Convergence theorems on viscosity approximation methods for a finite family of nonexpansive non-self-mappings

Mathematical and Computer Modelling 50 (2009) 1338–1347

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Convergence theorems on viscosity approximation methods for a finite family of nonexpansive non-self-mappingsI Joung-Hahn Yoon, Jong Soo Jung ∗ Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

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Article history: Received 17 October 2008 Received in revised form 23 June 2009 Accepted 2 July 2009 Keywords: Nonexpansive non-self-mapping Contraction Viscosity approximation method Common fixed points Strictly convex Weakly sequentially continuous duality mapping Variational inequality Sunny nonexpansive retraction Weakly asymptotically regular

abstract Let E be a Banach space, C a nonempty closed convex subset of E, f : C → C a contraction, TN and Ti : C → E nonexpansive mappings with nonempty F := i=1 Fix(Ti ), where N ≥ 1 is an integer and Fix(Ti ) is the set of fixed points of Ti . Assume that C is a sunny nonexpansive retract of E with Q as the sunny nonexpansive retraction. It is proved that an iterative scheme xn+1 = λn+1 f (xn ) + (1 − λn+1 )QTn+1 xn (n ≥ 0) converges strongly to a solution in F of a certain variational inequality provided E is strictly convex, reflexive and has a weakly sequentially continuous duality mapping and provided the sequence {λn } satisfies certain conditions and the sequence {xn } satisfies weak asymptotic regularity. An application of the main result is also given. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Let E be a real Banach space and C a nonempty closed convex subset of E. Recall that a mapping f : C → C is a contraction on C if there exists a constant k ∈ (0, 1) such that kf (x) − f (y)k ≤ kkx − yk for all x, y ∈ C . We use ΣC to denote the collection of mappings f verifying the above inequality. That is, ΣC = {f : C → C | f is a contraction with constant k}. Note that each f ∈ ΣC has a unique fixed point in C . Let T : C → C be a nonexpansive mapping (recall that a mapping T : C → C is nonexpansive if kTx − Tyk ≤ kx − yk, x, y ∈ C ) and Fix(T ) denote the set of fixed points of T ; that is, Fix(T ) = {x ∈ C : x = Tx}. We consider the iterative scheme: for N ≥ 1, T1 , T2 , . . . , TN : C → C nonexpansive mappings, f ∈ ΣC and λn ∈ (0, 1), xn+1 = λn+1 f (xn ) + (1 − λn+1 )Tn+1 xn ,

n ≥ 0,

(1.1)

where Tn := Tn mod N . As a special case of (1.1), we have the following scheme zn+1 = λn+1 u + (1 − λn+1 )Tn+1 zn ,

n ≥ 0,

(1.2)

where u, z0 ∈ C are arbitrary (but fixed). The iterative scheme (1.2) has been investigated by many authors: see, for example, Browder [1], Halpern [2], Lions [3], Reich [4], Shioji and Takahashi [5], Wittmann [6], Xu [7] for N = 1 and Bauschke [8], Jung [9], O’Hara et al. [10,11], Takahashi et al. [12] and Zhou et al. [13] for N > 1, respectively and so on. The authors above showed that the sequence {zn } generated by (1.2) converges strongly to a point in the fixed point set Fix(T ) for N = 1 and

I This study was supported by research funds from Dong-A University.

∗

Corresponding author. E-mail addresses: [email protected] (J.-H. Yoon), [email protected] (J.S. Jung).

0895-7177/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2009.07.005

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to a point in the common fixed point set i=1 Fix(Ti ) for N > 1 under appropriate conditions in either Hilbert spaces or certain Banach spaces. For N = 1, the viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping was proposed by Moudafi [14] in Hilbert space. In 2004, Xu [15] extended Theorem 2.2 of Moudafi [14] for the iterative scheme (1.1) to a uniformly smooth Banach space. In 2006, Chang [16] and Jung [17] improved the results of Xu [15] to the case of finite mappings T1 , . . . , Tn for N > 1. On the other hand, in 2005, Chidume et al. [18] considered the following iterative scheme for the nonexpansive non-selfmappings Ti : C → E, i = 1, . . . , N which are weakly inward:

TN

zn+1 = λn+1 u + (1 − λn+1 )QTn+1 zn ,

n ≥ 0,

(1.3)

where u, z0 ∈ C are arbitrary (but fixed) and Q is the sunny nonexpansive retraction form E onto C . Using the following control conditions: (C1) lim λn = 0; n→∞

(C2)

∞ X

λn = ∞;

(C3)

n=1

(C1) lim λn = 0; n→∞

(C2)

∞ X

∞ X

|λn+N − λn | < ∞,

or

n =1

λn = ∞;

λn = 1, n→∞ λn+N

(C4) lim

n=1

they proved convergence of the sequence {zn } generated by (1.3) in a reflexive Banach space E having a uniformly Gâteaux differentiable norm together with the assumption that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. In 2008, Matsushita and Takahashi [19] proved convergence of the sequence {zn } generated by (1.3) for N = 1 in a uniformly convex Banach space having a uniformly Gâteaux differentiable norm under the conditions (C1), (C2) and (C3) without the weak inwardness condition. In this paper, motivated and inspired by the above-mentioned results, we consider the following iterative scheme: for the nonexpansive non-self-mappings Ti : C → E, i = 1, . . . , N, f ∈ ΣC and λn ∈ (0, 1), xn+1 = λn+1 f (xn ) + (1 − λn+1 )QTn+1 xn ,

n ≥ 0,

(1.4)

where Q is the sunny nonexpansive retraction form E onto C . More precisely, we establish the strong convergence of the sequence {xn } generated by (1.4) in a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping provided {λn } satisfies the conditions (C1), (C2) and the sequence {xn } satisfies the weak asymptotic regularity. Moreover, we show that this strong limit is a solution of a certain variational inequality. The main results generalize the corresponding results of Chang [16], Chidume et al. [18] and Jung [20] and Xu [15] to the class of non-selfmappings which needn’t satisfy the weak inwardness condition. Our results also improve the corresponding results in [8,9, 17,19,10,11,13]. 2. Preliminaries and lemmas Let E be a real Banach space with norm k · k and let E ∗ be its dual. The value of f ∈ E ∗ at x ∈ E will be denoted by hx, f i. ∗

When {xn } is a sequence in E, then xn → x (resp., xn * x, xn * x) will denote strong (resp., weak, weak∗ ) convergence of the sequence {xn } to x. x +y A Banach space E is said to be strictly convex if k 2 k < 1 for all x, y ∈ E with kxk = kyk = 1. The norm of E is said to be Gâteaux differentiable if kx + tyk − kxk lim t →0 t exists for each x, y in its unit sphere U = {x ∈ E : kxk = 1}. Such an E is called a smooth Banach space. By a gauge function we mean a continuous strictly increasing function ϕ defined on R+ := [0, ∞) such that ϕ(0) = 0 ∗ and limr →∞ ϕ(r ) = ∞. The mapping Jϕ : E → 2E defined by Jϕ (x) = {f ∈ E ∗ : hx, f i = kxkkf k, kf k = ϕ(kxk)},

for all x ∈ E

is called the duality mapping with gauge function ϕ . In particular, the duality mapping with gauge function ϕ(t ) = t denoted by J, is referred to the normalized duality mapping. The following property of duality mapping is well known:

 ϕ(|λ| · kxk) J (x) for all x ∈ E \ {0}, λ ∈ R, kx k where R is the set of all real numbers; in particular, J (−x) = −J (x) for all x ∈ E [21]. Jϕ (λx) = sign λ



(2.1)

Following Browder [22], we say that a Banach space E has a weakly sequential continuous duality mapping if there exists a gauge function ϕ such that the duality mapping Jϕ is single-valued and continuous from the weak topology to the weak∗ ∗

topology, that is, for any {xn } ∈ E with xn * x, Jϕ (xn ) * Jϕ (x). For example, every lp space (1 < p < ∞) has a weakly sequentially continuous duality mapping with gauge function ϕ(t ) = t p−1 . Set

Φ (t ) =

t

Z

ϕ(τ )dτ , 0

for all t ∈ R+ .

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Then it is known [1] that Jϕ (x) is the subdifferential of the convex functional Φ (k · k) at x. Thus it is easy to see that the normalized duality mapping J (x) can also be defined as the subdifferential of the convex functional Φ (kxk) = kxk2 /2, that is, for all x ∈ E J (x) = ∂ Φ (kxk) = {f ∈ E ∗ : Φ (kyk) − Φ (kxk) ≥ hy − x, f i for all y ∈ E }. It is well known that if E is smooth, then the normalized duality mapping J is single-valued and norm to weak∗ continuous. We need the following lemmas for the proof of our main results. For these lemmas, we refer to [21,23,24]. Lemma 2.1. Let E be a real Banach space and ϕ a continuous strictly increasing function on R+ such that ϕ(0) = 0 and limr →∞ ϕ(r ) = ∞. Define

Φ (t ) =

t

Z

ϕ(τ )dτ ,

for all t ∈ R+ .

0

Then the following inequality holds

Φ (kx + yk) ≤ Φ (kxk) + hy, jϕ (x + y)i,

for all x, y ∈ E ,

where jϕ (x + y) ∈ Jϕ (x + y). In particular, if E is smooth, then one has

kx + yk2 ≤ kxk2 + 2hy, J (x + y)i,

for all x, y ∈ E .

Lemma 2.2. Let {sn } be a sequence of non-negative real numbers satisfying sn+1 ≤ (1 − λn )sn + λn βn + γn ,

n ≥ 0,

where {λn }, {βn } and {γn } satisfy the following conditions: (i) {λn } ⊂ [0, 1] and n=0P λn = ∞ or, equivalently, ∞ (ii) lim supn→∞ βn ≤P 0 or n=1 λn βn < ∞, ∞ (iii) γn ≥ 0 (n ≥ 0), n=0 γn < ∞.

P∞

Q∞

n=0

(1 − λn ) = 0,

Then limn→∞ sn = 0. Let µ be a continuous linear functional on l∞ and (a0 , a1 , . . .) ∈ l∞ . We write un (an ) instead of µ((a0 , a1 , . . .)). µ is said to be Banach limit [25] if µ satisfies kµk = µn (1) = 1 and un (an+1 ) = µn (an ) for all (a0 , a1 , . . .) ∈ l∞ . If µ is a Banach limit, the following are well known: (i) for all n ≥ 1, an ≤ cn implies µ(an ) ≤ µ(cn ), (ii) µ(an+N ) = µ(an ) for any fixed positive integer N, (iii) lim infn→∞ an ≤ µn (an ) ≤ lim supn→∞ an for all (a0 , a1 , . . .) ∈ l∞ . The following lemma was given in [13] as the revision of [5, Proposition 2]. Lemma 2.3. Let a ∈ R be a real number and a sequence {an } ∈ l∞ satisfy the condition µn (an ) ≤ a for all Banach limit µ. If lim supn→∞ (an+N − an ) ≤ 0 for N ≥ 1, then lim supn→∞ an ≤ a. Let C be a nonempty closed convex subset of E and D a subset of C . Then a mapping Q : C → D is said to be retraction from C onto D if Qx = x for all x ∈ D. A retraction Q : C → D is said to be sunny if Q (Qx + t (x − Qx)) = Qx for all t ≥ 0 and x + t (x − Qx) ∈ C . A sunny nonexpansive retraction is a sunny retraction which is also nonexpansive. Sunny nonexpansive retractions are characterized as follows [26, p. 48]: If E is a smooth Banach space, then Q : C → D is a sunny nonexpansive retraction if and only if the following condition holds:

hx − Qx, J (z − Qx)i ≤ 0,

x ∈ C , z ∈ D.

(2.2)

(Note that this fact still holds by (2.1) if the normalized duality mapping J is replaced by a general duality mapping Jϕ with gauge function ϕ .) We need the following lemmas which were given in [19]. Lemma 2.4 ([19]). Let C be a closed convex subset of a smooth Banach space E and T be a mapping from C into E. Suppose that C is a sunny nonexpansive retract of E. If T satisfies the nowhere-normal outward condition Tx ∈ Sxc ,

for all x ∈ C ,

where Sx = {y ∈ E : y 6= x, Qy = x} and

(2.3) Sxc

is the complement of Sx , then Fix(T ) = Fix(QT ).

Lemma 2.5 ([19]). Let C be a closed convex subset of a strictly convex Banach space E and T a nonexpansive mapping from C into E. Suppose that C is a sunny nonexpansive retract of E. If Fix(T ) 6= ∅, then T satisfies the nowhere-normal outward condition (2.3).

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Finally, the sequence {xn } in E is said to be weakly asymptotically regular if for N ≥ 1,

w − lim (xn+N − xn ) = 0, n→∞

that is, xn+N − xn * 0

and asymptotically regular if for N ≥ 1, lim kxn+N − xn k = 0,

n→∞

respectively. 3. Main results In this section, we study the strong convergence of the iterative scheme (1.4) for a family of finite nonexpansive nonself-mappings. Let T1 , T2 , . . . , TN be nonexpansive mappings of C into itself. Then for any n ≥ 1, Tn+N Tn+N −1 · · · Tn+1 : C → C is nonexpansive, and so, for any t ∈ (0, 1) and f ∈ ΣC , tf + (1 − t )Tn+N Tn+N −1 · · · Tn+1 : C → C is a contraction. Thus, by the Banach contraction principle, there exists a unique fixed point xnt (f ) satisfying xnt (f ) = tf (xnt (f )) + (1 − t )Tn+N Tn+N −1 · · · Tn+1 xnt (f ).

(3.1)

For simplicity we will write xnt for xnt (f ) provided no confusion occurs. The following result for the existence of q which solves a variational inequality

h(I − f )(q), Jϕ (q − p)i ≤ 0,

f ∈ ΣC , p ∈ F :=

N \

Fix(Ti ) 6= ∅

i =1

was obtained by Jung [20]. Theorem J ([20]). Let E be a reflexive Banach space having a weakly sequentially continuous duality mapping Jϕ with gauge function ϕ . Let C be a nonempty closed convex subset of E and T1 , . . . , TN nonexpansive mappings from C into itself with TN F := i=1 Fix(Ti ) 6= ∅ and F = Fix(TN TN −1 · · · T1 ) = Fix(T1 TN · · · T3 T2 ) = · · · = Fix(TN −1 TN −2 · · · T1 TN ). Then {xnt } defined by (3.1) converges strongly to a point in F . If we define P : ΣC → F by P (f ) := lim xnt , t →0

f ∈ ΣC ,

(3.2)

then P (f ) is independent of n and P (f ) solves the variational inequality

h(I − f )(P (f )), Jϕ (P (f ) − p)i ≤ 0,

f ∈ ΣC , p ∈ F .

Remark 3.1. (1) In Theorem J, if f (x) = u, x ∈ C , is a constant, then it follows from (2.2) that (3.2) is reduced to the sunny nonexpansive retraction from C onto F ,

hPu − u, Jϕ (Pu − p)i ≤ 0,

u ∈ C, p ∈ F.

(2) As a matter of fact, we point out that in Theorem 3.1, Lemma 3.1 and Theorem 3.2 of [20], for abbreviation, the normalized duality mapping J instead of the duality mapping Jϕ was assumed. Using Theorem J, we give the following result in a reflexive Banach space having a weakly sequentially continuous duality mapping. Proposition 3.1. Let E be a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping Jϕ with gauge function ϕ . Let C be a nonempty closed convex subset of E. Assume that C is a sunny nonexpansive retract of E with Q as the sunny nonexpansive retraction. Let T1 , . . . , TN be nonexpansive mappings from C into E with F := satisfying the following conditions: S := QTN QTN −1 · · · QT1 = QT1 QTN · · · QT3 QT2 = · · · = QTN −1 QTN −2 · · · QT1 QTN ; N \

Fix(QTi ) = Fix(S ).

TN

i=1

F (Ti ) 6= ∅

(i) (ii)

i =1

Let {λn } be a sequence in (0, 1) which satisfies the condition: (C1) limn→∞ αn = 0, f ∈ ΣC and x0 ∈ C may be chosen arbitrarily. Let {xn } be a sequence generated by xn+1 = λn+1 f (xn ) + (1 − λn+1 )QTn+1 xn ,

n ≥ 0,

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and µ a Banach limit. Then

µn (h(I − f )(q), Jϕ (q − xn )i) ≤ 0, where q = limt →0+ xt and xt = tf (xt ) + (1 − t )Sxt for S = QTN QTN −1 · · · QT1 and t ∈ (0, 1). Proof. Note that the definition of the weak sequential continuity of duality mapping Jϕ with gauge function ϕ implies that E is smooth. Let xt = tf (xt ) + (1 − t )Sxt for S = QTN QTN −1 · · · QT1 and t ∈ (0, 1). Then by Theorem J, {xt } strongly converges TN TN to a point in Fix(S ) = i=1 Fix(QTi ), which is denoted by q := limt →0+ xt . By Lemmas 2.4 and 2.5, q ∈ F = i=1 Fix(Ti ). Now, since xt − xn+N = (1 − t )(Sxt − xn+N ) + t (f (xt ) − xn+N ), by Lemma 2.1, we have

Φ (kxt − xn+N k) ≤ Φ ((1 − t )kSxt − xn+N k) + t hf (xt ) − xn+N , Jϕ (xt − xn+N )i.

(3.3)

Let p ∈ F . Then we have

kxt − pk ≤ t kf (xt ) − pk + (1 − t )kSxt − Spk ≤ t kf (xt ) − pk + (1 − t )kxt − pk. This gives that

kxt − pk ≤ kf (xt ) − pk ≤ kf (xt ) − f (p)k + kf (p) − P k ≤ kkxt − pk + kf (p) − pk and so

kx t − p k ≤

1 1−k

kf (p) − pk,

t ∈ (0, 1),

and hence {xt } is bounded. We also have

 kxn − pk ≤ max kx0 − pk,

1 1−k

kf (p) − pk



1 for all n ≥ 0 and all p ∈ F and so {xn } is bounded. Indeed, let p ∈ F and d = max{kx0 − pk, 1− kf (p) − pk}. Then by the k nonexpansivity of Tn and f ∈ ΣC ,

kx1 − pk ≤ ≤ ≤ ≤

(1 − λ1 )kQT1 x0 − pk + λ1 kf (x0 ) − pk (1 − λ1 )kx0 − pk + λ1 (kf (x0 ) − f (p)k + kf (p) − pk) (1 − (1 − k)λ1 )kx0 − pk + λ1 kf (p) − pk (1 − (1 − k)λ1 )d + λ1 (1 − k)d = d.

Using an induction, we obtain kxn+1 − pk ≤ d. Hence {xn } is bounded, and so are {QTn+1 xn } and {f (xn )}. As a consequence with the control condition (C1), we get

kxn+1 − QTn+1 xn k ≤ λn+1 kQTn+1 xn − f (xn )k → 0 (as n → ∞). By using the same method, we have

kxn+N − QTn+N · · · QTn+1 xn k → 0 (as n → ∞). Indeed, noting that each QTi is nonexpansive and using just above fact, we obtain the finite table xn+N − QTn+N xn+N −1 → 0, QTn+N xn+N −1 − QTn+N QTn+N −1 xn+N −2 → 0,

.. . QTn+N · · · QTn+2 xn+1 − QTn+N · · · QTn+1 xn → 0. Adding up this table yields xn+N − QTn+N · · · QTn+1 xn → 0 (as n → ∞). Moreover, we prove that xn+N − QTN QTN −1 · · · QT1 xn = xn+N − Sxn → 0

(as n → ∞).

(3.4)

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Indeed, we can see the following: If n mod N = 1, then QTn+N QTn+N −1 · · · QTn+1 = QT1 QTN · · · T2 ; If n mod N = 2, then QTn+N QTn+N −1 · · · QTn+1 = QT2 QT1 QTN · · · QT3 ;

.. .

If n mod N = N, then QTn+N QTn+N −1 · · · QTn+1 = QTN QTN −1 · · · QT1 . In view of the condition (i) QTN QTN −1 · · · QT1 = QT1 QTN · · · QT3 QT2 = · · · = QTN −1 QTN −2 · · · QT1 QTN , so we have QTN QTN1 · · · QT1 = QTn+N QTn+N −1 · · · QTn+1 ,

for all n ≥ 1.

This implies that xn+N − Sxn = xn+N − QTN QTN −1 · · · QT1 xn = xn+N − QTn+N QTn+N −1 · · · QTn+1 xn → 0 (as n → ∞). Observe also with (3.4) that

kSxt − xn+N k ≤ kxt − xn k + en , where en = kxn+N − Sxn k → 0 as n → ∞, and

hf (xt ) − xn+N , Jϕ (xt − xn+N )i = hf (xt ) − xt , Jϕ (xt − xn+N )i + kxt − xn+N kϕ(kxt − xn+N k). Thus it follows from (3.3) that

Φ (kxt − xn+N k) ≤ Φ ((1 − t )(kxt − xn k + en )) + t (hf (xt ) − xt , Jϕ (xt − xn+N )i + kxt − xn+N kϕ(kxt − xn+N k)). (3.5) Applying the Banach limit µ to (3.5), we have

µn (Φ (kxt − xn+N k)) ≤ µn (Φ ((1 − t )(kxt − xn k + en ))) + t µn (hf (xt ) − xt , Jϕ (xt − xn+N )i) + t µn (kxt − xn+N kϕ(kxt − xn+N k))

(3.6)

and it follows from (3.6) that 1

µn (Φ ((1 − t )kxt − xn k) − Φ (kxt − xn k)) + µn (kxt − xn+N kϕ(kxt − xn+N k)) Z kxt −xn k  1 ϕ(τ )dτ + µn (kxt − xn+N kϕ(kxt − xn+N k)) = − µn

µn (hxt − f (xt ), Jϕ (xt − xn )i) ≤

t

t

(1−t )kxt −xn k

= µn (kxt − xn k(ϕ(kxt − xn k) − ϕ(τn ))),

(3.7)

for some τn satisfying (1 − t )kxt − xn k ≤ τn ≤ kxt − xn k. Since ϕ is uniformly continuous on compact intervals of R , +

kxt − xn k − τn ≤ t kxt − xn k   2 kf (p) − pk + kx0 − pk → 0 (as t → 0), ≤t 1−k and q = limt →0 xt , we conclude from (3.7) that

µn (h(I − f )(q), Jϕ (q − xn )i) ≤ lim sup µn (hxt − f (xt ), Jϕ (xt − xn )i) ≤ 0.  t →0

Theorem 3.1. Let E be a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping Jϕ with gauge function ϕ . Let C be a nonempty closed convex subset of E. Assume that C is a sunny nonexpansive retract of E with TN Q as the sunny nonexpansive retraction. Let T1 , . . . , TN be nonexpansive mappings from C into E with F := i=1 Fix(Ti ) 6= ∅ satisfying the following conditions: S := QTN QTN −1 · · · QT1 = QT1 QTN · · · QT3 QT2 = · · · = QTN −1 QTN −2 · · · QT1 QTN ;

(i)

N

\

Fix(QTi ) = Fix(S ).

(ii)

i =1

Let {λn } be a sequence in (0, 1) which satisfies the following conditions: (C1) limn→∞ λn = 0; (C2) and x0 ∈ C may be chosen arbitrarily. Let {xn } be a sequence generated by xn+1 = λn+1 f (xn ) + (1 − λn+1 )QTn+1 xn ,

n ≥ 0.

P∞

n =1

λn = ∞. Let f ∈ ΣC (3.8)

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If {xn } is weakly asymptotically regular, then {xn } converges strongly to q ∈ F , where q is the unique solution of the variational inequality

h(I − f )(q), Jϕ (q − p)i ≤ 0,

f ∈ ΣC , p ∈ F .

Proof. Let xt = tf (xt ) + (1 − t )Sxt for S = QTN QTN −1 · · · QT1 and t ∈ (0, 1). Then by Theorem J, {xt } strongly TN TN converges to a point in i=1 Fix(QTi ), which is also denoted by q := limt →0+ xt ∈ i=1 Fix(QTi ). By Lemmas 2.4 and 2.5, TN q ∈ F = i=1 Fix(Ti ) and q is the unique solution of the variational inequality

h(I − f )(q), Jϕ (q − p)i ≤ 0,

f ∈ ΣC , p ∈ F .

We proceed with the following steps: 1 Step 1. We show that kxn − z k ≤ max{kx0 − z k, 1− kf (z )− z k} for all n ≥ 0 and all z ∈ F as in the proof of Proposition 3.1. k Hence {xn } is bounded and so are {QTn+1 xn } and {f (xn )}. Step 2. We show that lim supn→∞ h(I − f )(q), Jϕ (q − xn )i ≤ 0. To this end, put an := h(I − f )(q), Jϕ (q − xn )i,

n ≥ 1.

Then Proposition 3.1 implies that µn (an ) ≤ 0 for any Banach limit µ. Since {xn } is bounded, there exists a subsequence {xnj } of {xn } such that lim sup(an+N − an ) = lim (anj +N − anj ) j→∞

n→∞

and xnj * z ∈ E. This implies that xnj +N * z since {xn } is weakly asymptotically regular. From the weak sequential continuity of duality mapping Jϕ , we have

w − lim Jϕ (q − xnj +N ) = w − lim Jϕ (q − xnj ) = Jϕ (q − z ), j→∞

j→∞

and so lim sup(an+N − an ) = lim h(I − f )(q), Jϕ (q − xnj +N ) − Jϕ (q − xnj )i = 0. j→∞

n→∞

Then Lemma 2.3 implies that lim supn→∞ an ≤ 0, that is, lim suph(I − f )(q), Jϕ (q − xn )i ≤ 0. n→∞

Step 3. We show that limn→∞ kxn − qk = 0. By using (3.8), we have xn+1 − q = λn+1 (f (xn ) − q) + (1 − λn+1 )(QTn+1 xn − q). Applying Lemma 2.1, we obtain

kxn+1 − qk2 ≤ (1 − λn+1 )2 kQTn+1 xn − qk2 + 2λn+1 hf (xn ) − q, J (xn+1 ) − qi ≤ (1 − λn+1 )2 kxn − qk2 + 2kλn+1 kxn − qkkxn+1 − qk + 2λn+1 hf (q) − q, J (xn+1 ) − qi ≤ (1 − λn+1 )2 kxn − qk2 + kλn+1 (kxn − qk2 + kxn+1 − qk2 ) + 2λn+1 hf (q) − q, J (xn+1 ) − qi. Now, by using gauge function ϕ , we define for every n ≥ 0

  kq − x n k , θn := ϕ(kq − xn k) 0, From

kq−x k sup{ ϕ(kq−xn k) n

if q 6= xn if q = xn .

: q 6= xn } < ∞, we obtain limn→∞ sup θn < ∞. Also from (2.2), we have

J (q − xn+1 ) = θn+1 Jϕ (q − xn+1 ),

for all n ≥ 0.

It then follows that

kxn+1 − qk2 ≤ ≤

1 − (2 − k)λn + λ2n 1 − kλn 1 − (2 − k)λn 1 − kλn

kxn − qk2 +

kx n − q k2 +

2λn 1 − kλn

h(I − f )(q), J (q − xn+1 )i

λ2n 2λn M+ θn+1 h(I − f )(q), Jϕ (q − xn+1 )i, 1 − kλn 1 − kλn

where M = supn≥0 kxn − qk2 . Put

αn =

2(1 − k)λn 1 − kλn

and βn =

M λn 2(1 − k)

+

1 1−k

θn+1 h(I − f )(q), Jϕ (q − xn+1 )i.

(3.9)

J.-H. Yoon, J.S. Jung / Mathematical and Computer Modelling 50 (2009) 1338–1347

From (C1), (C2) and Step 2, it follows that αn → 0,

P∞

n=0

1345

αn = ∞ and lim supn→∞ βn ≤ 0. Since (3.9) reduces to

kxn+1 − qk ≤ (1 − αn )kxn − qk + αn βn , 2

2

from Lemma 2.2, we conclude that limn→∞ kxn − qk = 0. This completes the proof.



Corollary 3.1. Let E, Jϕ , C , T1 , . . . , TN , FP and Q be as in Theorem 3.1. Let {λn } be a sequence in (0, 1) which satisfies the following ∞ conditions: (C1) limn→∞ λn = 0; (C2) n=1 λn = ∞. Let f ∈ ΣC and x0 ∈ C may be chosen arbitrarily. Let {xn } be a sequence generated by xn+1 = λn+1 f (xn ) + (1 − λn+1 )QTn+1 xn ,

n ≥ 0.

(3.10)

If {xn } is asymptotically regular, then {xn } converges strongly to q ∈ F , where q is the unique solution of the variational inequality

h(I − f )(q), Jϕ (q − p)i ≤ 0,

f ∈ ΣC , p ∈ F .

Remark 3.2. (1) If {λn } in Corollary 3.1 satisfies the following conditions: (C1) (C2) (C3) (C4)

lim P∞n→∞ λn = 0; Pn=1 λn = ∞;

|λn+N − λn | < ∞; or λ −λ = 1 (or equivalently, limn→∞ nλn+nN+N = 0); or, n+N P∞ (C5) |λn+N − λn | ≤ o(λn+N ) + σn , n=1 σn < ∞ (the perturbed control condition), then the sequence {xn } generated by (3.10) is asymptotically regular. Indeed, by Step 1 in the proof of Theorem 3.1, there exists a constant L > 0 such that for all n ≥ 0, kf (xn )k + kQTn+1 xn k ≤ L. Since for all n ≥ 1, Tn+N = Tn , we have ∞ n =1

limn→∞ λ λn

kxn+N − xn k = k(1 − λn+N )(QTn+N xn+N −1 − QTn xn−1 ) + (λn+N − λn )(f (xn−1 ) − QTn xn−1 ) + λn+N (f (xn+N −1 ) − f (xn−1 ))k ≤ (1 − λn+N )kxn+N −1 − xn−1 k + |λn+N − λn |L + kλn+N kxn+N −1 − xn−1 k = (1 − (1 − k)λn+N )kxn+N −1 − xn−1 k + |λn+N − λn |L.

(3.11)

(a) In case of the conditions (C1), (C2) and (C5), from (3.11) and (C5) we obtain

kxn+N − xn k ≤ (1 − (1 − k)λn+N )kxn+N −1 − xn−1 k + |λn+N − λn |L ≤ (1 − (1 − k)λn+N )kxn+N −1 − xn−1 k + o(λn+N )L + σn L.

(3.12)

By taking sn+1 = kxn+N − xn k, αn = (1 − k)λn+N , αn βn = o(λn+N )L and γn = σn L, from (3.12) we have sn+1 ≤ (1 − αn )sn + αn βn + γn . Hence, by (C1), (C2), (C5) and Lemma 2.2, limn→∞ kxn+N − xn k = 0. (b) In case of the conditions (C1), (C2) and (C3), by taking sn+1 = kxn+N − xn k, αn = (1 − k)λn+N , βn = 0 and γn = |λn+N − λn |L, from (3.11) we have sn+1 ≤ (1 − αn )sn + γn . Hence, by (C1), (C2), (C3) and Lemma 2.2, limn→∞ kxn+N − xn k = 0. |λ −λ | (c) In case of the conditions (C1), (C2) and (C4), by sn+1 = kxn+N − xn k, αn = (1 − k)λn+N , βn = (1n−+kN)λ n L and γn = 0, n+N from (3.11) we have sn+1 ≤ (1 − αn )sn + αn βn . Hence, by (C1), (C2), (C4) and Lemma 2.2, limn→∞ kxn+N − xn k = 0. (2) If {λn } in Corollary 3.1 satisfies the condition (C1) limn→∞ λn = 0 and the sequence {xn } generated by (3.10) satisfies lim kxn − Sxn k = 0,

(H)

n→∞

where S = QTN QTn−1 · · · QT2 QT1 , then the sequence {xn } generated by (3.10) is asymptotically regular. Indeed, since limn→∞ kxn+N − Sxn k = 0 by (3.4), it follows from (H) that

kxn+N − xn k ≤ kxn+N − Sxn k + kSxn − xn k → 0 (as n → ∞). In view of these observations, we have the following results. Corollary 3.2. Let E, Jϕ , C , T1 , . . . , TN , F and Q be as in Corollary 3.1. Let {λn } be a sequence in (0, 1) which satisfies the conditions (C1), (C2) and (C5) (or the conditions (C1), (C2) and (C3), or the conditions (C1), (C2) and (C4)) in Remark 3.2, f ∈ ΣC and x0 ∈ C may chosen arbitrarily. Let {xn } be a sequence generated by xn+1 = λn+1 f (xn ) + (1 − λn+1 )QTn+1 xn ,

n ≥ 0.

1346

J.-H. Yoon, J.S. Jung / Mathematical and Computer Modelling 50 (2009) 1338–1347

Then {xn } converges strongly to q ∈ F , where q is the unique solution of the variational inequality

h(I − f )(q), Jϕ (q − p)i ≤ 0,

f ∈ ΣC , p ∈ F .

Corollary 3.3. Let E, Jϕ , C , T1 , . . . , TN and Q be as in Corollary 3.1. Let {λn } be a sequence in (0, 1) which satisfies the conditions (C1) and (C2) in Corollary 3.1, f ∈ ΣC and x0 ∈ C may chosen arbitrarily. Let {xn } be a sequence generated by xn+1 = λn+1 f (xn ) + (1 − λn+1 )QTn+1 xn ,

n ≥ 0.

If (H) limn→∞ kxn − Sxn k = 0 where S = QTN QTn−1 · · · QT2 QT1 , then {xn } converges strongly to q ∈ F , where q is the unique solution of the variational inequality

h(I − f )(q), Jϕ (q − p)i ≤ 0,

f ∈ ΣC , p ∈ F .

Remark 3.3. (1) Theorem 3.1 improves Theorem 3.1 in Chidume et al. [18] to the viscosity method without the weak inwardness condition in a different Banach space. (2) Theorem 3.1, Corollaries 3.2 and 3.3 also generalizes corresponding results in Chang [16] to the class of non-selfmappings in a different Banach space. Moreover, Theorem 3.1 improve partially Theorem 3.3 in Jung [20] to the class of non-self-mappings. (3) Corollaries 3.2 and 3.3 extend Theorem 4.1 in Matsushita and Takahashi [19] to the viscosity method for a family of finite nonexpansive non-self-mappings in a different Banach space. (4) Our results extend and improve the corresponding results of Bauschke [8], Jung [9,17], O’Hara et al. [11], Takahashi et al. [12] and Zhou et al. [13] to more general Banach spaces and to the viscosity method for the class of non-self-mappings. As in [18,20,13], we study an application of Theorem 3.1 in a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping. Using Lemmas 2.4 and 2.5 and the proof of Lemma 3.3 in [18], we have the following lemma. Lemma 3.1. Let E be a strictly convex Banach space and C a closed convex subset of E. Assume that C is a sunny nonexpansive retract of E with Q as the sunny nonexpansive retraction. Let T1 , . . . , TN be nonexpansive mappings from C into E with TN F := i=1 Fix(Ti ) 6= ∅. Let S1 , S2 , . . . , SN be nonexpansive mappings of C into E defined by Si = (1 − αi )I + αi Ti for any 0 < αi < 1, (i = 1, 2, . . . , N ) where I denotes the identity mapping on C . Then N \

TN

i=1

Fix(Ti ) =

TN

i=1

Fix(QSi ) and

Fix(Si ) = Fix(QSN · · · QS1 ) = Fix(QS1 QSN · · · QS2 )

i=1

= · · · = Fix(QSN −1 QSN −2 · · · QS1 QSN ). Using Lemma 3.1 and Theorem 3.1, we obtain the following result without the weak inwardness condition. Theorem 3.2. Let E be a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping Jϕ with gauge function ϕ . Let C be a nonempty closed convex subset of E. Assume that C is a sunny nonexpansive retract of E with TN Q as the sunny nonexpansive retraction. Let T1 , . . . , TN be nonexpansive mappings from C into E with F := i=1 Fix(Ti ) 6= ∅. Let S1 , S2 , . . . , SN be nonexpansive mappings of C into E defined by Si = (1 − αi )I + αi Ti for any 0 < αi < 1, (i = 1, 2, . . . , N ) which satisfies the condition: QSN QSN −1 · · · QS1 = QS1 QSN · · · QS3 QS2 = · · · = QSN −1 QSN −2 · · · QS1 QSN , where I denotes the identity mapping on C . For given f ∈ ΣC and x0 ∈ C , let {xn } be a sequence generated by xn+1 = λn+1 f (xn ) + (1 − λn+1 )QSn+1 xn ,

n ≥ 0,

where Sn := Sn mod N and {λn } satisfies the conditions (C1) and (C2) in Theorem 3.1. If {xn } is weakly asymptotically regular, then {xn } converges strongly to a point then {xn } converges strongly to q ∈ F , where q is the unique solution of the variational inequality

h(I − f )(q), Jϕ (q − p)i ≤ 0, Proof. By Lemma 3.1 we have

TN

i=1

f ∈ ΣC , p ∈ F . Fix(Ti ) =

TN

i=1

Fix(Si ) =

TN

i =1

Fix(QSi ) and

N

\

Fix(QSi ) = Fix(QSN · · · QS1 ) = Fix(QS1 QSN · · · QS3 QS2 )

i=1

= · · · = Fix(QSN −1 QSN −2 · · · QS1 QSN ). Thus, applying Theorem 3.1, we have the desired conclusion immediately.



J.-H. Yoon, J.S. Jung / Mathematical and Computer Modelling 50 (2009) 1338–1347

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As an immediate consequence of Remark 3.2 and Theorem 3.2, we have the following result without the weak inwardness condition: Corollary 3.4. Let E, Jϕ , C , Q , T1 , · · · , TN , F and S1 , . . . , SN be as in Theorem 3.2. For given f ∈ ΣC and x0 ∈ C , let {xn } be a sequence generated by xn+1 = λn+1 f (xn ) + (1 − λn+1 )QSn+1 xn ,

n ≥ 0,

and {λn } satisfies the conditions (C1), (C2) and (C5) (or the conditions (C1), (C2) and (C3), or the conditions (C1), (C2) and (C4)) TN in Remark 3.2. Then {xn } converges strongly to a point q ∈ i=1 Fix(Ti ), where q is the unique solution of the variational inequality

h(I − f )(q), Jϕ (q − p)i ≤ 0,

f ∈ ΣC , p ∈ F .

Remark 3.4. Theorem 3.2 and Corollary 3.4 improve Theorem 3.4 in Chidume et al. [18] to the viscosity method without the weak inwardness condition in a different Banach space. Theorem 3.2 also extends Theorem 3.5 in Jung [20] to the class of non-self-mappings. Acknowledgements The authors thank the referees for their valuable comments and suggestions along with careful reading of the manuscript, which improved the presentation of this manuscript. 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]

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