Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups

Nonlinear Analysis 67 (2007) 2306–2315 www.elsevier.com/locate/na

Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups Somyot Plubtieng ∗ , Kasamsuk Ungchittrakool Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Received 6 August 2006; accepted 11 September 2006

Abstract In this paper, we established strong convergence theorems for a common fixed point of two asymptotically nonexpansive mappings and for a common fixed point of two asymptotically nonexpansive semigroups by using the hybrid method in a Hilbert space. Moreover, we also proved a strong convergence theorem for a common fixed point of two nonexpansive mappings. Our results extend and improve the recent ones announced by Kim and Xu [T.W. Kim, H.W. Xu, Strong convergence of modified Mann iteration for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140–1152], Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379], and many others. c 2006 Elsevier Ltd. All rights reserved.

Keywords: Strong convergence; Asymptotically nonexpansive mappings; Asymptotically nonexpansive semigroups; Common fixed points

1. Introduction Let X be a real Banach space, C a nonempty closed convex subset of X and T : C → C a mapping. Recall that T is nonexpansive if kT x − T yk 6 kx − yk for all x, y ∈ C, and T is asymptotically nonexpansive [2] if there exists a sequence {kn } with kn > 1 for all n and limn→∞ kn = 1 and such that kT n x − T n yk 6 kn kx − yk for all n > 1 and x, y ∈ C. A point x ∈ C is a fixed point of T provided T x = x. Denote by Fix(T ) the set of fixed points of T ; that is, Fix(T ) = {x ∈ C : T x = x}. If S and T are two nonexpansive (asymptotically nonexpansive) mappings, then the point x ∈ Fix(S) ∩ Fix(T ) is called the common fixed point of S and T . Throughout this article, F := Fix(S) ∩ Fix(T ) is always assumed to be nonempty. Recall also that a one-parameter family T = {T (t) : 0 6 t < ∞} of self-mappings of a nonempty closed convex subset C of a Hilbert space H is said to be a (continuous) Lipschitzian semigroup on C (see, e.g., [12]) if the following conditions are satisfied: (i) T (0)x = x, x ∈ C, (ii) T (t + s)x = T (t)T (s)x, t, s > 0, x ∈ C, (iii) for each x ∈ C, the map t 7→ T (t)x is continuous on [0, ∞), ∗ Corresponding author. Tel.: +66 55261000; fax: +66 55261025.

E-mail address: [email protected] (S. Plubtieng). c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.09.023

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(iv) there exists a bounded measurable function L : (0, ∞) → [0, ∞) such that, for each t > 0, kT (t)x − T (t)yk 6 L t kx − yk, x, y ∈ C. A Lipschitzian semigroup T is called nonexpansive (or a contraction semigroup) if L t = 1 for all t > 0, and asymptotically nonexpansive if lim supt→∞ L t 6 1, respectively. We use Fix(T ) to denote the common fixed point set of the semigroup T ; that is, Fix(T ) = {x ∈ C : T (s)x = x, ∀s > 0}. Note that for an asymptotically nonexpansive semigroup T , we can always assume that the Lipschitzian constants {L t }t>0 are such that L t > 1 for all t > 0, L is nonincreasing in t, and limt→∞ L t = 1; otherwise we replace L t , for each t > 0, with L˜ t := max{sups >t L s , 1}. Construction of fixed points of nonexpansive mappings is an important subject in the theory of nonexpansive mappings and its applications. Fixed point iteration processes for nonexpansive mappings and asymptotically nonexpansive mappings in Hilbert spaces and Banach spaces including Mann and Ishikawa iteration processes have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequalities: see [6,3,9,10]. However, Mann and Ishikawa iterations processes have only weak convergence even in a Hilbert space: see [6,3]. Some attempts to modify the Mann iteration method so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [8] proposed the following modification of the Mann iteration method for a nonexpansive mapping T in a Hilbert space H :  x0 ∈ C chosen arbitrarily,     yn = αn xn + (1 − αn )T xn , Cn = {v ∈ C : kyn − vk 6 kxn − vk}, (1.1)    Q n = {v ∈ C : hxn − v, xn − x0 i 6 0},  xn+1 = PCn ∩Q n (x0 ), where PK denotes the metric projection from H onto a closed convex subset K of H . They proved that if the sequence {αn } is bounded above from one, then {xn } defined by (1.1) converges strongly to PFix(T ) (x0 ). Moreover they introduced and studied an iteration process of a nonexpansive semigroup T = {T (t) : 0 6 t < ∞} in a Hilbert space H :  x0 ∈ C chosen arbitrarily,  Z   1 tn   T (u)xn du,  yn = αn xn + (1 − αn ) tn 0 (1.2) Cn = {z ∈ C : kyn − zk 6 kxn − zk},       Q n = {z ∈ C : hxn − z, x0 − xn i > 0}, xn+1 = PCn ∩Q n (x0 ). Recently, Kim and Xu [4] adapted the iteration (1.1) to asymptotically nonexpansive mappings in a Hilbert space H:  x0 ∈ C chosen arbitrarily,    n   yn = αn xn + (1 − αn )T xn , Cn = {v ∈ C : kyn − vk2 6 kxn − vk2 + θn },   Q = {v ∈ C : hxn − v, x0 − xn i > 0},    n xn+1 = PCn ∩Q n (x0 ),

(1.3)

where θn = (1−αn )(kn2 −1)(diam C)2 → 0 as n → ∞. They also proved that if αn 6 a for all n and for some 0 < a < 1, then the sequence {xn } generated by (1.3) converges strongly to PFix(T ) (x0 ). Moreover, they modified an iterative method (1.2) to the case of an asymptotically nonexpansive semigroup T = {T (t) : 0 6 t < ∞} in a Hilbert space H :  x0 ∈ C chosen arbitrarily,  Z   1 tn    T (u)xn du,  yn = αn xn + (1 − αn ) tn 0 (1.4)  Cn = {z ∈ C : kyn − zk2 6 kxn − zk2 + θn },       Q n = {z ∈ C : hxn − z, x0 − xn i > 0}, xn+1 = PCn ∩Q n (x0 ).

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Rt where θn = (1 − αn )[( t1n 0n L u du)2 − 1](diam C)2 → 0 as n → ∞. Under the same condition of the sequences {αn } and {tn }, using boundedness of the nonempty closed convex subset C and the Lipschitzian constant L t of the mapping T (t) where L t is bounded and measurable, they proved that the sequence {xn } generated by (1.4) converges strongly to PFix(T ) (x0 ). Inspired and motivated by these facts, it is the purpose of this paper to develop the iteration (1.3) and (1.4) to the process for two asymptotically nonexpansive mappings and semigroups. More precisely, we introduce the modified Ishikawa iteration processes for two asymptotically nonexpansive mappings S and T , and two asymptotically nonexpansive semigroups T = {T (t) : 0 6 t < ∞} and S = {S(t) : 0 6 t < ∞}, with C a closed convex bounded subset of a Hilbert space H : x ∈ C chosen arbitrarily, 0   n  y  n = αn x n + (1 − αn )T z n ,  z = β x + (1 − β )S n x , n n n n n 2 C = {v ∈ C : ky − vk 6 kxn − vk2 + θn },  n n      Q n = {v ∈ C : hxn − v, x0 − xn i > 0}, xn+1 = PCn ∩Q n (x0 ),

(1.5)

where θn = (1 − αn )[(tn2 − 1) + (1 − βn )tn2 (sn2 − 1)](diam C)2 → 0 as n → ∞ (here {tn } and {sn } are two sequences from T and S, respectively.), and  x0 ∈ C chosen arbitrarily,  Z   1 tn   T (u)z n du, y = α x + (1 − α )  n n n n   tn 0   Z  1 sn z n = βn xn + (1 − βn ) S(u)xn du,  sn 0    2 2  ˜  Cn = {z ∈ C : kyn − zk 6 kxn − zk + θn },     Q n = {z ∈ C : hxn − z, x0 − xn i > 0}, xn+1 = PCn ∩Q n (x0 ),

(1.6)

Rt 2 2 where θ˜n = (1 − αn )[(t˜n − 1) + (1 − βn )t˜n (˜sn2 − 1)](diam C)2 → 0 as n → ∞ (here t˜n = t1n 0n L uT du and R s s˜n = s1n 0 n L uS du). We shall prove that both iterations (1.5) and (1.6) converge strongly to a common fixed point of two asymptotically nonexpansive mappings S and T , and two asymptotically nonexpansive semigroups S and T , respectively. Moreover, we study and prove a strong convergence theorem for a common fixed point of two nonexpansive mappings. Our results extend and improve the corresponding ones announced by Nakajo and Takahashi [8] and Kim and Xu [4]. We will use the notation: 1. * for weak convergence and → for strong convergence. 2. ωw (xn ) = {x : ∃xn j * x} denotes the weak ω-limit set of {xn }. 2. Preliminaries This section collects some lemmas which will be used in the proofs for the main results in the next section. Lemma 2.1. There holds the identity in a Hilbert space H : kλx + (1 − λ)yk2 = λkxk2 + (1 − λ)kyk2 − λ(1 − λ)kx − yk2 for all x, y ∈ H and λ ∈ [0, 1] Lemma 2.2 (Opail [9] and Goebel and Kirk [1]). Let C be a closed convex subset of a real Hilbert space H and let T : C → C be a nonexpansive mapping such that Fix(T ) 6= ∅. If {xn } is a sequence in C such that xn * z and xn − T xn → 0, then z = T z.

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Lemma 2.3 (Lin et al. [5]). Let T be an asymptotically nonexpansive mapping defined on a bounded closed convex subset C of a Hilbert space H . If {xn } is a sequence in C such that xn * z and T xn − xn → 0, then z ∈ Fix(T ). Lemma 2.4 ([7, Lemma 1.3]). Let H be a real Hilbert space. Given a closed convex subset C ⊂ H and points x, y, z ∈ H . Given also a real number a ∈ R. The set D := {v ∈ C : ky − vk2 6 kx − vk2 + hz, vi + a} is convex and closed. Lemma 2.5 ([4, Lemma 3.1]). Let C be a nonempty bounded closed convex subset of H and T = {T (t) : 0 6 t < ∞} be an asymptotically nonexpansive semigroup on C. If {xn } is a sequence in C satisfying the properties (a) xn * z; and (b) lim supt→∞ lim supn→∞ kT (t)xn − xn k = 0, then z ∈ Fix(T ). Lemma 2.6 ([4, Lemma 3.2]). Let C be a nonempty bounded closed convex subset of H and T = {T (t) : 0 6 t < ∞} be an asymptotically nonexpansive semigroup on C. Then it holds that

Z t  Z t 

1 1

lim sup lim sup sup T (u)xdu − T (s) T (u)xdu

= 0. t 0 s→∞ t→∞ x∈C t 0 3. Convergence to a common fixed point of two asymptotically nonexpansive and nonexpansive mappings In this section, we prove strong convergence theorems of a common fixed point for two asymptotically nonexpansive and nonexpansive mappings, respectively. Theorem 3.1. Let C be a bounded closed convex subset of a Hilbert space H and let S, T : C → C be two asymptotically nonexpansive mappings with sequences {sn } and {tn } respectively. Assume that αn 6 a for all n and for some 0 < a < 1 and βn ∈ [b, c] for all n and 0 < b < c < 1. If F := Fix(S) ∩ Fix(T ) 6= ∅, then the sequence {xn } generated by x ∈ C chosen arbitrarily, 0    yn = αn xn + (1 − αn )T n z n ,   z = β x + (1 − β )S n x , n n n n n Cn = {v ∈ C : kyn − vk2 6 kxn − vk2 + θn },     Q n = {v ∈ C : hxn − v, x0 − xn i > 0},  xn+1 = PCn ∩Q n (x0 ),

(3.1)

where θn = (1 − αn )[(tn2 − 1) + (1 − βn )tn2 (sn2 − 1)](diam C)2 → 0 as n → ∞, converges in norm to PF (x0 ). Proof. Firstly, we note that each mapping S and T has a fixed point in C (see [2]). Suppose that F 6= ∅. We observe that Cn is convex by Lemma 2.4. Putting t∞ = sup{tn : n > 1} < ∞ and s∞ = sup{sn : n > 1} < ∞. Next observe that F ⊂ Cn for all n. Indeed, we have, for all p ∈ F, kyn − pk2 6 αn kxn − pk2 + (1 − αn )tn2 kz n − pk2 = kxn − pk2 + (1 − αn )(tn2 kz n − pk2 − kxn − pk2 ).

(3.2)

By Lemma 2.1, we have kz n − pk2 = βn kxn − pk2 + (1 − βn )kS n xn − pk2 − βn (1 − βn )kS n xn − xn k2 6 kxn − pk2 + (1 − βn )(sn2 − 1)kxn − pk2 − βn (1 − βn )kS n xn − xn k2 6 kxn − pk2 + (1 − βn )(sn2 − 1)kxn − pk2 .

(3.3)

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Substituting (3.3) in (3.2) yields, kyn − pk2 6 kxn − pk2 + (1 − αn )(tn2 (kxn − pk2 + (1 − βn )(sn2 − 1)kxn − pk2 ) − kxn − pk2 ) = kxn − pk2 + [(1 − αn )(tn2 − 1) + (1 − αn )(1 − βn )tn2 (sn2 − 1)]kxn − pk2 6 kxn − pk2 + θn . By using the same argument as in the proof of [4, Theorem 2.2, pp. 1144–1145], we have F ⊂ Cn ∩ Q n for all n > 0 and kxn+1 − xn k → 0 as n → ∞. We now claim that kT xn −xn k → 0 and kSxn −xn k → 0. We observe that kxn −T n xn k → 0 and kxn −S n xn k → 0. Indeed, by definition of yn , we have 1 kyn − xn k 1 − αn 1 6 (kyn − xn+1 k + kxn+1 − xn k). 1 − αn

kT n z n − xn k =

Since xn+1 ∈ Cn , kyn − xn+1 k2 6 kxn − xn+1 k2 + θn → 0 as n → ∞, this implies that kT n z n − xn k → 0 as n → ∞. We now show that kS n xn − xn k → 0. Let {kS n k xn k − xn k k} be any subsequence of {kS n xn − xn k}. Since {xn k } is bounded, there is a subsequence {xn j } of {xn k } such that lim j→∞ kxn j − pk = lim supk→∞ kxn k − pk := a. We note that kxn j − pk 6 kxn j − T n j z n j k + kn j kz n j − pk, ∀ j > 1. This implies that a = lim inf kxn j − pk 6 lim inf kz n j − pk. j→∞

j→∞

1

By (3.3), we note that kz n j − pk 6 kxn j − pk + ((1 − βn j )(sn2 j − 1)) 2 kxn j − pk and hence lim sup j→∞ kz n j − pk 6 lim sup j→∞ kxn j − pk = a. Therefore lim kz n j − pk = a = lim kxn j − pk. j→∞

j→∞

Furthermore by (3.3) again, we observe that βn j (1 − βn j )kS n j xn j − xn j k2 6 kxn j − pk2 − kz n j − pk2 + (1 − βn j )(sn2 j − 1)kxn j − pk2 → 0 as j → ∞. This implies that lim j→∞ kS n j xn j − xn j k = 0 and hence limn→∞ kS n xn − xn k = 0. Next, we observe that kxn − T n xn k 6 kxn − T n z n k + kn kz n − xn k. Since kz n − xn k = (1 − βn )kS n xn − xn k → 0 and kT n z n − xn k → 0, we have limn→∞ kxn − T n xn k = 0 = limn→∞ kS n xn − xn k. It follows that kT xn − xn k 6 kT xn − T n+1 xn k + kT n+1 xn − T n+1 xn+1 k + kT n+1 xn+1 − xn+1 k + kxn+1 − xn k 6 t∞ kxn − T n xn k + kT n+1 xn+1 − xn+1 k + (1 + t∞ )kxn − xn+1 k → 0.

(3.4)

Similarly, we have kSxn − xn k → 0

as n → ∞.

(3.5)

By (3.4) and (3.5), Lemma 2.3 and the boundedness of {xn } we obtain that ∅ 6= ωw (xn ) ⊂ F. By the fact that kxn − x0 k 6 k p − x0 k for all n > 0 where p := PF (x0 ) and the weak lower semi-continuity of the norm, we have kw − x0 k 6 k p − x0 k for all w ∈ ωw (xn ). However, since ωw (xn ) ⊂ F, we must have w = p for all w ∈ ωw (xn ). Thus ωw (xn ) = { p} and then xn * p. Hence, xn → p = PF (x0 ) by kxn − pk2 = kxn − x0 k2 + 2 hxn − x0 , x0 − pi + kx0 − pk2 6 2(k p − x0 k2 + hxn − x0 , x0 − pi) → 0 as n → ∞. This completes the proof.



If S = I , sn = 1 for all n ∈ N ∪ {0} and then (1.5) reduces to the modified Mann iteration for an asymptotically nonexpansive mapping and so we obtain the following result:

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Corollary 3.2 ([4, Theorem 2.2]). Let C be a bounded closed convex subset of a Hilbert space H and let T : C → C be an asymptotically nonexpansive mapping. Assume that αn 6 a for all n and for some 0 < a < 1. Then the sequence {xn } generated by (1.3) converges in norm to PFix(T ) (x0 ). Moreover, we can prove a strong convergence theorem for a common fixed point of two nonexpansive mappings on C in a Hilbert space. Theorem 3.3. Let C be a closed convex subset of a Hilbert space H and let S, T : C → C be two nonexpansive mappings such that F = Fix(S) ∩ Fix(T ) 6= ∅. Assume that αn 6 1 − δ for all n and for some δ ∈ (0, 1] and βn ∈ [b, c] for all n and 0 < b < c < 1. Then the sequence {xn } generated by x ∈ C chosen arbitrarily, 0   y = α x + (1 − α )T z ,   n n n n n  z = β x + (1 − β )Sx , n n n n n Cn = {v ∈ C : kyn − vk2 6 kxn − vk2 },       Q n = {v ∈ C : hxn − v, xn − x0 i 6 0}, xn+1 = PCn ∩Q n (x0 ),

(3.6)

converges in norm to PF (x0 ). Proof. Using the same argument as in the proof of Theorem 3.1, we obtain kT xn − xn k → 0 and kSxn − xn k → 0 as n → ∞. Hence, by Lemma 2.2, ωw (xn ) ⊂ F and therefore {xn } converges strongly to PF (x0 ).  The following corollary follows from Theorem 3.3. Corollary 3.4 ([8, Theorem 3.4]). Let C be a closed convex subset of a Hilbert space H and let T : C → C be a nonexpansive mapping such that Fix(T ) is not empty. Assume that αn 6 1 − δ for all n and for some δ ∈ (0, 1]. Then the sequence {xn } generated by (1.1) converges in norm to the fixed point PFix(T ) (x0 ). 4. Convergence to a common fixed point of two asymptotically nonexpansive and nonexpansive semigroups Suppose that T = {T (t) : 0 6 t < ∞} and S = {S(t) : 0 6 t < ∞} are two asymptotically nonexpansive semigroups defined on a nonempty closed convex bounded subset C of a Hilbert space H . Recall that we use L tT and L tS to denote the Lipschitzian constant of the mapping T (t) and S(t), respectively. In the rest of this section, we put L ∞ = sup{L tT , L tS : 0 < t < ∞} and we use Fix(T ) and Fix(S ) to denote the common fixed point set of T and S , respectively. Furthermore we use F := Fix(T ) ∩ Fix(S ) to denote the set of common fixed points of two asymptotically nonexpansive semigroups T and S . Note that the boundedness of C implies that Fix(T ) and Fix(S ) are nonempty (see [11]) and we assume throughout in this section that the set of two common fixed point F is nonempty. Theorem 4.1. Let C be a nonempty bounded closed convex subset of H and T = {T (t) : 0 6 t < ∞} and S = {S(t) : 0 6 t < ∞} be two asymptotically nonexpansive semigroups on C. Assume also that 0 < αn 6 a < 1 and 0 < b 6 βn 6 c < 1 for all n ∈ N ∪ {0} and {tn } and {sn } are two positive real divergent sequences. If F = Fix(S ) ∩ Fix(T ) 6= ∅, then the sequence {xn } generated by  x0 ∈ C chosen arbitrarily,  Z   1 tn   T (u)z n du, y = α x + (1 − α )  n n n n   tn 0   Z  1 sn z n = βn xn + (1 − βn ) S(u)xn du,  sn 0     Cn = {z ∈ C : kyn − zk2 6 kxn − zk2 + θ˜n },       Q n = {z ∈ C : hxn − z, x0 − xn i > 0}, xn+1 = PCn ∩Q n (x0 ),

(4.1)

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2 2 where θ˜n = (1 − αn )[(t˜n − 1) + (1 − βn )t˜n (˜sn2 − 1)](diam C)2 → 0 as n → ∞ (here t˜n = Rs s˜n = s1n 0 n L uS du), converges in norm to PF (x0 ).

1 tn

R tn 0

L uT du and

Proof. First observe that F ⊂ Cn for all n. Indeed, we have for all p ∈ F,

Z tn

2

1

2 2

kyn − pk 6 αn kxn − pk + (1 − αn ) T (u)z n du − p

tn 0 2  Z tn 1 6 αn kxn − pk2 + (1 − αn ) kT (u)z n − pkdu tn 0 2  Z tn 1 2 T 6 αn kxn − pk + (1 − αn ) L u du kz n − pk2 tn 0 2

= kxn − pk2 + (1 − αn )(t˜n kz n − pk2 − kxn − pk2 ).

(4.2)

By Lemma 2.1, we have

Z sn

2

2 Z

1

1 sn

kz n − pk2 = βn kxn − pk2 + (1 − βn ) − β (1 − β ) x − S(u)x du − p S(u)x du n n n n n

s sn 0 n 0

2  Z sn 2 Z

1 1 sn 2

6 βn kxn − pk + (1 − βn ) kS(u)xn − pkdu − βn (1 − βn ) xn − S(u)xn du

sn 0 sn 0

2 2  Z sn Z sn

1 1 6 βn kxn − pk2 + (1 − βn ) L uS du kxn − pk2 − βn (1 − βn ) xn − S(u)xn du

sn 0 sn 0

2 Z sn

1 6 kxn − pk2 + (1 − βn )(˜sn2 − 1)kxn − pk2 − βn (1 − βn ) S(u)xn du

xn − s

n

6 kxn − pk

2

+ (1 − βn )(˜sn2

0

− 1)kxn − pk . 2

(4.3)

Substituting (4.3) in (4.2) yields, 2

kyn − pk2 6 kxn − pk2 + (1 − αn )(t˜n (kxn − pk2 + (1 − βn )(˜sn2 − 1)kxn − pk2 ) − kxn − pk2 ) 2

2

= kxn − pk2 + [(1 − αn )(t˜n − 1) + (1 − αn )(1 − βn )t˜n (˜sn2 − 1)]kxn − pk2 6 kxn − pk2 + θn . This implies that p ∈ Cn and hence F ⊂ Cn for all n. Again, by using the same argument as in the proof of [4, Theorem 2.2, pp. 1144–1145], we have F ⊂ Cn ∩ Q n for all n, and kxn+1 − xn k → 0 as n → ∞. We now claim that lim sup lim sup kT (r )xn − xn k = 0 = lim sup lim sup kS(r )xn − xn k. r →∞

n→∞

r →∞

n→∞

Indeed, by definition of yn and xn+1 ∈ Cn we have

Z tn

1

1

T (u)z n du − xn

= 1 − α kyn − xn k

t n 0 n 1 6 (kyn − xn+1 k + kxn+1 − xn k) 1−a p 1 6 (2kxn+1 − xn k + θn ) → 0 as n → ∞. 1−a R sn We now show that lim supr →∞ lim supn→∞ kS(r )xn − xn k = 0. Let {k sn1 0 k S(u)xn k du − xn k k} be any k Rs subsequence of {k s1n 0 n S(u)xn du − xn k}. Since {xn k } is bounded, there is a subsequence {xn j } of {xn k } such that lim kxn j − pk = lim sup kxn k − pk := a.

j→∞

k→∞

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We observe that kxn j



Z tn

1 Z tn j

j 1



− pk 6 xn j − T (u)z n j du + T (u)z n j du − p

tn j 0

tn j 0

Z Z

tn j tn j 1 1

T (u)z n j du + kT (u)z n j − pkdu 6 xn j −

tn j 0

tn j 0

Z tn

j 1

T (u)z n j du + t˜n j kz n j − pk. 6 xn j −

tn j 0

This implies that a = lim inf j→∞ kxn j − pk 6 lim inf j→∞ kz n j − pk. By (4.3), we note that kz n j − pk 6 1

kxn j − pk + ((1 − βn j )(˜sn2 j − 1)) 2 kxn j − pk and hence lim sup kz n j − pk 6 lim sup kxn j − pk = a. j→∞

j→∞

Therefore lim j→∞ kz n j − pk = a = lim j→∞ kxn j − pk. Furthermore, by (4.3) again, we observe that

2

Z sn

j 1

S(u)xn j du 6 kxn j − pk2 − kz n j − pk2 + (1 − βn j )(˜sn2 j − 1)kxn j − pk2 βn j (1 − βn j ) xn j −

sn j 0 → 0, as j → ∞. This implies that lim j→∞ k sn1

Z

1 lim n→∞ s n

0

sn

R sn j j

0

S(u)xn du − xn

= 0.

S(u)xn j du − xn j k = 0 and hence (4.4)

For all 0 6 r < ∞, we note that

 Z sn  Z sn   Z



1 1 1 sn

kS(r )xn − xn k 6 S(r )xn − S(r ) S(u)xn du + S(r ) S(u)xn du − S(u)xn du

sn 0 sn 0 sn 0

Z sn

1

+ S(u)xn du − xn

s

n 0



Z sn  Z sn  Z



1 1 1 sn

S(u)xn du − xn + S(r ) S(u)xn du − S(u)xn du 6 (L ∞ + 1)

sn 0 sn 0 sn 0 := (L ∞ + 1)AnS + BnS (r ).

(4.5)

By (4.4) and Lemma 2.6, we have limn→∞ AnS = 0 = lim supr →∞ lim supn→∞ BnS (r ). Moreover, we observe that



Z tn

Z tn Z Z



1

1 tn 1 tn

xn − 1



T (u)x du 6 x − T (u)z du + T (u)z du − T (u)x du n n n n n

tn 0 tn 0 tn 0 tn 0

Z Z

1 tn 1 tn

6 x − T (u)z du + kT (u)z n − T (u)xn kdu n

n t tn 0 n 0

Z

1 tn

6 x − T (u)z du n + t˜n kz n − x n k.

n t n 0 Rs Rt Since kz n − xn k = (1 − βn )k s1n 0 n S(u)xn du − xn k → 0 and kxn − t1n 0n T (u)z n duk → 0, we obtain

Z

1 tn

lim xn − T (u)xn du

= 0. n→∞ tn 0 We can deduce that for all 0 6 r < ∞,

(4.6)

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S. Plubtieng, K. Ungchittrakool / Nonlinear Analysis 67 (2007) 2306–2315

  Z tn

1

kT (r )xn − xn k 6 T (r )xn − T (r ) T (u)xn du

tn 0

Z tn

  Z tn Z tn

1

1 1

+ T (r ) T (u)xn du − T (u)xn du + T (u)xn du − xn

tn tn 0 tn 0

Z0 tn

1 T (u)xn du − xn 6 (L ∞ + 1)

t n 0

  Z tn Z

1 1 tn

+ T (r ) T (u)xn du − T (u)xn du

tn 0 tn 0 := (L ∞ + 1)AnT + BnT (r ).

(4.7)

By (4.6) and Lemma 2.6, we have limn→∞ AnT = 0 = lim supr →∞ lim supn→∞ BnT (r ). From (4.5) and (4.7), we obtain lim sup lim sup kT (r )xn − xn k = 0 = lim sup lim sup kS(r )xn − xn k. r →∞

n→∞

r →∞

n→∞

We note by Lemma 2.5 that every weak limit point of {xn } is a member of F. Repeating the last part of the proof of [4, Theorem 2.2], we can prove that ωw (xn ) = {PF (x0 )}. Hence {xn } weakly converges to PF (x0 ), and therefore the convergence is strong.  If the semigroup R s S = {S(t) : 0 6 t < ∞} = I := {I (t) : 0 6 t < ∞}, then S(t)xn = xn for all n and for all t > 0. Hence s1n 0 n S(u)xn du = xn for all n, z n = xn , and therefore Theorem 4.1 reduces to the following corollary. Corollary 4.2 ([4, Theorem 3.3]). Let C be a nonempty bounded closed convex subset of H and T = {T (t) : 0 6 t < ∞} be an asymptotically nonexpansive semigroup on C. Assume also that 0 < αn 6 a < 1 for all n ∈ N ∪ {0} and {tn } is a positive real divergent sequence. Then, the sequence {xn } generated by (1.4) converges in norm to PFix(T ) (x0 ). Finally, we prove a strong convergence theorem for a common fixed point of two nonexpansive semigroups on C in a Hilbert space H . Theorem 4.3. Let C be a nonempty closed convex subset of H and T = {T (t) : 0 6 t < ∞} and S = {S(t) : 0 6 t < ∞} be two nonexpansive semigroups on C. Assume also that 0 < αn 6 a < 1 and 0 < b 6 βn 6 c < 1 for all n ∈ N ∪ {0} and {tn } and {sn } are two positive real divergent sequences. If F = Fix(S ) ∩ Fix(T ) 6= ∅, then the sequence {xn } generated by  x0 ∈ C chosen arbitrarily,  Z   1 tn    T (u)z n du, y = α x + (1 − α ) n n n n   tn 0   Z  1 sn (4.8) S(u)xn du, z n = βn xn + (1 − βn )  sn 0     Cn = {z ∈ C : kyn − zk 6 kxn − zk},       Q n = {z ∈ C : hxn − z, x0 − xn i > 0}, xn+1 = PCn ∩Q n (x0 ), converges in norm to PF (x0 ). Proof. Put q := PF (x0 ) and let Ω = {ω ∈ C : kω − qk 6 2kq − x0 k}. Then, Ω is a bounded closed convex subset of C which is T (r )-invariant and S(r )-invariant for each 0 6 r < ∞ and contains {xn }. By [8, Lemma 2.1] and using the same argument as in the proof of Theorem 4.1, we obtain lim kT (r )xn − xn k = 0 = lim kS(r )xn − xn k

n→∞

n→∞

for all 0 6 r < ∞.

Hence, by Lemma 2.5, ωw (xn ) ⊂ F and therefore {xn } converges strongly to PF (x0 ). The following corollary follows from Theorem 4.3.



S. Plubtieng, K. Ungchittrakool / Nonlinear Analysis 67 (2007) 2306–2315

2315

Corollary 4.4 ([8, Theorem 4.1]). Let C be a nonempty closed convex subset of H and T = {T (t) : 0 6 t < ∞} be a nonexpansive semigroup on C. Assume also that 0 < αn 6 a < 1 for all n ∈ N ∪ {0} and {tn } is a positive real divergent sequence. If Fix(T ) 6= ∅, then the sequence {xn } generated by (1.2) converges in norm to PFix(T ) (x0 ). Acknowledgements The authors would like to thank The Thailand Research Fund for financial support. Moreover, K. Ungchittrakool is also supported by the Royal Golden Jubilee Program under Grant PHD/0086/2547, Thailand. References [1] K. Goebel, W.A. Kirk, Topic in Metric Fixed Point Theory, in: Cambridge Studies in Advanced Mathematics, vol. 28, Cambrige University Press, Cambrige, 1990. [2] K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972) 171–174. [3] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974) 147–150. [4] T.W. Kim, H.K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140–1152. [5] P.K. Lin, K.K. Tan, H.K. Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal. 24 (1995) 929–946. [6] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506–510. [7] C. Martinez-Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400–2411. [8] K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379. [9] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967) 591–597. [10] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991) 153–159. [11] K.K. Tan, H.K. Xu, Fixed point theorems for Lipschitzian semigroups in Banach spaces, Nonlinear Anal. 20 (1993) 395–404. [12] H.K. Xu, Strong asymptotic behavior of almost-robits of nonlinear semigroups, Nonlinear Anal. 46 (2001) 135–151.