DEMI-CLOSED PRINCIPLE AND WEAK CONVERGENCE ... - CiteSeerX

J. Korean Math. Soc. 38 (2001), No. 6, pp. 1245–1260

DEMI-CLOSED PRINCIPLE AND WEAK CONVERGENCE PROBLEMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou Abstract. A demi-closed theorem and some new weak convergence theorems of iterative sequences for asymptotically nonexpansive and nonexpansive mappings in Banach spaces are obtained. The results presented in this paper improve and extend the corresponding results of [1], [8]-[10], [12], [13], [15], [16], and [18].

1. Introduction and preliminaries Throughout this paper, we assume that E is a real Banach space and E ∗ is the dual space of E. Let D be a nonempty subset of E and F (T ) denote the set of fixed points of T . Definition 1.1. Let T : D → D be a mapping. (1) T is said to be asymptotically nonexpansive ([7]) if there exists a sequence {kn } ⊂ [1, ∞) with limn→∞ kn = 1 such that (1.1)

||T n x − T n y|| ≤ kn ||x − y||

for all x, y ∈ D and n ≥ 1. (2) T is said to be uniformly L-Lipschitzian if (1.2)

||T n x − T n y|| ≤ L||x − y||

Received September 17, 2001. 2000 Mathematics Subject Classification: 47H05, 47H10, 47H15. Key words and phrases: demi-closed principle, asymptotically nonexpansive mapping, nonexpansive mapping, fixed point, Ishikawa iterative sequence, Mann iterative sequence. The first and second authors were supported financially by the National Natural Science Foundation of China and the Korea Research Foundation (KRF-2000DP0013), respectively.

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for all x, y ∈ D and n ≥ 1, where L is a positive constant. Remark 1.1. (1) It is easy to see that, if T : D → D is a nonexpansive mapping, then T is an asymptotically nonexpansive mapping with a constant sequence {1}. (2) If T : D → D is an asymptotically nonexpansive mapping with a sequence {kn } ⊂ [1, ∞) such that kn → 1, then it must be uniformly L-Lipschitzian with L = supn≥1 {kn }. Definition 1.2. Let E be a real Banach space and D be a closed subset of E. A mapping T : D → D is said to be demi-closed at the origin if, for any sequence {xn } in D, the conditions xn → x0 weakly and T xn → 0 strongly imply T x0 = 0. Recall that E is said to satisfy Opial’s condition if, for each sequence {xn } in E, the condition that the sequence xn → x weakly implies that (1.3)

lim sup kxn − xk < lim sup kxn − yk n→∞

n→∞

for all y ∈ E with y 6= x, and also that E is said to have the Fr´echet differentiable norm if, for each x ∈ S(E), the limit kx + tyk − kxk t→0 t exists and is attained uniformly in y ∈ S(E), where S(E) denotes the unit sphere of E. The weak and strong convergence problems to the fixed points for nonexpansive and asymptotically nonexpansive mappings have been studied by many authors (for example, see [1], [3]-[18] and the references therein). In 1972, Goebel and Kirk [7] proved that, if D is a bounded closed convex subset of a uniformly convex Banach space E, then every asymptotically nonexpansive mapping T : D → D has a fixed point in D. Subsequently, Bose [1] first proved that, if D is a bounded closed convex subset of a uniformly convex Banach space E satisfying Opial’s condition and T : D → D is an asymptotically nonexpansive mapping, then the sequence {T n x} converges weakly to a fixed point of T provided T is asymptotically regular at x ∈ D, i.e., lim

lim ||T n x − T n+1 x|| = 0.

n→∞

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Furthermore, Passty [9] and Xu [18] proved that, if Opial’s condition of E is replaced by the condition that E has the Fr´echet differentiable norm, then this conclusion still also holds. In addition, Tan and Xu [15], [16] proved that, in both cases, the asymptotic regularity of T at x can be weakened to the weakly asymptotic regularity of T at x, i.e., w − lim (T n x − T n+1 x) = 0. n→∞

Recently, Tan and Xu [13] proved the following theorem: Theorem A. Let E be a real uniformly convex Banach space which satisfies Opial’s condition or has the Fr´echet differentiable norm, D be a nonempty bounded closed convex subset of E and T : D → D be an asymptotically nonexpansive mapping with a sequence {kn } ⊂ [1, ∞) P∞ and n=0 (kn − 1) < ∞. Let {αn } and {βn } be two sequences in [0, 1] satisfying the following conditions: (i) 0 < a1 ≤ αn ≤ a2 < 1, n ≥ 0, (ii) 0 ≤ βn ≤ b, n ≥ 0, where a1 , a2 , b ∈ (0, 1) are some constants. For any given x0 ∈ D, let {xn } ⊂ D be a sequence defined by ( xn+1 = (1 − αn )xn + αn T n yn , n ≥ 0, (1.4) yn = (1 − βn )xn + βn T n xn , n ≥ 0. Then the sequence {xn } converges weakly to a fixed point of T in D. The purpose of this paper is to prove a new demi-closed principle and then, by using this principle, to prove some convergence theorems for asymptotically nonexpansive mappings and nonexpansive mappings in Banach spaces without assuming any one of the following conditions: (1) E satisfies Opial’s condition or E has the Fr´echet differentiable norm, (2) T is asymptotically regular or weakly asymptotically regular, (3) D is a bounded subset of E. Our main results presented in this paper improve and generalize the corresponding results of [1], [8]-[10], [12], [13], [15], [16] and [18]. The following four theorems are the main results of this paper:

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Theorem 1 (Demi-closed Principle). Let E be a uniformly convex Banach space, D be a nonempty closed convex subset of E and T : D → D be an asymptotically nonexpansive mapping with a sequence {kn } ⊂ [1, ∞) and kn → 1. Then I − T is semi-closed at zero, i.e., for each sequence {xn } in D, if the sequence {xn } converges weakly to q ∈ D and {(I − T )xn } converges strongly to 0, then (I − T )q = 0. Theorem 2. Let E be a real uniformly convex Banach space, D be a nonempty closed convex subset of E, T : D → D be an asymptotically nonexpansive mapping with a sequence {kn } ⊂ [1, ∞), kn → 1. Suppose that the set F (T ) of fixed points of T in D is nonempty and {αn }, {βn } are two sequences in [0, 1] satisfying the following conditions: (i) There exist positive integers n0 , n1 and ² > 0, 0 < b < min{1, L1 }, where L = supn≥0 kn , such that 0 < ² ≤ αn ≤ 1 − ², n ≥ n0 , 0 ≤ βn ≤ b, n ≥ n1 ,

(1.5)

P∞ 2 (ii) n=0 (kn − 1) < ∞. Then the Ishikawa iterative sequence {xn } defined by (1.4) converges weakly to some fixed point x∗ of T in D. Theorem 3. Let E be a real uniformly convex Banach space, D be a nonempty closed convex subset of E and T : D → D be an asymptotically nonexpansive mapping with a sequence {kn } ⊂ [1, ∞) and kn → 1. Suppose that the set F (T ) of fixed points of T in D is nonempty and {αn } is a sequence in [0, 1] satisfying the following conditions: (i) There exist positive integers n0 and ² > 0 such that 0 < ² ≤ αn ≤ 1 − ²,

n ≥ n0 ,

P∞ 2 (ii) n=0 (kn − 1) < ∞. Then the Mann iterative sequence {xn } defined by ½ (1.6)

x0 ∈ D, xn+1 = (1 − αn )xn + αn T n xn ,

n ≥ 0,

converges weakly to some fixed point x∗ of T in D.

Asymptotically nonexpansive mapping

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Theorem 4. Let E be a real uniformly convex Banach space, D be a nonempty closed convex subset of E and T : D → D be an nonexpansive mapping. Suppose that the set F (T ) of fixed points of T in D is nonempty and {αn } and {βn } are two sequences in [0, 1] satisfying the following conditions: There exist positive integers n0 , n1 and ² > 0, 0 < b < 1 such that (1.7)

0 < ² ≤ αn ≤ 1 − ², n ≥ n0 , 0 ≤ βn ≤ b, n ≥ n1 .

Then the Ishikawa iterative sequence {xn } defined by (1.4) converges weakly to some fixed point x∗ of T in D. The following lemmas play an important role in proving our main results: Lemma 1 ([17]). Let p > 1 and r > 0 be two fixed real numbers. Then a Banach space E is uniformly convex if and only if there exists a continuous strictly increasing convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that ||λx + (1 − λ)y||p ≤ λ||x||p + (1 − λ)||y||p − ωp (λ)g(||x − y||) for all x, y ∈ B(0, r) and 0 ≤ λ ≤ 1, where B(0, r) is the closed ball of E with center zero and radius r and (1.8)

ωp (λ) = λp (1 − λ) + λ(1 − λ)p .

Lemma 2 ([11]). Let E be a real Banach space, D be a nonempty closed convex subset of E and T : D → D be an asymptotically nonexpansive mapping with a sequence {kn } ⊂ [1, ∞) and limn→∞ kn = 1. Let {xn } be the Ishikawa iterative sequence defined by (1.4). Then the condition ||xn − T n xn || → 0 as n → ∞ implies that ||xn − T xn || → 0 as n → ∞. Lemma 3. P Let {an } and {bn } be two sequences of nonnegative real ∞ numbers with n=1 bn < ∞. If one of the following conditions is satisfied: (i) an+1 ≤ an + bn , n ≥ 1, (ii) an+1 ≤ (1 + bn )an , n ≥ 1,

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then the limit limn→∞ an exists. Proof. If the condition (i) is satisfied, then the conclusion of Lemma 3 can be obtained from Tan-Xu [14, Lemma 1]. Next, if the condition (ii) is satisfied, then we have an+1 ≤ (1 + bn )an ≤ (1 + bn )(1 + bn−1 )an−1 ≤ ··· ≤

n Y

(1 + bj )a1 ≤

j=1

∞ Y

(1 + bj )a1

j=1

= M < ∞, which implies that {an } is a bounded sequence and so we have an+1 ≤ (1 + bn )an ≤ an + M bn ,

n ≥ 1.

Therefore, the conclusion (ii) can be obtained from the conclusion (i). This completes the proof. ¤ Lemma 4. Let E be a normed space, D be a nonempty closed convex subset of E and T : D → D be an asymptotically P∞ nonexpansive mapping with a sequence {kn } ⊂ [1, ∞), kn → 1 and n=1 (kn2 − 1) < ∞. If F (T ) is nonempty in D and {αn }, {βn } are two sequences in [0, 1]. Then the limit limn→∞ ||xn − q|| exists for all q ∈ F (T ), where {xn } is the Ishikawa iterative sequence defined by (1.4). Proof. Taking q ∈ F (T ), by (1.4), we have ||xn+1 − q|| = ||(1 − αn )(xn − q) + αn (T n yn − q)|| ≤ (1 − αn )||xn − q|| + αn kn ||yn − q|| = (1 − αn )||xn − q|| + αn kn ||(1 − βn )(xn − q) + βn (T n xn − q)|| ≤ (1 − αn )||xn − q|| + αn kn {(1 − βn )||xn − q|| + βn kn ||xn − q||} = {1 + αn (kn − 1)(kn βn + 1)}||xn − q|| ≤ {1 + (kn2 − 1)}||xn − q||,

n ≥ 1.

Taking an = ||xn − q|| and bn = kn2 − 1 in Lemma 3, the conclusion of Lemma 4 can be obtained from Lemma 3 immediately. This completes the proof. ¤

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Lemma 5 ([2]). Let E be a uniformly convex Banach space, C be a nonempty bounded closed convex subset of E. Then there exists a strictly increasing continuous convex function f : [0, ∞) → [0, ∞) with f (0) = 0 such that, for any Lipschitzian mapping T : C → E with n the Lipschitz constant L ≥ 1, any finite many elements Pn{xi }i=1 in C n and any finite many nonnegative numbers {ti }i=1 with i=1 ti = 1, the following inequality holds: n n ° ³X ° ´ X ° ° ti xi − ti T xi ° °T

≤ Lf

i=1 −1

i=1

{ max (||xi − xj || − L−1 ||T xi − T xj ||)}. 1≤i,j≤n

2. The proofs of Theorems 1∼4 Now we are ready to prove our main results in this paper. Proof of Theorem 1. Since {xn } converges weakly to q ∈ D, {xn } is a bounded sequence in D. Therefore, there exists r > 0 such that {xn } ⊂ C =: D ∩ B(0, r), where B(0, r) is a closed ball of E with center 0 and radius r and so C is a nonempty bounded closed convex subset in D. Next, we prove that, as n → ∞, T n q → q.

(2.1)

In fact, since {xn } converges weakly to q, by Mazur’s theorem, for each positive integer n ≥ 1, there exists a convex combination yn = Pm(n) (n) Pm(n) (n) (n) ≥ 0 and i=1 ti = 1 such that i=1 ti xi+n with ti (2.2)

||yn − q||
0 and positive integer j ≥ 1, there exists a positive integer N = N (², j) such that 1 N < ² and ||(I − T )xn || ≤

1 < ², (1 + k1 + · · · + kj−1 )

n ≥ N.

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Hence, for any n ≥ N, we have ||(I − T j )xn || ≤ ||(I − T )xn || + ||(T − T 2 )xn || + · · · + ||(T j−1 − T j )xn ||

(2.3)

≤ (1 + k1 + · · · + kj−1 )||(I − T )xn || < ².

Since T : D → D is asymptotically nonexpansive, T : C → D is also an asymptotically nonexpansive mapping. Therefore, T j : C → D is a Lipschitzian mapping with the Lipschitz constant kj ≥ 1. Next, we consider the following inequality: m(n) ° X (n) ° ||T j yn − yn || = °T j yn − ti T j xi+n i=1 m(n)

+

X

m(n) (n) ti T j xi+n

−

i=1

(2.4)

X

° ° (n) ti xi+n °

i=1 m(n)

≤ ||T j yn −

X

(n)

ti T j xi+n ||

i=1 m(n)

+

X

(n)

ti ||T j xi+n − xi+n ||.

i=1

By (2.3), we know that m(n)

X

(2.5)

(n)

ti ||T j xi+n − xi+n || < ²,

n ≥ N.

i=1

Moreover, by Lemma 5 and (2.3), we have m(n) ° ° X (n) ° j ° ti T j xi+n ° °T yn − i=1

(2.6)

≤ kj f

−1

≤ kj f

−1

{ max (||xi+n − xk+n || − kj−1 ||T j xi+n − T j xk+n ||)} 1≤i,k≤n

{ max [||xi+n − T j xi+n || + ||T j xi+n − T j xk+n || 1≤i,k≤n

+ ||T j xk+n − xk+n ||] − kj−1 ||T j xi+n − T j xk+n ||} ≤ kj f −1 { max [2² + (1 − kj−1 )kj ||xi+n − xk+n ||]} 1≤i,k≤n

≤ kj f

−1

(2² + 2r(1 − kj−1 )kj ),

n ≥ N,

Asymptotically nonexpansive mapping

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since xi+n and xk+n both are in C. Substituting (2.5) and (2.6) into (2.4), we have ||T j yn − yn || ≤ kj f −1 [2² + 2rkj (1 − kj−1 )] + ². Taking the superior limit as n → ∞ in the above inequality and noting the arbitrariness of ² > 0, we have (2.7)

lim sup ||T j yn − yn || ≤ kj f −1 [2rkj (1 − kj−1 )]. n→∞

On the other hand, for any given j ≥ 1, it follows from (2.2) that ||T j q − q|| ≤ ||T j q − T j yn || + ||T j yn − yn || + ||yn − q|| ≤ (kj + 1)||yn − q|| + ||T j yn − yn || 1 ≤ (kj + 1) + ||T j yn − yn ||. n Taking the superior limit as n → ∞ in the above inequality, it follows from (2.7) that ||T j q − q|| ≤ kj f −1 (2rkj (1 − kj−1 )). Taking the superior limit as j → ∞ in the above inequality, we have lim sup ||T j q − q|| ≤ f −1 (0) = 0, which implies that ||T j q − q|| → 0 as j → ∞. Therefore, (2.1) is proved. By the continuity of T, we have limj→∞ T T j q = T q = q. This completes the proof. ¤ Proof of Theorem 2. By the assumption, F (T ) is nonempty. Take q ∈ F (T ). It follows from Lemma 4 that the limit limn→∞ ||xn − q|| exists. Therefore, {xn − q} is a bounded sequence in E. Let L = supn≥1 kn . Then we have ||T n xn − q|| ≤ kn ||xn − q|| ≤ L||xn − q||, ||yn − q|| = ||(1 − βn )(xn − q) + βn (T n xn − q)|| ≤ L||xn − q||, ||T n yn − q|| ≤ kn ||yn − q|| ≤ L2 ||xn − q||,

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Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou

which imply that {xn }, {yn }, {T n xn } and {T n yn } all are bounded sequences in D. Therefore, there exists r > 0 such that {xn } ∪ {yn } ∪ {T n yn } ∪ {T n xn } ⊂ B(q, r) ∩ D =: C, where B(q, r) is a closed ball of E with center q and radius r and so C is a nonempty bounded closed convex subset of D. By Lemma 1 with p = 2 and λ = αn and (1.4), we have ||xn+1 − q||2 = ||(1 − αn )(xn − q) + αn (T n yn − q)||2 ≤ (1 − αn )||xn − q||2 + αn ||T n yn − q||2

(3.1)

− ω2 (αn )g(||xn − T n yn ||). It follows from (1.8) that ω2 (αn ) = αn2 (1 − αn ) + αn (1 − αn )2 = αn (1 − αn ). Substituting the above expressions into (3.1) and simplifying, we have ||xn+1 − q||2 ≤ (1 − αn )||xn − q||2 + αn ||T n yn − q||2 (3.2)

− αn (1 − αn )g(||xn − T n yn ||) © ª = ||xn − q||2 + αn ||T n yn − q||2 − ||yn − q||2 ¯ © + αn ||yn − q||2 − ||xn − q||2 ¯|} − αn (1 − αn )g(||xn − T n yn ||).

First, we consider the third term on the right side of (3.2). By Lemma 1 with p = 2, we have ||yn − q||2 − ||xn − q||2 = ||(1 − βn )(xn − q) + βn (T n xn − q)||2 − ||xn − q||2 (3.3)

≤ (1 − βn )||xn − q||2 + βn ||T n xn − q||2 − ω2 (βn )g(||xn − T n xn ||) − ||xn − q||2 ≤ (1 − βn )||xn − q||2 + βn ||T n xn − q||2 − ||xn − q||2 ,

which implies that (3.4)

© ª ||yn − q||2 − ||xn − q||2 ≤ βn ||T n xn − q||2 − ||xn − q||2 ≤ βn (kn2 − 1)||xn − q||2 .

Asymptotically nonexpansive mapping

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Substituting (3.4) into (3.2) and simplifying, we have ||xn+1 − q||2

© ª ≤ ||xn − q||2 + αn (kn2 − 1)||yn − q||2 + αn βn (kn2 − 1)||xn − q||2 ) − αn (1 − αn )g(||xn − T n yn ||) ≤ ||xn − q||2 + αn (kn2 − 1){||yn − q||2 + ||xn − q||2 } − αn (1 − αn )g (||xn − T n yn ||) . Since {xn } and {yn } both belong to C, we have ||xn − q|| ≤ r and ||yn − q|| ≤ r. Besides, by the condition (1.5), 0 < ² ≤ αn and ² ≤ 1 − αn for all n ≥ n0 . Hence we have (3.5)

||xn+1 − q||2 ≤||xn − q||2 + 2αn (kn2 − 1)r2 − ²2 g (||xn − T n yn ||) ,

n ≥ n0 .

Therefore, we have ²2 g (||xn − T n yn ||) ≤ ||xn − q||2 − ||xn+1 − q||2 + 2(kn2 − 1)r2 ,

n ≥ n0 .

For any m > n0 , we have ²2

m X

g (||xn − T n yn ||)

n=n0

≤ ||xn0 − q||2 − ||xm+1 − q||2 + 2r2

m X

(kn2 − 1)

n=n0

≤ ||xn0 − q||2 + 2r2

m X

(kn2 − 1).

n=n0

Letting m → ∞, by the condition (ii), we have (3.6) ²2

∞ X

g (||xn − T n yn ||) ≤ ||xn0 − q||2 + 2r2

n=n0

n=n0

which implies that, as n → ∞, (3.7)

∞ X

g (||xn − T n yn ||) → 0.

(kn2 − 1) < ∞,

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Since g : [0, ∞) → [0, ∞) is continuous and strictly increasing with g(0) = 0, it follows from (3.7) that, as n → ∞, ||xn − T n yn || → g −1 (0) = 0.

(3.8) From (1.4), we have

||xn − yn || = ||βn (xn − T n xn )|| ≤ βn {||xn − T n yn || + ||T n yn − T n xn ||}

(3.9)

≤ βn {||xn − T n yn || + L||yn − xn ||}, i.e., it follows that (1 − Lβn )||xn − yn || ≤ βn ||xn − T n yn || ≤ ||xn − T n yn ||. By the condition (ii), we have 1 − L · βn > 0 for all n ≥ n1 . Therefore, from (3.8), we have (3.10)

lim ||xn − yn || = 0

n→∞

and so it follows from (3.10) and (3.8) that, as n → ∞, (3.11)

||T n xn − xn || ≤ ||T n xn − T n yn || + ||T n yn − xn || ≤ L||xn − yn || + ||T n yn − xn || → 0.

Therefore, by Lemma 2, as n → ∞, (3.12)

||T xn − xn || → 0.

Next, we prove that ωw (xn ) is a nonempty set and (3.13)

ωw (xn ) ⊂ F (T ),

where ωw (xn ) is the weak ω−limit set of {xn } defined by ωw (xn ) = {y ∈ E : y = w − lim xnk for some nk → ∞}. k→∞

Indeed, since E is uniformly convex and C is a nonempty bounded closed convex subset of D and so C is a weakly compact and weakly closed subset of D. This implies that there exists a subsequence {xni } of {xn } such

Asymptotically nonexpansive mapping

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that {xni } converges weakly to a point p ∈ ωw (xn ), which shows that ωw (xn ) is nonempty. For any p ∈ ωw (xn ), there exists a subsequence {xnk } of {xn } such that xnk → p weakly. Again, by (3.12), we have limk→∞ ||xnk − T xnk || = 0. Therefore, it follows from Theorem 1 that p − T p = 0, i.e., p ∈ F (T ). The conclusion (3.13) is proved. Taking any p ∈ ωw (xn ) ⊂ F (T ), there exists a subsequence {xni } of {xn } such that, as ni → ∞, (3.14)

xni → p weakly.

Hence, from (3.11) and (3.14), it follows that, as ni → ∞, T ni xni = (T ni xni − xni ) + xni → p weakly. Again, from (1.4), (3.11) and (3.14), it follows that, as ni → ∞, yni = xni − βni (xni − T ni xni ) → p weakly. On the other hand, from (3.8) and (3.14), we have, as ni → ∞, (3.15)

T ni yni = (T ni yni − xni ) + xni → p weakly.

By the same way, from (1.4), (3.8) and (3.14), as ni → ∞, (3.16)

xni +1 = xni − αni (xni − T ni yni ) → p weakly.

Therefore, it follows from (3.11) and (3.16) that, as ni → ∞, (3.17)

T ni +1 xni +1 = (T ni +1 xni +1 − xni +1 ) + xni +1 → p weakly.

By (1.4), (3.11) and (3.16), we have, as ni → ∞, ¡ ¢ (3.18) yni +1 = xni +1 − βni +1 T ni +1 xni +1 − xni +1 → p weakly. Therefore, from (3.8) and (3.16), it follows that, as ni → ∞, (3.19)

T ni +1 yni +1 = (T ni +1 yni +1 − xni +1 ) + xni +1 → p weakly.

Continuing in this way, by induction, we can prove that, for any m ≥ 0, xni +m → p weakly, yni +m → p weakly (ni → ∞), T

ni +m

xni +m → p weakly, T ni +m yni +m → p weakly (ni → ∞).

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Next, we prove that xn → p weakly as n → ∞. In fact, it is easy to see that (3.20)

{xn }∞ n=n1 = lim

k→∞

k [

{xni +m }∞ i=1 .

m=0

Since the sequences {xni } and {xni +1 } both converge weakly to p as S1 ni → ∞, the sequence m=0 {xni +m }∞ i=1 converges weakly to p. By induction, we can prove that, for any positive integer k, k [

{xni +m }∞ i=1 → p weakly.

m=0

Letting k → ∞, it follows from (3.20) that the sequence {xn } converges weakly to p. Similarly, we can also prove that yn → p weakly. This completes the proof. ¤ Proof of Theorem 3. In Theorem 2, taking βn = 0 for all n ≥ 0, then the conclusion of Theorem 3 can be obtained from Theorem 2 immediately. ¤ Proof of Theorem 4. It follows from Remark 1.1 that, if T : D → D is a nonexpansive mapping, then T is asymptotically nonexpansive with a constant sequence {1}. Therefore, the conclusion of Theorem 4 follows from Theorem 2 immediately. ¤ Remark 2.1. Theorems 2-4 improve and extend the corresponding results of Bose [1], Gornicki [8], Passty [9], Reich [10], Schu [12], Tan and Xu [13], [15], [16], [18]. References [1] [2] [3]

S. C. Bose, Weak convergence to the fixed point of an asymptotically nonexpansive map, Proc. Amer. Math. Soc. 68 (1978), 305–308. R. E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Isreal J. Math. 32 (1979), 107–116. S. S. Chang, On Chidume’s open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces, J. Math. Anal. Appl. 216 (1997), 94–111.

Asymptotically nonexpansive mapping [4] [5] [6]

[7] [8]

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Shih-sen Chang Department of Mathematics Sichuan University Chengdu, Sichuan 610064 People’s Republic of China Yeol Je Cho Department of Mathematics Education College of Education Gyeongsang National University Chinju 660-701, Korea E-mail : [email protected]

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Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou

Haiyun Zhou Department of Mathematics Shijiazhuang Mechanical Engineering College Shijiazhuang, Hebei 050003, P. R. China