776–787. [3] A. Bnouhachem, M. Aslam Noor, Z. Hao, Some new extragradient iterative methods for variational inequalities, Nonlinear Anal. 70 (2009) 1321–1329. [4] Y. Yao, et al., Modified extragradient methods for a system of variational inequalities in Banach spaces, Acta Appl. Math. 110 (2010) 1211–1224. [5] Y. Yao, Y.-C. Liou, R. Chen, Convergence theorems for fixed point problems and variational inequality problems in Hilbert spaces, Math. Nachr. 282 (2009) 1827–1835. [6] Y. Yao, S. Maruster, Strong convergence of an iterative algorithm for variational inequalities in Banach spaces, Math. Comput. Modelling 54 (2011) 325–329. [7] S. Reich, Asymptotic behavior of contractions in Banach spaces, J. Math. Anal. Appl. 44 (1973) 57–70. [8] S. Kitahara, W. Takahashi, Image recovery by convex combinations of sunny nonexpansive retractions, Topol. Methods Nonlinear Anal. 2 (1993) 333–342. [9] T. Suzuki, Strong convergence of Krasnoselskii and Manns type sequences for one parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305 (2005) 227–239. [10] H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291. [11] K. Aoyama, et al., Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350–2360. [12] H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991) 1127–1138. [13] S. Kamimura, W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (2002) 938–945. [14] R.E. Bruck, Properties of fixed point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math. Soc. 179 (1973) 251–262. [15] S.S. Chang, On Chidumes open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces, J. Math. Anal. Appl. 216 (1997) 94–111.