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Convergence theorem for I-asymptotically quasi-nonexpansive mapping in Hilbert space
J. Math. Anal. Appl. 329 (2007) 759–765 www.elsevier.com/locate/jmaa
Convergence theorem for I -asymptotically quasi-nonexpansive mapping in Hilbert space Seyit Temir ∗ , Ozlem Gul Department of Mathematics, Arts and Science Faculty, Harran University, 63200 Sanliurfa, Turkey Received 16 September 2005 Available online 4 August 2006 Submitted by L. Debnath
Abstract Let H be a Hilbert space with inner product (·,·) and · norm, and let K be weakly compact a subset of H. Let T : K → K be nonlinear mapping and I : K → K be a nonlinear bounded mapping. In this paper, we define the I -asymptotically quasi-nonexpansive mapping in Hilbert space. If T is an I -asymptotically i quasi-nonexpansive mapping, then we prove that n1 n−1 i=0 T u, for u ∈ K as n → ∞, is weakly almost convergent to its asymptotic center. © 2006 Elsevier Inc. All rights reserved. Keywords: Asymptotic center; Asymptotically quasi-nonexpansive mapping; Nonlinear ergodic theorems
1. Introduction Throughout this paper, we assume that H is a real Hilbert space with inner product (·,·) and · norm. The basic concepts and properties in the theory of Hilbert spaces are given in [1]. Let K be a nonempty subset of H real Hilbert space and T : K → K be a nonlinear mapping. Let F (T ) = {u ∈ K: T u = u} be denoted as the set of fixed points of a mapping T . A mapping T : K → K is called nonexpansive provided k T u − T k v u − v for all u, v ∈ K and k 1. T is called asymptotically nonexpansive mapping if there exists a sequence {λk } ⊂ [0, ∞) with limk→∞ λk = 0 such that * Corresponding author.
E-mail addresses: [email protected] (S. Temir), [email protected] (O. Gul). 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.07.004
760
S. Temir, O. Gul / J. Math. Anal. Appl. 329 (2007) 759–765
k T u − T k v (λk + 1)u − v for all u, v ∈ K and k 1. The asymptotically nonexpansive mapping was defined as original by Goebel and Kirk [5]. In [5], they proved that every asymptotically nonexpansive mapping T : K → K has a fixed point when E is uniformly convex Banach space. Let T , I : K → K. Then T is called I -nonexpansive on K if T u − T v I u − I v for all u, v ∈ K. T is called I -asymptotically nonexpansive on K if there exists a sequence {λk } ⊂ [0, ∞) with limk→∞ λk = 0 such that k T u − T k v λ + 1 I k u − I k v k for all u, v ∈ K and k = 1, 2, . . . . The concept of quasi-nonexpansive mapping was initiated by Tricomi in 1916 for real functions. The quasi-nonexpansive mapping in Banach spaces was studied by Diaz and Metcalf [2] and Dotson [3]. Recently, this concept was given by Kirk [6] in metric spaces which we adapt to Hilbert space as following: T is called quasi-nonexpansive mapping provided k T u − f u − f for all u ∈ K and f ∈ F (T ) and k 1. T is called asymptotically quasi-nonexpansive mapping if there exists a sequence {λk } ⊂ [0, ∞) with limk→∞ λk = 0 such that k T u − f (λk + 1)u − f for all u ∈ K and f ∈ F (T ) and k 1. Remark 1.1. From above definitions, it is easy to see that if F (T ) is nonempty, a nonexpansive mapping must be quasi-nonexpansive, and an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive. But the converse does not hold. There have been a number of recent results on fixed points on nonexpansive and quasinonexpansive mappings in Banach spaces and metric spaces. Recently, concerning the problem of iterative sequences for asymptotically nonexpansive mappings or asymptotically quasinonexpansive mappings converging to some fixed points have been considered by many authors. For example of these papers, the strong and weak convergence of the sequence of certain iterates to a fixed point of quasi-nonexpansive maps were studied by Petryshyn and Williamson [9]. In particular, the weak and strong convergence of iterates sequence for various special cases of quasi-nonexpansive and one-set and/or one-ball contractive mappings were discussed by Petryshyn and Williamson [9]. Their analysis was connected with the convergence of Mann iterates studied by Dotson [3]. Subsequently, the convergence of Ishikawa iterates of quasinonexpansive mappings in Banach spaces was discussed by Ghosh and Debnath [4]. Their paper was concerned with the convergence of Ishikawa iterates of quasi-nonexpansive mappings in a Banach space. The above results and obtained some necessary and sufficient conditions for Ishikawa iterative sequence to converge to a fixed point for asymptotically quasi-nonexpansive mappings was extended by Liu [7]. The aim of our study is to prove weakly convergence theorem for I -asymptotically quasinonexpansive mapping in Hilbert space. We will benefit from the following condition (1) having been given in [8] with a different approximation from above results improved by various authors. In order to prove weak nonlinear ergodic theorem for I -asymptotically quasi-nonexpansive
S. Temir, O. Gul / J. Math. Anal. Appl. 329 (2007) 759–765
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mapping, we shall give definition of I -asymptotically quasi-nonexpansive mapping and use the condition (1) as a tool to prove our main result. Furthermore, we also give stronger condition than R-weakly commuting stated in Shahzad’s study [10] to prove our main result. If T is I asymptotically quasi-nonexpansive mapping and I is bounded, then using condition (1), we prove that {T n u} almost convergent to its asymptotic center. For every u ∈ K, the is weakly i u coincides with the asymptotic center of {T n u}. T weak limit of n1 n−1 i=0 We consider our main result in two case. In condition (1), we consider the case c = 0 and F (T ) ∩ F (I ) = ∅ and the case c > 0 and F (T ) ∩ F (I ) = ∅. Therefore, under these cases, it is sufficient to prove that {T n u} is weakly almost convergent to its asymptotic center. First, we give the following some definitions and statements to use in our weak convergence theorem and then, we present our main result. Let {xn } be a bounded sequence in Hilbert space H. There exists a unique element y ∈ H such that limn→∞ xn − y < limn→∞ xn − z for every z ∈ H \ {y}. The element y is called the asymptotic center of {xn }. In [8], it was showed that the following condition (1) for every u ∈ K, {T n u} is weakly almost convergent, as n → ∞, to its asymptotic center. Assume that for every bounded set B ⊂ K and integer k 0 there exists a δk (B) 0 with limk→∞ δk (B) = 0 such that k T u − T k v p λk u − vp + c λk up − T k up + λk vp − T k v p + δk (B) (1) for u, v ∈ B and k 0, where λk , c and p are nonnegative constants such that limk→∞ λk = 1 and p 1. We introduce some definitions and statements presented in [10]: Let T , I : K → K be nonlinear mappings. Then: (i) the mappings T and I are called commuting on K if I T u = T I u for all u ∈ K; (ii) the mappings T and I are called R-weakly commuting on K if there exists a real number R > 0 such that T I u − I T u RT u − I u for all u ∈ K. We need the following well-known lemma to prove our main theorem. Lemma 1.1. [8] Let {xn } be a sequence in H and let {xn } be convergent, then the following (i)–(iii) are mutually equivalent: (i) limm→∞ limn→∞ limi→∞ [(xm+i , xm ) − (xn+i , xn )] 0; (ii) limm→∞ limn→∞ limi→∞ [xm+i + xm 2 − xn+i + xn 2 ] 0; (iii) limm→∞ limn→∞ limi→∞ [xn+i − xn 2 − xm+i − xm 2 ] 0. Remark 1.2. If {xn } satisfies the equivalent conditions above and {xn } is convergent, then {xn } is weakly almost convergent to its asymptotic center. The main result in this paper is stated as follows.
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2. The main result Let H be a Hilbert space with inner product (·,·) and · norm, and let K be weakly compact a subset of H. Definition 2.1. Let T , I : K → K. T is called I -asymptotically quasi-nonexpansive on K if there exists a sequence {λk } ⊂ [0, ∞) with limk→∞ λk = 0 such that k T u − f λ + 1 I k u − f k for all u, v ∈ K and for every f ∈ F (T ) ∩ F (I ) and k 0. Theorem 2.1. Let K be a weakly compact subset of H. Let I : K → K be a nonlinear bounded mapping, T : K → K be a nonlinear mapping. Assume that for every bounded set B ⊂ K and integer k 0 there exists a δk (B) = 0 such that (q1)
k T u − T k v p λ + 1 I k u − I k v p + c λ + 1 up − T k up k k p + λk + 1 vp − T k v + δk (B) ∈ B, where λk , c
are nonnegative constants such that limk→∞ λk
and p for u, v and (q2) there exists Rm 0 with limm→∞ Rm = 1 such that k m T I u − I m T k v Rm T k u − I m v
(2) = 0, p 1;
for all u, v ∈ K. If the above conditions satisfy, i.e. let c = 0 and F (T ) ∩ F (I ) = ∅ or let c > 0 be in (q1) condition, then for all u ∈ K, {T n u} is weakly almost convergent, as n → ∞ to its asymptotic center. Proof. We consider the case c = 0 and F (T ) ∩ F (I ) = ∅. Let f ∈ F (T ) ∩ F (I ) and B be bounded subset of K. We have from condition (2) the following: There exists a sequence {λk } ⊂ [0, ∞) with limk→∞ λk = 0 and for every bounded set B ⊂ K there exists a δk (B) 0 with limk→∞ δk (B) = 0 such that k T u − T k v λ + 1 I k u − I k v + δk (B) (3) k for all u, v ∈ K and k 0. For every bounded set B ⊂ K, u, v ∈ B and integer k 0, where λk and p are nonnegative constants with limk→∞ λk = 0, and p 1. Set B = {x, f }. We rewrite in condition (3) u = x, v = f ∈ F (T ) ∩ F (I ) then we have k T x − T k f p λ + 1 I k x − I k f p + δk (B) k for k 0. Since f ∈ F (T ) ∩ F (I ), we write k T x − f p λ + 1 I k x − f p + δk (B). k
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Since I is bounded, {T n x} is bounded. Now, assume B = {T n x: n 0} ∪ {f }. Then we rewrite condition (3) the following n+k p p T x − I k f λk + 1 I k T n x − T n I k f + δk (B) p p λk + 1 Rk T n x − I k f + δk (B) p p λk + 1 Rk T n x − f + δk (B). Then we have from last inequality p p n+k p T x − I k f λk + 1 Rk T n x − f + δk (B). Taking limit we have the following p p lim T n+k x − I k f lim T n x − f , k→∞
k→∞
k→∞
k→∞
p p lim T n+k x − I k f − lim T n x − f 0,
for n 0, since limk→∞ λk = 0, limk→∞ Rk = 1 and limk→∞ δk (B) = 0, which shows that {T n x − f } is convergent. By condition (3), if we take k = n − m (n > m 0), u = T m+i x and v = T m x, then we obtain p p n+i T x − T n x λn−m + 1 I n−m T m+i x − I n−m T m x + δn−m (B) p p = λn−m + 1 I n−m T m+i x − I n−m T m x + T m+i x − T m x p p − T m+i x − T m x + T m+i I n−m x − I n−m T m+i x p − T m+i I n−m x − I n−m T m+i x + δn−m (B). p
Then we obtain p p n+i T x − T n x − T m+i x − T m x p p λn−m + 1 I n−m T m+i x − I n−m T m x +T m+i I n−m x − I n−m T m+i x p p − T m+i I n−m x − I n−m T m+i x − T m+i x − T m x + δn−m (B). By the condition (q2), p p n+i T x − T n x − T m+i x − T m x p p p λn−m + 1 I n−m T m+i x − I n−m T m x + Rn−m T m+i x − I n−m x p p − T m+i I n−m x − I n−m T m+i x − T m+i x − T m x + δn−m (B), then, for n > m 0 and i 0 we write p λn−m + 1 − 1 2p C 2p + Rn−m − 1 2p C p + δn−m (B), max{M, N } = C, where M = sup0 {T x: x ∈ K}, N = supj 0 {I j x: x ∈ K}. Taking limit we have p p lim lim supT n+i x − T n x − T m+i x − T m x 0, m→∞ n→∞ i0
since limn,m→∞ λn−m = 0, limn,m→∞ Rn−m = 1 and limn,m→∞ δn−m (B) = 0. p
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If we take p = 2, then setting xn = T n x − f for n 0, {xn } satisfies the condition (iii) in Lemma 1.1. So, {T n x − f } is weakly almost convergent to its asymptotic center z so that for n 0, {T n x} is weakly almost convergent to its asymptotic center z + f . Now, we consider the case c > 0 and F (T ) ∩ F (I ) = ∅. Set B = {x}. We rewrite the condition (2) as following: p 0 c 2 λk + 1 xp − 2T k x + δk {x} , k p [(λk + 1)xp + δk ({x})] T x . 2c Then, for n 0, {T n x} is bounded. Assume B = {T n x: n 0}. For u = v = T n x and k, n 0, we have n+k p [(λk + 1)T n xp + δk (B)] T . x 2c Taking limit, it is easy to see that {T n x} is convergent. Consequently, if we take k = n − m (n > m 0), u = T m+i x and v = T m x in the condition (2), then we obtain p n+i T x − T nx p λn−m + 1 I n−m T m+i x − I n−m T m x p p p p + c λ + 1 T m+i x −T n+i x + λ + 1 T m x − T n x n−m
n−m
+ δn−m (B) p p = λn−m + 1 I n−m T m+i x − I n−m T m x + T m+i I n−m x − I n−m T m+i x p − T m+i I n−m x − I n−m T m+i x p p p p + c λn−m + 1 T m+i x − T n+i x + λn−m + 1 T m x − T n x p p + T m+i x − T m x − T m+i x − T m x + δn−m (B). By the condition (q2), p p n+i T x − T n x − T m+i x − T m x p p λn−m + 1 I n−m T m+i x − I n−m T m x − T m+i I n−m x − I n−m T m+i x p p p p + c λn−m + 1 T m+i x − T n+i x + λn−m + 1 T m x − T n x p p m+i p T +R x − I n−m x − T m+i x − T m x + δn−m (B). n−m
Then, for n > m 0 and i 0 we write p λn−m + 1 − 1 2p C 2p + Rn−m − 1 2p C p p p + c λ + 1 T m+i x − T n+i x + λ n−m
n−m
p p + 1 T m x − T n x
+ δn−m (B), max{M, N} = C, where M = sup0 {T x: x ∈ K}, N = supj 0 {I j x: x ∈ K}. Taking limit we have p p lim lim supT n+i x − T n x − T m+i x − T m x 0, m→∞ n→∞ i0
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since for n 0, {T n x} is convergent, limn,m→∞ λn−m = 0, limn,m→∞ Rn−m = 1 and limn,m→∞ δn−m (B) = 0. Since for p = 2 it is satisfied (iii) in Lemma 1.1, then for all u ∈ K, {T n u} is weakly almost convergent, as n → ∞ to its asymptotic center. 2 p
Corollary 2.1. In particularly, if T is an I -asymptotically quasi-nonexpansive mapping, then for all u ∈ K, {T n u} is weakly almost convergent, as n → ∞ to its asymptotic center. References [1] L. Debnath, P. Mikusinski, Introduction to Hilbert Space with Application, third ed., Elsevier Academic, New York, 2005. [2] J.B. Diaz, F.T. Metcalf, On the set of sequential limit points of successive approximations, Trans. Amer. Math. Soc. 135 (1969) 459–485. [3] W.G. Dotson Jr., On the Mann iterative process, Trans. Amer. Math. Soc. 149 (1970) 65–73. [4] M.K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl. 207 (1997) 96–103. [5] K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972) 171–174. [6] W.A. Kirk, Remarks on approximation and approximate fixed points in metric fixed point theory, Ann. Univ. Mariae Curie-Sklodowska Sect. A LI.2 15 (1997) 167–178. [7] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 259 (2001) 1–7. [8] I. Miyadera, Nonlinear mean ergodic theorems, Taiwanese J. Math. 1 (4) (1997) 433–449. [9] W.V. Petryshyn, T.E. Williamson, Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl. 43 (1973) 459–497. [10] N. Shahzad, Generalized I -nonexpansive maps and best approximations in Banach spaces, Demonstratio Math. XXXVII (3) (2004) 597–600.
Convergence theorem for I -asymptotically quasi-nonexpansive mapping in Hilbert space Seyit Temir ∗ , Ozlem Gul Department of Mathematics, Arts and Science Faculty, Harran University, 63200 Sanliurfa, Turkey Received 16 September 2005 Available online 4 August 2006 Submitted by L. Debnath
Abstract Let H be a Hilbert space with inner product (·,·) and · norm, and let K be weakly compact a subset of H. Let T : K → K be nonlinear mapping and I : K → K be a nonlinear bounded mapping. In this paper, we define the I -asymptotically quasi-nonexpansive mapping in Hilbert space. If T is an I -asymptotically i quasi-nonexpansive mapping, then we prove that n1 n−1 i=0 T u, for u ∈ K as n → ∞, is weakly almost convergent to its asymptotic center. © 2006 Elsevier Inc. All rights reserved. Keywords: Asymptotic center; Asymptotically quasi-nonexpansive mapping; Nonlinear ergodic theorems
1. Introduction Throughout this paper, we assume that H is a real Hilbert space with inner product (·,·) and · norm. The basic concepts and properties in the theory of Hilbert spaces are given in [1]. Let K be a nonempty subset of H real Hilbert space and T : K → K be a nonlinear mapping. Let F (T ) = {u ∈ K: T u = u} be denoted as the set of fixed points of a mapping T . A mapping T : K → K is called nonexpansive provided k T u − T k v u − v for all u, v ∈ K and k 1. T is called asymptotically nonexpansive mapping if there exists a sequence {λk } ⊂ [0, ∞) with limk→∞ λk = 0 such that * Corresponding author.
E-mail addresses: [email protected] (S. Temir), [email protected] (O. Gul). 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.07.004
760
S. Temir, O. Gul / J. Math. Anal. Appl. 329 (2007) 759–765
k T u − T k v (λk + 1)u − v for all u, v ∈ K and k 1. The asymptotically nonexpansive mapping was defined as original by Goebel and Kirk [5]. In [5], they proved that every asymptotically nonexpansive mapping T : K → K has a fixed point when E is uniformly convex Banach space. Let T , I : K → K. Then T is called I -nonexpansive on K if T u − T v I u − I v for all u, v ∈ K. T is called I -asymptotically nonexpansive on K if there exists a sequence {λk } ⊂ [0, ∞) with limk→∞ λk = 0 such that k T u − T k v λ + 1 I k u − I k v k for all u, v ∈ K and k = 1, 2, . . . . The concept of quasi-nonexpansive mapping was initiated by Tricomi in 1916 for real functions. The quasi-nonexpansive mapping in Banach spaces was studied by Diaz and Metcalf [2] and Dotson [3]. Recently, this concept was given by Kirk [6] in metric spaces which we adapt to Hilbert space as following: T is called quasi-nonexpansive mapping provided k T u − f u − f for all u ∈ K and f ∈ F (T ) and k 1. T is called asymptotically quasi-nonexpansive mapping if there exists a sequence {λk } ⊂ [0, ∞) with limk→∞ λk = 0 such that k T u − f (λk + 1)u − f for all u ∈ K and f ∈ F (T ) and k 1. Remark 1.1. From above definitions, it is easy to see that if F (T ) is nonempty, a nonexpansive mapping must be quasi-nonexpansive, and an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive. But the converse does not hold. There have been a number of recent results on fixed points on nonexpansive and quasinonexpansive mappings in Banach spaces and metric spaces. Recently, concerning the problem of iterative sequences for asymptotically nonexpansive mappings or asymptotically quasinonexpansive mappings converging to some fixed points have been considered by many authors. For example of these papers, the strong and weak convergence of the sequence of certain iterates to a fixed point of quasi-nonexpansive maps were studied by Petryshyn and Williamson [9]. In particular, the weak and strong convergence of iterates sequence for various special cases of quasi-nonexpansive and one-set and/or one-ball contractive mappings were discussed by Petryshyn and Williamson [9]. Their analysis was connected with the convergence of Mann iterates studied by Dotson [3]. Subsequently, the convergence of Ishikawa iterates of quasinonexpansive mappings in Banach spaces was discussed by Ghosh and Debnath [4]. Their paper was concerned with the convergence of Ishikawa iterates of quasi-nonexpansive mappings in a Banach space. The above results and obtained some necessary and sufficient conditions for Ishikawa iterative sequence to converge to a fixed point for asymptotically quasi-nonexpansive mappings was extended by Liu [7]. The aim of our study is to prove weakly convergence theorem for I -asymptotically quasinonexpansive mapping in Hilbert space. We will benefit from the following condition (1) having been given in [8] with a different approximation from above results improved by various authors. In order to prove weak nonlinear ergodic theorem for I -asymptotically quasi-nonexpansive
S. Temir, O. Gul / J. Math. Anal. Appl. 329 (2007) 759–765
761
mapping, we shall give definition of I -asymptotically quasi-nonexpansive mapping and use the condition (1) as a tool to prove our main result. Furthermore, we also give stronger condition than R-weakly commuting stated in Shahzad’s study [10] to prove our main result. If T is I asymptotically quasi-nonexpansive mapping and I is bounded, then using condition (1), we prove that {T n u} almost convergent to its asymptotic center. For every u ∈ K, the is weakly i u coincides with the asymptotic center of {T n u}. T weak limit of n1 n−1 i=0 We consider our main result in two case. In condition (1), we consider the case c = 0 and F (T ) ∩ F (I ) = ∅ and the case c > 0 and F (T ) ∩ F (I ) = ∅. Therefore, under these cases, it is sufficient to prove that {T n u} is weakly almost convergent to its asymptotic center. First, we give the following some definitions and statements to use in our weak convergence theorem and then, we present our main result. Let {xn } be a bounded sequence in Hilbert space H. There exists a unique element y ∈ H such that limn→∞ xn − y < limn→∞ xn − z for every z ∈ H \ {y}. The element y is called the asymptotic center of {xn }. In [8], it was showed that the following condition (1) for every u ∈ K, {T n u} is weakly almost convergent, as n → ∞, to its asymptotic center. Assume that for every bounded set B ⊂ K and integer k 0 there exists a δk (B) 0 with limk→∞ δk (B) = 0 such that k T u − T k v p λk u − vp + c λk up − T k up + λk vp − T k v p + δk (B) (1) for u, v ∈ B and k 0, where λk , c and p are nonnegative constants such that limk→∞ λk = 1 and p 1. We introduce some definitions and statements presented in [10]: Let T , I : K → K be nonlinear mappings. Then: (i) the mappings T and I are called commuting on K if I T u = T I u for all u ∈ K; (ii) the mappings T and I are called R-weakly commuting on K if there exists a real number R > 0 such that T I u − I T u RT u − I u for all u ∈ K. We need the following well-known lemma to prove our main theorem. Lemma 1.1. [8] Let {xn } be a sequence in H and let {xn } be convergent, then the following (i)–(iii) are mutually equivalent: (i) limm→∞ limn→∞ limi→∞ [(xm+i , xm ) − (xn+i , xn )] 0; (ii) limm→∞ limn→∞ limi→∞ [xm+i + xm 2 − xn+i + xn 2 ] 0; (iii) limm→∞ limn→∞ limi→∞ [xn+i − xn 2 − xm+i − xm 2 ] 0. Remark 1.2. If {xn } satisfies the equivalent conditions above and {xn } is convergent, then {xn } is weakly almost convergent to its asymptotic center. The main result in this paper is stated as follows.
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2. The main result Let H be a Hilbert space with inner product (·,·) and · norm, and let K be weakly compact a subset of H. Definition 2.1. Let T , I : K → K. T is called I -asymptotically quasi-nonexpansive on K if there exists a sequence {λk } ⊂ [0, ∞) with limk→∞ λk = 0 such that k T u − f λ + 1 I k u − f k for all u, v ∈ K and for every f ∈ F (T ) ∩ F (I ) and k 0. Theorem 2.1. Let K be a weakly compact subset of H. Let I : K → K be a nonlinear bounded mapping, T : K → K be a nonlinear mapping. Assume that for every bounded set B ⊂ K and integer k 0 there exists a δk (B) = 0 such that (q1)
k T u − T k v p λ + 1 I k u − I k v p + c λ + 1 up − T k up k k p + λk + 1 vp − T k v + δk (B) ∈ B, where λk , c
are nonnegative constants such that limk→∞ λk
and p for u, v and (q2) there exists Rm 0 with limm→∞ Rm = 1 such that k m T I u − I m T k v Rm T k u − I m v
(2) = 0, p 1;
for all u, v ∈ K. If the above conditions satisfy, i.e. let c = 0 and F (T ) ∩ F (I ) = ∅ or let c > 0 be in (q1) condition, then for all u ∈ K, {T n u} is weakly almost convergent, as n → ∞ to its asymptotic center. Proof. We consider the case c = 0 and F (T ) ∩ F (I ) = ∅. Let f ∈ F (T ) ∩ F (I ) and B be bounded subset of K. We have from condition (2) the following: There exists a sequence {λk } ⊂ [0, ∞) with limk→∞ λk = 0 and for every bounded set B ⊂ K there exists a δk (B) 0 with limk→∞ δk (B) = 0 such that k T u − T k v λ + 1 I k u − I k v + δk (B) (3) k for all u, v ∈ K and k 0. For every bounded set B ⊂ K, u, v ∈ B and integer k 0, where λk and p are nonnegative constants with limk→∞ λk = 0, and p 1. Set B = {x, f }. We rewrite in condition (3) u = x, v = f ∈ F (T ) ∩ F (I ) then we have k T x − T k f p λ + 1 I k x − I k f p + δk (B) k for k 0. Since f ∈ F (T ) ∩ F (I ), we write k T x − f p λ + 1 I k x − f p + δk (B). k
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Since I is bounded, {T n x} is bounded. Now, assume B = {T n x: n 0} ∪ {f }. Then we rewrite condition (3) the following n+k p p T x − I k f λk + 1 I k T n x − T n I k f + δk (B) p p λk + 1 Rk T n x − I k f + δk (B) p p λk + 1 Rk T n x − f + δk (B). Then we have from last inequality p p n+k p T x − I k f λk + 1 Rk T n x − f + δk (B). Taking limit we have the following p p lim T n+k x − I k f lim T n x − f , k→∞
k→∞
k→∞
k→∞
p p lim T n+k x − I k f − lim T n x − f 0,
for n 0, since limk→∞ λk = 0, limk→∞ Rk = 1 and limk→∞ δk (B) = 0, which shows that {T n x − f } is convergent. By condition (3), if we take k = n − m (n > m 0), u = T m+i x and v = T m x, then we obtain p p n+i T x − T n x λn−m + 1 I n−m T m+i x − I n−m T m x + δn−m (B) p p = λn−m + 1 I n−m T m+i x − I n−m T m x + T m+i x − T m x p p − T m+i x − T m x + T m+i I n−m x − I n−m T m+i x p − T m+i I n−m x − I n−m T m+i x + δn−m (B). p
Then we obtain p p n+i T x − T n x − T m+i x − T m x p p λn−m + 1 I n−m T m+i x − I n−m T m x +T m+i I n−m x − I n−m T m+i x p p − T m+i I n−m x − I n−m T m+i x − T m+i x − T m x + δn−m (B). By the condition (q2), p p n+i T x − T n x − T m+i x − T m x p p p λn−m + 1 I n−m T m+i x − I n−m T m x + Rn−m T m+i x − I n−m x p p − T m+i I n−m x − I n−m T m+i x − T m+i x − T m x + δn−m (B), then, for n > m 0 and i 0 we write p λn−m + 1 − 1 2p C 2p + Rn−m − 1 2p C p + δn−m (B), max{M, N } = C, where M = sup0 {T x: x ∈ K}, N = supj 0 {I j x: x ∈ K}. Taking limit we have p p lim lim supT n+i x − T n x − T m+i x − T m x 0, m→∞ n→∞ i0
since limn,m→∞ λn−m = 0, limn,m→∞ Rn−m = 1 and limn,m→∞ δn−m (B) = 0. p
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If we take p = 2, then setting xn = T n x − f for n 0, {xn } satisfies the condition (iii) in Lemma 1.1. So, {T n x − f } is weakly almost convergent to its asymptotic center z so that for n 0, {T n x} is weakly almost convergent to its asymptotic center z + f . Now, we consider the case c > 0 and F (T ) ∩ F (I ) = ∅. Set B = {x}. We rewrite the condition (2) as following: p 0 c 2 λk + 1 xp − 2T k x + δk {x} , k p [(λk + 1)xp + δk ({x})] T x . 2c Then, for n 0, {T n x} is bounded. Assume B = {T n x: n 0}. For u = v = T n x and k, n 0, we have n+k p [(λk + 1)T n xp + δk (B)] T . x 2c Taking limit, it is easy to see that {T n x} is convergent. Consequently, if we take k = n − m (n > m 0), u = T m+i x and v = T m x in the condition (2), then we obtain p n+i T x − T nx p λn−m + 1 I n−m T m+i x − I n−m T m x p p p p + c λ + 1 T m+i x −T n+i x + λ + 1 T m x − T n x n−m
n−m
+ δn−m (B) p p = λn−m + 1 I n−m T m+i x − I n−m T m x + T m+i I n−m x − I n−m T m+i x p − T m+i I n−m x − I n−m T m+i x p p p p + c λn−m + 1 T m+i x − T n+i x + λn−m + 1 T m x − T n x p p + T m+i x − T m x − T m+i x − T m x + δn−m (B). By the condition (q2), p p n+i T x − T n x − T m+i x − T m x p p λn−m + 1 I n−m T m+i x − I n−m T m x − T m+i I n−m x − I n−m T m+i x p p p p + c λn−m + 1 T m+i x − T n+i x + λn−m + 1 T m x − T n x p p m+i p T +R x − I n−m x − T m+i x − T m x + δn−m (B). n−m
Then, for n > m 0 and i 0 we write p λn−m + 1 − 1 2p C 2p + Rn−m − 1 2p C p p p + c λ + 1 T m+i x − T n+i x + λ n−m
n−m
p p + 1 T m x − T n x
+ δn−m (B), max{M, N} = C, where M = sup0 {T x: x ∈ K}, N = supj 0 {I j x: x ∈ K}. Taking limit we have p p lim lim supT n+i x − T n x − T m+i x − T m x 0, m→∞ n→∞ i0
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since for n 0, {T n x} is convergent, limn,m→∞ λn−m = 0, limn,m→∞ Rn−m = 1 and limn,m→∞ δn−m (B) = 0. Since for p = 2 it is satisfied (iii) in Lemma 1.1, then for all u ∈ K, {T n u} is weakly almost convergent, as n → ∞ to its asymptotic center. 2 p
Corollary 2.1. In particularly, if T is an I -asymptotically quasi-nonexpansive mapping, then for all u ∈ K, {T n u} is weakly almost convergent, as n → ∞ to its asymptotic center. References [1] L. Debnath, P. Mikusinski, Introduction to Hilbert Space with Application, third ed., Elsevier Academic, New York, 2005. [2] J.B. Diaz, F.T. Metcalf, On the set of sequential limit points of successive approximations, Trans. Amer. Math. Soc. 135 (1969) 459–485. [3] W.G. Dotson Jr., On the Mann iterative process, Trans. Amer. Math. Soc. 149 (1970) 65–73. [4] M.K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl. 207 (1997) 96–103. [5] K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972) 171–174. [6] W.A. Kirk, Remarks on approximation and approximate fixed points in metric fixed point theory, Ann. Univ. Mariae Curie-Sklodowska Sect. A LI.2 15 (1997) 167–178. [7] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 259 (2001) 1–7. [8] I. Miyadera, Nonlinear mean ergodic theorems, Taiwanese J. Math. 1 (4) (1997) 433–449. [9] W.V. Petryshyn, T.E. Williamson, Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl. 43 (1973) 459–497. [10] N. Shahzad, Generalized I -nonexpansive maps and best approximations in Banach spaces, Demonstratio Math. XXXVII (3) (2004) 597–600.