Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications

Nonlinear Analysis: Hybrid Systems 4 (2010) 755–765

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications Jinlong Kang a,b , Yongfu Su a,∗ , Xin Zhang a a

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

b

Department of Foundation, Xi’an Communication of Institute, Xi’an 710106, China

article

abstract

info

Article history: Received 7 September 2009 Accepted 3 May 2010

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a weak relatively nonexpansive mapping, the set of solutions of the variational inequality for the monotone mapping and the set of solutions of an equilibrium problem in a 2-uniformly convex and uniformly smooth Banach space. Then we show that the iterative sequence converges strongly to a common element of the three sets. In this paper, we also give an example which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping in Banach space l2 . Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved.

Keywords: Weak relatively nonexpansive mapping Monotone mapping Variational inequality Equilibrium problem Fixed point

1. Introduction Let E be a real Banach space and let E ∗ be the dual space of E. Let C be a closed convex subset of E. Let F be a bifunction from C × C → R, where R is the set of real numbers. The equilibrium problem is to find xˆ ∈ C such that F (ˆx, y) ≥ 0,

∀y ∈ C .

(1.1)

The set of solutions of (1.1) is denoted by EP (F ). Numerous problems in physics, optimization and economics can be reduced to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem in Hilbert spaces, see, for instance, Flam–Antipin [1], Blum–Oettli [2], Combettes–Hirstoaga [3], Moudafi [4] and Takahashi–Takahashi [5]. Recall that a mapping A : D(A) ⊂ E → E ∗ is called monotone if for all x, y ∈ D(A),

hx − y, Ax − Ayi ≥ 0.

(1.2)

Recall that a mapping A : D(A) ⊂ E → E ∗ is called inverse-strongly monotone if there exists a positive real number α such that

hx − y, Ax − Ayi ≥ αkAx − Ayk2

(1.3)

for all x, y ∈ D(A); see [6,4,7]. For such a case, A is called α -inverse-strongly monotone. If A is an α -inverse-strongly monotone mapping of D(A) into E ∗ , then it is obvious that A is 1/α -Lipschitz continuous, i.e. kAx − Ayk ≤ 1/αkx − yk for all x, y ∈ D(A). If A is called strongly monotone for each x, y ∈ D(A) there exists k ∈ (0, 1) such that

hx − y, Ax − Ayi ≥ kkx − yk2 .

(1.4)

A monotone mapping A is said to be maximal if its graph G(A) = {(x, f ) : f ∈ Ax} is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping A is maximal if and only if for

∗

Corresponding author. E-mail address: [email protected] (Y. Su).

1751-570X/$ – see front matter Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2010.05.002

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(x, f ) ∈ E × E ∗ ,hx − y, f − g i ≥ 0 for every (y, g ) ∈ G(A) implies f ∈ Ax. Suppose that A is monotone mapping from C into E. The classical variational inequality problem is to find a u ∈ C such that hv − u, Aui ≥ 0

for all v ∈ C .

(1.5)

We denoted by VI (C , A) the set of solutions of the variational inequality problem. Such a problem is connected with the convex minimization problem, the complementarity problem, the problem of finding point u ∈ C satisfying VI (C , A). The variational inequality has been extensively studied in the literature. See [8–13] and the references therein. Let E be a smooth, strictly convex and reflexive Banach space and let C be a nonempty closed convex subset of E. Throughout this paper, we denote by φ the function defined by

φ(x, y) = kxk2 − 2hx, Jyi + kyk2 for x, y ∈ E . Following Alber [14], the generalized projection ΠC from E onto C is defined by

ΠC (x) = arg min φ(y, x), y∈C

∀x ∈ E .

The generalized projection ΠC from E onto C is well defined, single-valued and satisfies

(kxk − kyk)2 ≤ φ(x, y) ≤ (kxk + kyk)2 for x, y ∈ E . If E is a Hilbert space, then φ(y, x) = ky − xk2 and ΠC is the metric projection of E onto C . Let C be a closed convex subset of E, and let T be a mapping from C into itself. We denote by F (T ) the set of fixed points of T . A point p in C is said to be an asymptotic fixed point of T [15] if C contains a sequence {xn } which converges weakly to p such that limn→∞ kTxn − xn k = 0. The set of asymptotic fixed point of T will be denoted by b b F (T ). A mapping S : C → C is called nonexpansive if

kSx − Syk ≤ kx − yk for all x, y ∈ C . Following Matsushita and Takahashi [16,4], a mapping T of C into itself is said to be relatively nonexpansive (see also [17]) if the following conditions are satisfied: (1) F (T ) is nonempty; (2) φ(u, Tx) ≤ φ(u, x), ∀u ∈ F (T ), x ∈ C ; F (T ) = F (T ). (3) b A point p in C is said to be a strong asymptotic fixed point of T [13] if C contains a sequence {xn } which converges strongly to p such that limn→∞ kTxn − xn k = 0. The set of strong asymptotic fixed points of T will be denoted by e F (T ). A mapping T from C into itself is called weak relatively nonexpansive if (1) F (T ) is nonempty; (2) φ(u, Tx) ≤ φ(u, x), ∀u ∈ F (T ), x ∈ C ; F (T ) = F (T ). (3) e If E is strictly convex and reflexive Banach space, and A ⊂ E × E ∗ is a continuous monotone mapping with A−1 (0) 6= ∅ then it is proved in [18,16] that Jr := (J + rA)−1 J, for r > 0 is relatively nonexpansive. Moreover, if T : E → E is relatively nonexpansive then using the definition of φ one can show that F (T ) is closed and convex. It is obvious that relatively nonexpansive mapping is weak relatively nonexpansive mapping. In fact, for any mapping T : C → C , we have F (T ) ⊂ e F (T ) ⊂ b F (T ). Therefore, if T is relatively nonexpansive mapping, then F (T ) = e F (T ) = b F (T ). In this paper, we give an example which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping. Example. Let E = l2 and x0 = (1, 0, 0, 0, . . .) x1 = (1, 1, 0, 0, . . .) x2 = (1, 0, 1, 0, 0, . . .) x3 = (1, 0, 0, 1, 0, 0, . . .)

... xn = (1, 0, 0, 0, . . . , 0, 1, 0, 0, 0, . . .) .... It is obvious that {xn }∞ n=1 converges weakly to x0 . On the other hand, we have kxn − xm k = mapping T : E → E as follows

( n xn if x = xn (∃ n ≥ 1), T (x) = n + 1 −x if x 6= xn (∀ n ≥ 1).

√ 2 for any n 6= m. Define a

J. Kang et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 755–765

757

It is also obvious that F (T ) = {0} and

kTx − 0k = kTxk ≤ kxk = kx − 0k,

∀ x ∈ E.

2

Since E = L is a Hilbert space, we have

φ(Tx, 0) = kTx − 0k2 = kTxk2 ≤ kxk2 = kx − 0k2 = φ(x, 0),

∀ x ∈ E.

We first claim that T is a weak relatively nonexpansive mapping. In fact that, for any strong convergent sequence {zn } ⊂ E such that zn → z0 and kzn − Tzn k → 0 as n → ∞, then there exist sufficiently large nature number N such that zn 6= xm , for any n, m > N. Then Tzn = −zn for n > N, it follows from kzn − Tzn k → 0 that 2zn → 0 and hence zn → z0 = 0. Since z0 ∈ F (T ), then T is a weak relatively nonexpansive mapping. We second claim that, T is not relatively nonexpansive mapping. In fact that, though xn * x0 and



n

1

kTxn − xn k = x − x kx n k → 0 n =

n + 1 n n+1 as n → ∞, but x0 ∈F (T ). For finding an element of F (S ) and

T

VI (C , A), Takahashi and Toyoda [19] introduced the following iterative scheme: x1 ∈ C

xn+1 = αn xn + (1 − αn )SPC (xn − λn Axn ),

n≥1

(1.6)

and obtain a weak convergence theorem in a Hilbert space. Recently, to fond a point of F (T ) studied the following iterative scheme

T

VI (C , A) Zegeye, Shahzad [20]

 y = Π (J −1 (Jxn − αn Axn ))  z n = TyC   n n  H = {v ∈ C : φ(v, z ) ≤ φ(v, y ) ≤ φ(v, x )}  0 0 0 0 Hn = {v ∈ Hn−1 ∩ Wn−1 : φ(v, zn ) ≤ φ(v, yn ) ≤ φ(v, xn )}   W0 = C     Wn = {v ∈ Wn−1 ∩ Hn−1 : hxn − v, Jx0 − Jxn i ≥ 0} xn+1 = ΠHn ∩Wn (x0 ), n ≥ 1,

(1.7)

where J is the normalized duality mapping on E. And they proved that {xn } strong convergence to a common element of the set of fixed points of weak relatively nonexpansive mapping and the set of solution of a certain variational problem under appropriate conditions on inverse-strongly monotone mapping A in a 2-uniformly convex and uniformly smooth Banach space E. Remark. In [20], the weak relatively nonexpansive mapping is also called the relatively weak nonexpansive mapping. On the other hand, for finding a element of EP (F ) ∩ F (T ), Takahashi, Zembayashi [21] introduced the following iterative scheme in a uniformly smooth and uniformly convex Banach space: x0 ∈ C and

 yn = J −1 (αn Jxn + (1 − αn )JTxn )     1  un ∈ C such that F (un , y) + hy − un , Jun − Jyn i ≥ 0, rn

∀y ∈ C ,

Hn = {v ∈ C : φ(v, un ) ≤ φ(v, xn )}      Wn = {v ∈ C : hxn − v, Jx0 − Jxn i ≥ 0} xn+1 = ΠHn ∩Wn (x0 ), n ≥ 1,

(1.8)

for all n ∈ N ∪ {0}, where {αn } ⊂ [0, 1] and {rn } ⊂ [a, ∞) satisfy some appropriate conditions. Furthermore, they proved {xn } and {un } converge strongly to z ∈ ΠF (S )∩EP (F ) , where z = PF (S )∩EP (F ) (x0 ). In this paper, motivated and inspired by the above results, we introduce a new following iterative scheme: x0 ∈ C and

 yn = ΠC (J −1 (Jxn − βn Axn ))     zn = J −1 (αn Jxn + (1 − αn )JSyn )    1    un ∈ C such that F (un , y) + rn hy − un , Jun − Jzn i ≥ 0, ∀y ∈ C , H0 = {v ∈ C : φ(v, u0 ) ≤ φ(v, z0 ) ≤ α0 φ(v, x0 ) + (1 − α0 )φ(v, y0 ) ≤ φ(v, x0 )}   H   n = {v ∈ Hn−1 ∩ Wn−1 : φ(v, un ) ≤ φ(v, zn ) ≤ αn φ(v, xn ) + (1 − αn )φ(v, yn ) ≤ φ(v, xn )}   W0 = C     Wn = {v ∈ Wn−1 ∩ Hn−1 : hxn − v, Jx0 − Jxn i ≥ 0} xn+1 = ΠHn ∩Wn (x0 ), n ≥ 1,

(1.9)

for finding a common element of the set of fixed points of a weak relatively nonexpansive mapping, the set of solutions of a variational inequality for an α -inverse-strongly monotone mapping and the set of solutions of an equilibrium problem

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J. Kang et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 755–765

a 2-uniformly convex and uniformly smooth Banach space E. Furthermore, we show the iterative sequences {xn } and {un } converge strongly to z ∈ F (S ) ∩ VI (C , A) ∩ EP (F ), where z = PF (S )∩VI (C ,A)∩EP (F ) x0 . The results of this paper extended and improved the results of Takahashi and Toyoda [19] and Takahashi, Zembayashi [21] and others. 2. Preliminaries Throughout this paper, all the Banach spaces are real. We denote by N and R the sets of positive integers and real numbers, respectively. Let E be a Banach space and let E ∗ be the topological dual of E. For all x ∈ E and x∗ ∈ E ∗ , we denote the value ∗ of x∗ at x by hx, x∗ i. Let J denote the normalized duality mapping from E into the dual space 2E given by J (x) = {x∗ ∈ E ∗ : hx, x∗ i = kxk2 = kx∗ k2 },

x ∈ E.

By the Hahn–Banach theorem, J (x) is nonempty; see [22] for more details. Let C be a subset of E and T be a self-mapping of ∗

C . Let F (T ) denote the set of fixed points of T , →, * and *, denote the strong, weak convergence and weak* convergence, respectively. ωω (xn ) = {x : ∃xnj * x} denotes the weak ω-limit set of {xn }. A Banach space E is said to be strictly convex if 2 < 1 for x, y ∈ E with kxk = kyk = 1 and x 6= y. It is said to be uniformly convex if for each  > 0 there is a δ > 0 such that for x, y ∈ E with kxk, kyk ≤ 1 and kx − yk ≥  , kx + yk ≤ 2(1 −δ) holds. A uniformly convex Banach space has the Kadec–Klee property, that is, xn * x and kxn k → kxk imply xn → x. The modulus of convexity of E is the function δE : [0, 2] → [0, 1] defined by kx+yk

 

1

: kxk, kyk ≤ 1, kx − yk ≥  , δE () := inf 1 − ( x + y )

2

for all ε ∈ [0, 2]. E is said to uniformly convex if and only if δE (0) = 0, and δE () > 0 for all 0 <  ≤ 2. Let p > 1, then E is said to be p-uniformly convex if there exists a constant c > 0 such that δ() ≥ c  p for all  ∈ [0, 2]. Observe that every p-uniformly convex space is uniformly convex. It is well known (see for example [23]) that Lp (lp ) or Wmp is



p-uniformly convex if p ≥ 2; 2-uniformly convex if 1 < p ≤ 2,

and Hilbert space H is 2-uniformly convex. The space E is said to be smooth if the limit lim

kx + tyk − kxk

t →0

t

exists for each x, y ∈ S (E ) = {x ∈ E : kxk = 1}. It is also said to be uniformly smooth if the limit exists uniformly in x, y ∈ S (E ). The modulus of smoothness of E is the function ρE : [0, ∞) → [0, ∞) defined by

ρE (t ) := sup



1 2



(kx + yk + kx − yk) − 1 : x ∈ S (E ), kyk ≤ t . ρ (t )

A Banach space E is said to be uniformly smooth if Et → 0 as t → 0. We know that if E is smooth, strictly convex and reflexive, then the duality mapping J is single-valued, one-to-one and onto. The duality mapping J is said to be weakly ∗

sequentially continuous if xn * x implies that Jxn * Jx; see [6] for more details. Lemma 2.1 ([24,23]). Let E be a 2-uniformly convex Banach space. Then, for all x, y ∈ E, we have

kx − y k ≤

2 c2

kJx − Jyk,

(2.1)

where J is the normalized duality mapping of E and 0 < c ≤ 1. Lemma 2.2 ([14,25]). Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Then

φ(x, ΠC y) + φ(ΠC y, y) ≤ φ(x, y),

∀x ∈ C and y ∈ E .

Lemma 2.3 ([14,25]). Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E, let x ∈ E and z ∈ C . Then z = ΠC x ⇐⇒ hy − z , Jx − Jz i ≤ 0,

∀y ∈ C .

Lemma 2.4 ([25]). Let E be a smooth and uniformly convex Banach space and let {xn } and {yn } be two sequences in E. If either {xn } or {yn } is bounded and φ(xn , yn ) → 0 as n → ∞, then kxn − yn k → 0, as n → ∞.

J. Kang et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 755–765

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Lemma 2.5 ([23,26,7]). Let E be a 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

ktx + (1 − t )yk2 ≤ t kxk2 + (1 − t )kyk2 − t (1 − t )g (kx − yk) for all x, y ∈ Br and t ∈ [0, 1], where Br = {z ∈ E : kxk ≤ r }. Lemma 2.6 ([25]). Let E be a 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 (kx − yk) ≤ φ(x, y) for all x, y ∈ Br . We denote by NC v the normal cone for C at a point v ∈ C , that is NC v = {w ∈ E ∗ : hv − u, wi ≥ 0, ∀u ∈ C }. In what follows we shall use the following lemma. Lemma 2.7 ([27,28]). Let C be nonempty closed convex subset of a Banach space E and let A be a monotone and hemicontinuous operator of C into E ∗ with C = D(A). Let B be an operator defined as follows: Bv =



Av + NC v, ∅, v ∈¯ C .

v ∈ C,

Then B is maximal monotone and 0 ∈ Bv if and only if v ∈ VI (C , A). It is well known that monotone and hemicontinuous operator A with D(A) = E is maximal (see, eg, [25]). Note that Lemma 2.7 is for monotone and hemicontinuous operators. For solving the equilibrium problem for a bifunction F : C ×C → R, let us assume that F satisfies the following conditions: (A1) F (x, x) = 0 for all x ∈ C ; (A2) F is monotone, i.e., F (x, y) + F (y, x) ≤ 0 for all x, y ∈ C ; (A3) for each x, y, z ∈ C , lim F (tz + (1 − t )x, y) ≤ F (x, y); t ↓0

(A4) for each x ∈ C , y 7→ F (x, y) is convex and lower semicontinuous. Lemma 2.8 ([2]). 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 → R satisfying (A1)–(A4), and let r > 0 and x ∈ E. Then, there exists z ∈ C such that 1

F (z , y) +

r

hy − z , Jz − Jxi ≥ 0,

∀y ∈ C .

Lemma 2.9 ([21]). Let C be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E, let F be a bifunction from C × C → R satisfying (A1)–(A4), and let r > 0 and x ∈ E, define a mapping Tr : E → 2C as follows: Tr (x) =



z ∈ C : F (z , y) +

1 r

hy − z , Jz − Jxi ≥ 0, ∀y ∈ C



for all x ∈ E. Then, the following hold: (1) Tr is single-valued; (2) Tr is a firmly nonexpansive-type mapping [29], i.e., for any x, y ∈ E,

hTr x − Tr y, JTr x − JTr yi ≤ hTr x − Tr y, Jx − Jyi; (3) F (Tr ) = b F (Tr ) = EP (F ); (4) EP (F ) is closed and convex. Lemma 2.10 ([21]). 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 → R satisfying (A1)–(A4). Then for r > 0, x ∈ E and q ∈ F (Tr ),

φ(q, Tr x) + φ(Tr x, x) ≤ φ(q, x). We make use of the function V : E × E ∗ → R defined by V (x, x∗ ) = kxk2 − 2hx, x∗ i + kx∗ k2 ,

for all x ∈ E and x∗ ∈ E ∗ ,

studied by Alber [14]. That is, V (x, x∗ ) = φ(x, J −1 x∗ ) for all x ∈ E and x∗ ∈ E ∗ . We know the following lemma. Lemma 2.11 ([14]). Let E be reflexive strictly convex and smooth Banach space with E ∗ as its dual. Then V (x, x∗ ) + 2hJ −1 x∗ − x, y∗ i ≤ V (x, x∗ + y∗ ), for all x ∈ E and x∗ , y∗ ∈ E ∗ .

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J. Kang et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 755–765

3. Main results In this section, we introduce an iterative method by the hybrid method for finding a common element of the set of fixed points of a relatively weak nonexpansive mapping, the set of solutions of the variational inequality for the monotone mapping and the set of solutions of an equilibrium problem in a real uniformly smooth and 2-uniformly convex Banach space. We show that the iterative sequence converges strongly to a common element of the three sets. Theorem 3.1. Let E be a real uniformly smooth and 2-uniformly convex Banach space with dual space E ∗ and let C be a nonempty closed convex subset of E. Let F be a bifunction from C × C to R satisfying (A1)–(A4) and S be a weak relatively nonexpansive mapping of C into E and let A be an α -inverse-strongly monotone mapping of C into E ∗ such that F (S ) ∩ VI (C , A) ∩ EP (F ) 6= ∅. Assume that kAxk ≤ kAx − Apk for all x ∈ C and p ∈ VI (C , A). Let 0 < a0 ≤ βn ≤ b0 := let {xn } and {un } be sequences generated by

c2α , 2

where c is the constant in (2.1).

 yn = ΠC (J −1 (Jxn − βn Axn ))     zn = J −1 (αn Jxn + (1 − αn )JSyn )    1    un ∈ C such that F (un , y) + rn hy − un , Jun − Jzn i ≥ 0, ∀y ∈ C , H0 = {v ∈ C : φ(v, u0 ) ≤ φ(v, z0 ) ≤ α0 φ(v, x0 ) + (1 − α0 )φ(v, y0 ) ≤ φ(v, x0 )}   Hn = {v ∈ Hn−1 ∩ Wn−1 : φ(v, un ) ≤ φ(v, zn ) ≤ αn φ(v, xn ) + (1 − αn )φ(v, yn ) ≤ φ(v, xn )}     W0 = C    Wn = {v ∈ Wn−1 ∩ Hn−1 : hxn − v, Jx0 − Jxn i ≥ 0} xn+1 = ΠHn ∩Wn (x0 ), n ≥ 1,

(3.1)

for every n ∈ N ∪ {0}, where J is the duality mapping on E, {αn } ⊂ [0, 1] satisfies lim infn→∞ αn (1 − αn ) > 0 and {rn } ⊂ [a, ∞) for some a > 0. Then {xn } and {un } converge strongly to ΠF (S )∩VI (C ,A)∩EP (F ) (x0 ), where ΠF (S )∩VI (C ,A)∩EP (F ) is the generalized projection from E onto F (S ) ∩ VI (C , A) ∩ EP (F ). Proof. We first show that Hn and Wn are closed and convex for each n ≥ 0. From the definitions of Hn and Wn , it is obvious that Wn closed and convex and Hn is closed for each n ≥ 0. Moreover, since φ(v, un ) ≤ φ(v, zn ) is equivalent to 2hv, Jzn − Jun i − kzn k2 + kun k2 ≤ 0, φ(v, zn ) ≤ αn φ(v, xn ) + (1 − αn )φ(v, yn ) is equivalent to 2hv, αn Jxn + (1 − αn )Jyn − Jzn i − αn kxn k2 − (1 − αn )kyn k2 + kzn k2 ≤ 0 and αn φ(v, xn ) + (1 − αn )φ(v, yn ) ≤ φ(v, xn ) is equivalent to 2hv, Jxn − Jyn i − kxn k2 + kyn k2 ≤ 0, it follows that Hn is convex for each n ≥ 0. So, Hn ∩ Wn is a closed convex subset of E for all n ∈ N ∪ {0}. Let u ∈ F := F (S ) ∩ VI (C , A) ∩ EP (F ). Putting un = Trn zn for all n ∈ N ∪ {0}, we have from Lemma 2.9 that Trn are relatively nonexpansive. Since S is weak relatively nonexpansive, Lemma 2.11, (2.1) and α -inverse-strongly monotonicity of A, we have

φ(u, Syn ) ≤ = ≤ = ≤ = = ≤

φ(u, yn ) φ(u, ΠC (J −1 (Jxn − βn Axn ))) φ(u, J −1 (Jxn − βn Axn )) V (u, Jxn − βn Axn ) V (u, (Jxn − βn Axn ) + βn Axn ) − 2hJ −1 (Jxn − βn Axn ) − u, βn Axn i V (u, Jxn ) − 2βn hJ −1 (Jxn − βn Axn ) − u, Axn i φ(u, xn ) − 2βn hxn − u, Axn i − 2βn hJ −1 (Jxn − βn Axn ) − J −1 (Jxn ), Axn i φ(u, xn ) − 2βn hxn − u, Axn − Aui − 2βn hxn − u, Aui + 2βn kJ −1 (Jxn − βn Axn ) − J −1 (Jxn )kkAxn k 4β 2 ≤ φ(u, xn ) − 2βn αkAxn − Auk2 + 2n kAxn k2 c 4βn2

≤ φ(u, xn ) − 2βn αkAxn − Auk2 + 2 kAxn − Auk2 c   2βn = φ(u, xn ) + 2βn − α kAxn − Auk2 2 c

≤ φ(u, xn ). Since Trn relatively nonexpansive, we have

φ(u, un ) = φ(u, Trn zn ) ≤ φ(u, zn ) ≤ φ(u, J −1 (αn Jxn + (1 − αn )JSyn ))

J. Kang et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 755–765

= ≤ = ≤ ≤ =

761

kuk2 − 2hu, αn Jxn + (1 − αn )JSyn i + kαn Jxn + (1 − αn )JSyn k2 kuk2 − 2αn hu, Jxn i − 2(1 − αn )hu, JSyn i + αn kxn k2 + (1 − αn )kSyn k2 αn φ(u, xn ) + (1 − αn )φ(u, Syn ) αn φ(u, xn ) + (1 − αn )φ(u, yn ) αn φ(u, xn ) + (1 − αn )φ(u, xn ) φ(u, xn ).

Hence we have u ∈ Hn . This implies that F ⊂ Hn , ∀n ≥ 0. Next we show by induction that F ⊂ Hn ∩ Wn for all n ∈ N ∪ {0}. From W0 = C , it is clear that F ⊂ H0 ∩ W0 and hence x1 = ΠH0 ∩W0 (x0 ) is well defined. Suppose that F ⊂ Hk−1 ∩ Wk−1 . Then there exists xk ∈ Hk−1 ∩ Wk−1 such that xk = ΠHk−1 ∩Wk−1 (x0 ). From the definition of xk and Lemma 2.3, we have, for all u ∈ Hk−1 ∩ Wk−1 ,

hu − xk , Jxk − Jx0 i ≥ 0 which implies that u ∈ Wk . So, we have F ⊂ Wk . Hence we have F ⊂ Hk ∩ Wk . Therefore, we get that F ⊂ Hn ∩ Wn for all n ∈ N ∪ {0}. This means that {xn } is well defined. From the definition of Wn , we have xn = ΠWn (x0 ). Using xn = ΠWn (x0 ), from Lemma 2.2, we have

φ(xn , x0 ) = φ(ΠWn (x0 ), x0 ) ≤ φ(u, x0 ) − φ(u, ΠWn (x0 )) ≤ φ(u, x0 ), for all u ∈ F ⊂ Wn . Then φ(xn , x0 ) is bounded. Therefore, {xn } and {Sxn } are bounded. Since xn+1 = ΠHn ∩Wn (x0 ) ∈ Hn ∩ Wn ⊂ Wn and xn = ΠWn (x0 ), by Lemma 2.2, we have

φ(xn+1 , xn ) + φ(xn , x0 ) ≤ φ(xn+1 , x0 ), which implies that φ(xn , x0 ) is nondecreasing and hence the limit of {φ(xn , x0 )} exists. From xn = ΠWn (x0 ) and Lemma 2.2, for any positive integer m, we have

φ(xn+m , xn ) = φ(xn+m , ΠWn (x0 )) ≤ φ(xn+m , x0 ) − φ(ΠWn (x0 ), x0 ) = φ(xn+m , x0 ) − φ(xn , x0 ), for all n ∈ N ∪ {0}. The existence of limn→∞ φ(xn , x0 ) implies that φ(xn+m , xn ) → 0 as n → ∞. Thus, from Lemma 2.4 we get that lim kxn+m − xn k = 0.

(3.2)

n→∞

Therefore, {xn } is Cauchy sequence and there exists a point p ∈ C such that xn → p as n → ∞. From xn+1 = ΠHn ∩Wn (x0 ) ∈ Hn , we have φ(xn+1 , un ) ≤ φ(xn+1 , zn ) ≤ φ(xn+1 , xn ) and φ(xn+1 , yn ) ≤ φ(xn+1 , xn ). Therefore, by Lemma 2.4 and (3.2), we get that lim kxn+1 − un k = lim kxn+1 − zn k = lim kxn+1 − yn k = 0

n→∞

n→∞

n→∞

and hence kxn − un k ≤ kxn − xn+1 k + kxn+1 − un k → 0 as n → ∞ and

kxn − yn k ≤ kxn − xn+1 k + kxn+1 − yn k → 0,

as n → ∞.

(3.3)

Furthermore, since J is uniformly norm-to-norm continuous on bounded sets and limn→∞ kxn − un k = 0, we have limn→∞ kJxn − Jun k = 0. Let r = supn∈N {kxn k, kSxn k}. Since E is a uniformly smooth Banach space, we know that E ∗ is a uniformly convex Banach space. Therefore, from Lemma 2.5, there exists a continuous, strictly increasing and convex function g with g (0) = 0 such that

kα x∗ + (1 − α)y∗ k2 ≤ αkx∗ k2 + (1 − α)ky∗ k2 − α(1 − α)g (kx∗ − y∗ k), for x∗ , y∗ ∈ B∗r and α ∈ [0, 1]. So, for u ∈ F , we have

φ(u, un ) = ≤ ≤ = ≤ = ≤ =

φ(u, Trn zn ) φ(u, zn ) φ(u, J −1 (αn Jxn + (1 − αn )JSyn )) kuk2 − 2hu, α0 Jxn + (1 − αn )JSyn i + kαn Jxn + (1 − αn )JSyn k2 kuk2 − 2αn hu, Jxn i − 2(1 − αn )hu, JSyn i + αn kxn k2 + (1 − αn )kSyn k2 − αn (1 − αn )g (kJxn − JSyn k) αn φ(u, xn ) + (1 − αn )φ(u, Syn ) − αn (1 − αn )g (kJxn − JSyn k) αn φ(u, xn ) + (1 − αn )φ(u, xn ) − αn (1 − αn )g (kJxn − JSyn k) φ(u, xn ) − αn (1 − αn )g (kJxn − JSyn k).

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Therefore, we have

αn (1 − αn )g (kJxn − JSyn k) ≤ φ(u, xn ) − φ(u, un ),

∀n ∈ N ∪ {0}.

(3.4)

Since,

φ(u, xn ) − φ(u, un ) = ≤ ≤ ≤

kxn k2 − kun k2 − 2hu, Jxn − Jun i |kxn k2 − kun k2 | + 2|hu, Jxn − Jun i| |kxn k − kun k|(kxn k + kun k) + 2kukkJxn − Jun k kxn − un k(kxn k + kun k) + 2kukkJxn − Jun k,

we get that lim (φ(u, xn ) − φ(u, un )) = 0.

(3.5)

n→∞

From the (3.4) and lim infn→∞ αn (1 − αn ) > 0, we have lim g (kJxn − JSyn k) = 0.

n→∞

Since J −1 is uniformly norm-to-norm continuous on bounded sets, we obtain lim kxn − Syn k = 0.

(3.6)

n→∞

Therefore, by (3.3), (3.6) and kyn − Syn k ≤ kyn − xn k + kxn − Syn k, we get that kyn − Syn k → 0 as n → ∞. This together with the facts that {xn } (and hence {yn }) converges strongly to p ∈ E and the definition of weak relatively nonexpansive mapping imply that p ∈ e F (S ) = F (S ). Now, we show that p ∈ VI (C , A). Define B ⊂ E × E ∗ be an operator as follows: Bv =

Av + NC v, ∅, v ∈¯ C .



v ∈ C,

(3.7)

By Lemma 2.7, B is maximal monotone and B−1 (0) = VI (C , A). Let (v, ω) ∈ G(B). Since ω ∈ Bv = Av + NC v , we have ω − Av ∈ NC v . Since also yn ∈ C , we have

hv − yn , ω − Avi ≥ 0.

(3.8)

On the other hand, from yn = ΠC J (Jxn − βn Axn ) and Lemma 2.3, we have hv − yn , Jyn − (Jxn − βn Axn )i ≥ 0 for ∀v ∈ C Jx −Jy and hence hv − yn , nβ n − Axn i ≤ 0. Therefore, from (3.7) and (3.8), we have −1

n

hv − yn , ωi ≥ hv − yn , Avi   Jxn − Jyn − Axn ≥ hv − yn , Avi + v − yn , βn   Jxn − Jyn = v − yn , Av − Axn + βn 

= hv − yn , Av − Ayn i + hv − yn , Ayn − Axn i + v − yn ,   Jxn − Jyn ≥ hv − yn , Ayn − Axn i + v − yn , βn

Jxn − Jyn

, ≥ −kv − yn kkAyn − Axn k − kv − yn k

β

n

Jxn − Jyn



βn

which together with kxn − yn k → 0 by the uniform continuity of J and A is Lipschitz continuous implies that hv − p, ωi ≥ 0. Since B is maximal monotone, we have p ∈ B−1 0 and hence p ∈ VI (C , A). Next, we show that p ∈ EP (F ). From un = Trn zn (3.1) and Lemma 2.10, we have

φ(un , zn ) = ≤ ≤ ≤

φ(Trn zn , zn ) φ(u, zn ) − φ(u, Trn zn ) φ(u, xn ) − φ(u, Trn zn ) φ(u, xn ) − φ(u, un ).

So, we have from (3.5) that

φ(un , zn ) = 0.

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Since E is 2-uniformly convex and uniformly smooth and {un } is bounded, we have from Lemma 2.4 that lim kun − zn k = 0.

(3.9)

n→∞

From xn → p, (3.9) and kxn − un k → 0, we have zn → p. Since J is uniformly norm-to-norm continuous on bounded sets and (3.9), we get lim kJun − Jzn k = 0.

n→∞

Since rn ≥ a, we have lim

kJun − Jzn k

n→∞

rn

= 0.

(3.10)

By un = Trn zn , F (un , y) +

1 rn

hy − un , Jun − Jzn i ≥ 0 for all y ∈ C .

Replacing n by nk , we have from (A2) that 1 rnk

hy − unk , Junk − Jznk i ≥ −F (unk , y) ≥ F (y, unk ) for all y ∈ C .

Since F (x, ·) is convex and lower semicontinuous, it is also weakly lower semicontinuous. Therefore, let k → ∞, we have from (3.10) and (A4) that F (y, p) ≤ 0,

for all y ∈ C .

For t with 0 < t ≤ 1 and y ∈ C , let yt = ty + (1 − t )p. Since y ∈ C and p ∈ C , we have yt ∈ C and hence F (yt , p) ≤ 0. So, from (A1) and (A4), we have 0 = F (yt , yt ) ≤ tF (yt , y) + (1 − t )F (yt , p) ≤ tF (yt , y) and hence 0 ≤ F (yt , y). From (A3), we have 0 ≤ F (p, y) for all y ∈ C and hence p ∈ EP (F ). Therefore p ∈ F := F (S ) ∩ EP (F ) ∩ VI (C , A). Finally we show that p = ΠF (x0 ). From Lemma 2.2, we obtain that

φ(p, ΠF (x0 )) + φ(ΠF (x0 ), x0 ) ≤ φ(p, x0 ).

(3.11)

On the other hand, since xn+1 = ΠHn ∩Wn (x0 ) and F ⊂ Hn ∩ Wn for all n ≥ 0, we have that

φ(ΠF (x0 ), xn+1 ) + φ(xn+1 , x0 ) ≤ φ(ΠF (x0 ), x0 ).

(3.12)

Moreover, by the definition of φ(x, y), we get that lim φ(xn+1 , x0 ) = φ(p, x0 ).

(3.13)

n→∞

Combining (3.11)–(3.13) we obtain that φ(p, x0 ) = φ(ΠF x0 , x0 ). Therefore, it follows from the uniqueness of ΠF (x0 ) that p = ΠF (x0 ). This completes the proof.  Remark 3.1. We remark that Theorem 3.1 hold for Lipschitz strongly monotone mappings. In fact, we have the following corollary. Corollary 3.1. Let E be a real uniformly smooth and 2-uniformly convex Banach space with dual space E ∗ and let C be a nonempty closed convex subset of E. Let F be a bifunction from C × C to R satisfying (A1)–(A4), A : C → E ∗ be a strongly monotone mapping with constant k, Lipschitz with constant L > 0 and let S be a weak relatively nonexpansive mapping of C into E such that F (S ) ∩ A−1 (0) ∩ EP (F ) 6= ∅. Assume that 0 < a0 ≤ βn ≤ b0 := c 2α , {αn } ⊂ [0, 1] satisfies lim infn→∞ αn (1 − αn ) > 0, where c is the constant in (2.1). Then the sequences {xn } and {un } generated by (3.1) converge strongly to ΠF (S )∩A−1 (0)∩EP (F ) (x0 ), where ΠF (S )∩A−1 (0)∩EP (F ) is the generalized projection from E onto F (S ) ∩ A−1 (0) ∩ EP (F ). 2

Proof. Note that every Lipschitz strongly monotone mapping is α -inverse-strongly monotone with α :=

kAx − Ayk ≤ Lkx − yk for x, y ∈ D(A), implies that kx − yk ≥ kAx − Ayk, for x, y ∈ D(A) and hence 1 L

k . L

In fact,

k

hAx − Ay, x − yi ≥ kkx − yk2 ≥ kAx − Ayk2 , L

and hence A is α -inverse-strongly monotone with α := kL . Let A−1 (0) = VI (C , A), then the conditions of the Theorem 3.1 are satisfied and hence the conclusion follows from Theorem 3.1.  Now we prove the strong convergence theorem for non-Lipschitz monotone mapping in Banach spaces.

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Theorem 3.2. Let E be a real uniformly smooth and 2-uniformly convex Banach space with dual space E ∗ and let C be a nonempty closed convex subset of E. Let F be a bifunction from C × C to R satisfying (A1)–(A4) and S be a weak relatively nonexpansive mapping of C into E and let A be a maximal monotone mapping of C into E ∗ with J −1 T weak relatively nonexpansive such that F (S ) ∩ VI (C , A) ∩ EP (F ) 6= ∅, where Tx := Jx − Ax. Assume that kAxk ≤ kAx − Apk for all x ∈ C and p ∈ VI (C , A). Let 0 < a0 ≤ βn ≤ 1, where c is the constant in (2.1). Let {xn } and {un } be sequences generated by

 yn = ΠC (J −1 (Jxn − βn Axn ))    zn = J −1 (αn Jxn + (1 − αn )JSyn )    1    un ∈ C such that F (un , y) + rn hy − un , Jun − Jzn i ≥ 0, ∀y ∈ C , H0 = {v ∈ C : φ(v, u0 ) ≤ φ(v, z0 ) ≤ α0 φ(v, x0 ) + (1 − α0 )φ(v, y0 ) ≤ φ(v, x0 )}   Hn = {v ∈ Hn−1 ∩ Wn−1 : φ(v, un ) ≤ φ(v, zn ) ≤ αn φ(v, xn ) + (1 − αn )φ(v, yn ) ≤ φ(v, xn )}     W0 = C   Wn = {v ∈ Wn−1 ∩ Hn−1 : hxn − v, Jx0 − Jxn i ≥ 0}   xn+1 = ΠHn ∩Wn (x0 ), n ≥ 1,

(3.14)

for every n ∈ N ∪ {0}, where J is the duality mapping on E, {αn } ⊂ [0, 1] satisfies lim infn→∞ αn (1 − αn ) > 0 and {rn } ⊂ [a, ∞) for some a > 0. Then {xn } and {un } converge strongly to ΠF (S )∩VI (C ,A)∩EP (F ) (x0 ), where ΠF (S )∩VI (C ,A)∩EP (F ) is the generalized projection from E onto F (S ) ∩ VI (C , A) ∩ EP (F ). Proof. Following the method of proof of Theorem 3.1 we have that Hn and Wn are both closed and convex for each n ≥ 0. Next, we show that F := F (S ) ∩ VI (C , A) ∩ EP (F ) ⊂ Hn ∩ Wn for each n ≥ 0 and hence {xn } is well defined. Let u ∈ F then relatively weak nonexpansiveness of S and J −1 T , definition of φ and βn ∈ [a0 , 1] give that

φ(u, Syn ) ≤ = = ≤ ≤ = = ≤ =

φ(u, yn ) φ(u, ΠC (J −1 (Jxn − βn Axn ))) φ(u, ΠC (J −1 (Jxn − βn (Jxn − Txn )))) φ(u, J −1 (Jxn − βn (Jxn − Txn ))) kuk2 − 2(1 − βn )hu, Jxn i − 2βn hu, Txn i + (1 − βn )kxn k2 + βn kTxn k2 (1 − βn )(kuk2 − 2hu, Jxn i + kxn k2 ) + βn (kuk2 − 2hu, Txn i + kTxn k2 ) (1 − βn )φ(u, xn ) + βn φ(u, J −1 Txn ) (1 − βn )φ(u, xn ) + βn φ(u, xn ) φ(u, xn ).

Putting un = Trn zn for all n ∈ N ∪ {0}, we have from Lemma 2.9 that Trn are relatively nonexpansive.

φ(u, un ) = ≤ ≤ = ≤ = ≤ ≤ =

φ(u, Trn zn ) φ(u, zn ) φ(u, J −1 (αn Jxn + (1 − αn )JSyn )) kuk2 − 2hu, αn Jxn + (1 − αn )JSyn i + kαn Jxn + (1 − αn )JSyn k2 kuk2 − 2αn hu, Jxn i − 2(1 − αn )hu, JSyn i + αn kxn k2 + (1 − αn )kSyn k2 αn φ(u, xn ) + (1 − αn )φ(u, Syn ) αn φ(u, xn ) + (1 − αn )φ(u, yn ) αn φ(u, xn ) + (1 − αn )φ(u, xn ) φ(u, xn ).

Hence we have u ∈ Hn . This implies that F ⊂ Hn , ∀n ≥ 0. Then following the method of proof of Theorem 3.1 we get that {xn } is well defined and converge strongly to ΠF (S )∩VI (C ,A)∩EP (F ) x0 .  Remark 3.2. Since every relatively nonexpansive mapping is weak relatively nonexpansive, Theorems 3.1, 3.2 also hold when S is relatively nonexpansive. 4. Applications In this section, put F (x, y) = 0 for all x, y ∈ C and rn = 1 for all n ∈ N in Theorems 3.1 and 3.2. Then, we have un = ΠC zn . So, by using Theorems 3.1 and 3.2, we can obtain two corollaries. Corollary 4.1. Let E be a real uniformly smooth and 2-uniformly convex Banach space with dual space E ∗ and let C be a nonempty closed convex subset of E. Let S be a weak relatively nonexpansive mapping of C into E and let A be an α -inverse-strongly

J. Kang et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 755–765

765

monotone mapping of C into E ∗ such that F (S ) ∩ VI (C , A) 6= ∅. Assume that kAxk ≤ kAx − Apk for all x ∈ C and p ∈ VI (C , A). Let 0 < a0 ≤ βn ≤ b0 :=

c2α , 2

where c is the constant in (2.1). Let {xn } be sequence generated by

 yn = ΠC (J −1 (Jxn − βn Axn ))   zn = J −1 (αn Jxn + (1 − αn )JSyn )    H0 = {v ∈ C : φ(v, z0 ) ≤ α0 φ(v, x0 ) + (1 − α0 )φ(v, y0 ) ≤ φ(v, x0 )} Hn = {v ∈ Hn−1 ∩ Wn−1 : φ(v, zn ) ≤ αn φ(v, xn ) + (1 − αn )φ(v, yn ) ≤ φ(v, xn )}   W0 = C     Wn = {v ∈ Wn−1 ∩ Hn−1 : hxn − v, Jx0 − Jxn i ≥ 0} xn+1 = ΠHn ∩Wn (x0 ), n ≥ 1,

(4.1)

for every n ∈ N ∪ {0}, where J is the duality mapping on E, {αn } ⊂ [0, 1] satisfies lim infn→∞ αn (1 − αn ) > 0. Then {xn } converges strongly to ΠF (S )∩VI (C ,A) (x0 ), where ΠF (S )∩VI (C ,A) is the generalized projection from E onto F (S ) ∩ VI (C , A). Corollary 4.2. Let E be a real uniformly smooth and 2-uniformly convex Banach space with dual space E ∗ and let C be a nonempty closed convex subset of E. Let S be a weak relatively nonexpansive mapping of C into E and let A be a maximal monotone mapping of C into E ∗ with J −1 T relatively weak nonexpansive such that F (S ) ∩ VI (C , A) 6= ∅, where Tx := Jx − Ax. Assume that kAxk ≤ kAx − Apk for all x ∈ C and p ∈ VI (C , A). Let 0 < a0 ≤ βn ≤ 1, where c is the constant in (2.1) and {αn } ⊂ [0, 1] satisfies lim infn→∞ αn (1 − αn ) > 0 and {rn } ⊂ [a, ∞) for some a > 0. Then the sequence {xn } generated by (4.1) converges strongly to ΠF (S )∩VI (C ,A) (x0 ), where ΠF (S )∩VI (C ,A) is the generalized projection from E onto F (S ) ∩ VI (C , A). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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