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A new sufficient condition for the strong convergence of Halpern type iterations
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Applied Mathematics and Computation 198 (2008) 721–728 www.elsevier.com/locate/amc
A new sufficient condition for the strong convergence of Halpern type iterations Yisheng Song College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, PR China
Abstract The aim of this work is to give a new sufficient condition of the strong convergence of the Halpern type iteration for a non-expansive self-mapping defined on a Banach space with a uniformly Gaˆteaux differentiable norm. Several examples satisfying our condition are presented. Our results not only remove the restriction of the space with the fixed point property for non-expansive self-mappings, but also get rid of the dependence on the convergence of the implicit anchor-like continuous path zt in the proof. 2007 Elsevier Inc. All rights reserved. Keywords: Non-expansive mappings; Halpern type iteration; Uniformly Gaˆteaux differentiable norm
1. Introduction Let E be a real Banach space with dual E* and K a non-empty closed convex subset of E. Let J denote the normalized duality mapping from E into 2E given by J(x) = {f 2 E*, hx, fi = kxkkfk, kxk = kfk} "x 2 E, where hÆ, Æi denotes the generalized duality pairing. We say F(T) = {x 2 E; Tx = x}, the set of all fixed point for a mapping T and N, the set of all positive integer. We write xn N x (respectively xn * x) to indicate that the sequence xn weakly (respectively weak*) converges to x; as usual xn ! x will symbolize strong convergence. Let T be a non-expansive mappings from K into itself (a mapping T : K ! K is said to be non-expansive if kTx Tyk 6 kx yk "x, y 2 K). In 1967, Halpern [3] was the first who introduced the following Halpern type iteration in a Hilbert spaces: for any initialization x0 2 K and any anchor u 2 K, an 2 [0, 1], xnþ1 ¼ an u þ ð1 an ÞTxn ;
n P 0: P1
ð1:1Þ
He pointed out that the control conditions ðC1Þlimn!1 an ¼ 0 and ðC2Þ n¼1 an ¼ 1 are necessary for the convergence of the Halpern type iteration (1.1) to a fixed point of T. At the same time, he put forth the following open problem, which was put forward by Reich in Refs. [8,12] also. Question 1. Are sufficient the control conditions (C1) and (C2) for the convergence of the Halpern type iteration to a fixed point of T?
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Subsequently, many mathematic workers carefully researched the problem. For example, Lions [5] investigated the problem and established a strong convergence theorem under the control conditions (C1), (C2) and nþ1 (C3) limn!1 anaa ¼ 0. In 1980, Reich [11] studied the iteration scheme (1.1) with a specific sequence of param2 nþ1
eters {an} = {na}, where a 2 (0, 1). However, both Lions’ and Reich’s control conditions excluded the natural choice of an = 1/(n + 1). In 1992, Wittmann [14] studied the iterative scheme (1.1) in a Hilbert space, and obtained the strong convergence of the iteration. In particular,P he proved a strong convergence theorem [14, Theorem 2] under the control conditions (C1), (C2) and (C4) 1 n¼1 jan anþ1 j < 1. The main advantage of Wittmann’s convergence theorem lies in the fact that it allows 1/(n + 1) to be a candidate of an. An analogy of Wittmann’s result was first proved outside Hilbert space by Reich in [13] for spaces with a weakly sequentially continuous duality mapping in 1994. In 1997, Shioji and Takahashi [15] also extended Wittmann’s result to a real Banach space with a uniformly Gaˆteaux differentiable norm. In 2002, Xu [23] showed Wittmann’s result in a uniformly smooth Banach space using the conditions (C1), (C2) and (C5) limn!1 aan1 ¼ 1. We observe n from above results that the aim of the additive conditions (C4) (or (C5)) is to guarantee limn!1 kxn xn+1k = 0, and that limn!1 kxn Txnk = 0. In nature, they proved the following result. Theorem 1.1. Let K be a non-empty closed convex subset of a Banach space E, and T : K ! K be a non-expansive mapping. Assumed that {xn} is given by (1.1) with {an} (0, 1). If F(T) 5 ;, then (i) {xn} is bounded. If, in addition, {an} satisfies (C2) together with one of (C4) and (C5), then (ii) limn!1 kxn xn+1k = 0. Suppose that, {an} fulfils (C1), then (iii) limn!1 kxn+1 Txnk = 0. By the means of Theorem 1.1 along with the fact that the implicit anchor-like continuous path zt, defined by zt = tu + (1t)Tzt, strongly converges to a fixed point of T, the strong convergence of {xn} is obtained. That is, the strong convergence of {xn} depends upon the convergence of {zt} even if the conditions (C1) and (C2) together with one of the conditions (C4) and (C5) are satisfied. Thus, it will be a very interesting to establish a sufficient condition to assure the strong convergence of the explicit iteration {xn} which is independent of the strong convergence of zt. In this paper, we will study and explore the condition reached the above desire. Let {xn} be a bounded sequence defined by (1.1) and ln be any Banach limit (see Preliminaries). Set 2
uðyÞ ¼ ln kxn yk ; then u(y) is convex and continuous, and u(y) ! 1 as kyk ! 1. If E is reflexive, there exists z 2 K such that u(z) = infy2Ku(y) (see [21, Theorem 1.3.11]). So the set K min ¼ z 2 K; uðzÞ ¼ inf uðyÞ 6¼ ;: y2K
Clearly, Kmin is closed convex by the convexity and continuity of u(y). It is easily seen that if xn ! x* 2 K (n ! 1), then x* 2 Kmin since u(x*) = 0. In addition, suppose that the condition (C1) is satisfied, then x* 2 F(T) as kxnþ1 Tx k 6 an ku Tx k þ ð1 an Þkxn x k ! 0: Namely, x* 2 Kmin \ F(T) is necessary for the convergence of the Halpern type iteration (1.1) to x*. Then we spontaneously put forth the following problem. Question 2. Are sufficient the control conditions Kmin \ F(T) 5 ; and (C2) for the convergence of the Halpern type iteration to a fixed point of T? Our main aim in this paper is to not only give a sufficient condition of the strong convergence of the Halpern type iteration {xn} for a non-expansive self-mapping defined on a Banach space with a uniformly Gaˆteaux differentiable norm, but also make the strong convergence of the Halpern type iteration {xn} to shake off the restriction and dependence on the implicit anchor-like continuous path zt. Several examples satisfying our condition Kmin \ F(T) 5 ; are presented. As application, we also investigate the modified Mann’s iteration (1.2) for finding a fixed point of non-expansive mapping T xnþ1 ¼ an u þ an ðð1 bn ÞTxn þ bn xn Þ:
ð1:2Þ
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2. Preliminaries Let S(E): = {x 2 E; kxk = 1} denote the unit sphere of a Banach space E. The space E is said to have (i) a Gaˆteaux differentiable norm (we also say that E is smooth), if the limit ðÞ
lim t!0
kx þ tyk kxk t
exists for each x, y 2 S(E); (ii) a uniformly Gaˆteaux differentiable norm, if for each y in S(E), the limit (*) is uniformly attained for x 2 S(E); (iii) a Fre´chet differentiable norm, if for each x 2 S(E), the limit (*) is attained uniformly for y 2 S(E); (iv) a uniformly Fre´chet differentiable norm (we also say that E is uniformly smooth), if the limit (*) is attained uniformly for (x, y) 2 S(E) · S(E). (v) fixed point property for non-expansive self-mappings, if each non-expansive self-mapping defined on any bounded closed convex subset K of E has at least a fixed point. A Banach space E is said to be (vi) strictly convex if kxk ¼ kyk ¼ 1; x 6¼ y implies
kx þ yk < 1; 2
(vii) uniformly convex if for all e 2 [0, 2], $de > 0 such that kxk ¼ kyk ¼ 1 implies
kx þ yk < 1 de whenever kx yk P e: 2
The following results is well known which are found in Refs. [2,21,6]: the normalized duality mapping J in a Banach space E with a uniformly Gaˆteaux differentiable norm is single-valued and strong-weak* uniformly continuous on any bounded subset of E; each uniformly convex Banach space E is reflexive and strictly convex and has fixed point property for non-expansive self-mappings; every uniformly smooth Banach space E is a reflexive Banach space with a uniformly Gaˆteaux differentiable norm and has fixed point property for nonexpansive self-mappings. In the proof of our main theorems, we also need the following definitions and results. Let l be a continuous linear functional on l1 satisfying klk = 1 = l(1). Then we know that l is a mean on N if and only if inffan ; n 2 N g 6 lðaÞ 6 supfan ; n 2 N g for every a = (a1, a2, . . .)2 l1. According to time and circumstances, we ln(an) instead of l(a). A mean l on N is called a Banach limit if ln ðan Þ ¼ ln ðanþ1 Þ for every a = (a1, a2, . . .) 2 l1. Lemma 2.1 [22, Lemma 1]. Let C be a nonempty closed convex subset of a Banach space E with a uniformly Gaˆteaux differentiable norm. Let {xn} be a bounded sequence of E and let ln be a Banach limit and z 2 C. Then 2
ln kxn zk ¼ min ln kxn yk
2
y2C
if and only if ln hy z; J ðxn zÞi 6 0;
8y 2 C:
Lemma 2.2 [15, Proposition 2]. Let a is a real number and (x0, x1, . . .) 2 l1 such that lnxn 6 a for all Banach Limits. If lim supn!1 (xn+1 xn) 6 0, then lim supn!1 xn 6 a. In the sequel, we shall make use of the following lemma which can be found in the existing literature [23]. Furthermore, a variant of Lemma 2.3 has already been used by Reich in [10, Theorem 1].
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Lemma 2.3 [23, Lemma 2.5]. Let {an} be a sequence of non-negative real numbers satisfying the property anþ1 6 ð1 cn Þan þ cn bn ; n P 0: Suppose that {cn} (0, 1) and {bn} are real number sequence such that (i) Then {an} converges to zero, as n ! 1.
P1
n¼0 cn
¼ 1; (ii) lim supn!1 bn 6 0.
3. Strong convergence of Halpern type iteration In this section, let K be a non-empty closed convex subset of a real Banach space E, and T : K ! K be a non-expansive mapping with a fixed point. For arbitrary initial value x0 2 K and fixed anchor u 2 K, define iteratively a sequence {xn} as follows: xnþ1 ¼ an u þ ð1 an ÞTxn ;
ð3:1Þ
where {an} is a sequence in (0, 1). Clearly, following Theorem 1.1, {xn} is bounded. Then there exists r > 0 sufficiently large such that xn 2 Br ¼ fx; kxk 6 rg8n 2 N: Furthermore, the set K \ Br is a bounded closed and convex non-empty subset of E. If we define a map u : E ! R by 2
uðyÞ ¼ ln kxn yk ; then u(y) is convex and continuous, and u(y) ! 1 as kyk ! 1. If E is reflexive, there exists z 2 K such that u(z) = infy2Ku(y) (see [21, Theorem 1.3.11]). So the set K min ¼ z 2 K; uðzÞ ¼ inf uðyÞ 6¼ ;: y2K
Clearly, Kmin is closed convex by the convexity and continuity of u(y). Theorem 3.1. Let K be a non-empty closed convex subset of a real Banach space E with a uniformly Gaˆteaux differentiable norm, and T : K ! K be a non-expansive mapping with a fixed point. Suppose that P Kmin ˙ F(T) 5 ;, and {an} is a sequence in (0, 1) satisfying the conditions ðC2Þ þ1 n¼1 an ¼ þ1 and limn!1 kxn+1 xnk = 0. Then as n ! 1, {xn} given by (3.1) converges strongly to some fixed point p of T. Proof. Take p 2 Kmin ˙ F(T). Then ln kxn pk2 ¼ inf ln kxn yk2
and
y2K
p ¼ Tp:
It follows from Lemma 2.1 that for u 2 K, ln hu p; J ðxn pÞi 6 0: Since limn!1 kxn+1 xnk = 0, then it follows from the norm-weak* uniformly continuity of the duality mapping J that lim ðhu p; J ðxnþ1 pÞi hu p; J ðxn pÞiÞ ¼ 0:
n!1
Hence, the sequence {hu p, J(xn p)i} satisfies the conditions of Lemma 2.2. As a result, we must have lim suphu p; J ðxnþ1 pÞi 6 0:
ð3:2Þ
n!1
Using Eq. (3.1), we make the following estimates: 2
kxnþ1 pk ¼ ð1 an ÞhTxn p; J ðxnþ1 pÞi þ an hu p; J ðxnþ1 pÞi 6 ð1 an ÞkTxn pkkxnþ1 pk þ an hu p; J ðxnþ1 pÞi 2
6 ð1 an Þ
2
kxn pk þ kxnþ1 pk þ an hu p; J ðxnþ1 pÞi: 2
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Thus, 2
2
kxnþ1 pk 6 ð1 an Þkxn pk þ 2an hu p; J ðxnþ1 pÞi:
ð3:3Þ
Hence, noting Eq. (3.2), an application of Lemma 2.3 onto Eq. (3.3), the conclusion desired is reached.
h
Using Theorem 1.1, we can obtain the following result. Theorem 3.2. Let E, K, T, {xn} are as Theorem 3.1. Suppose that {an} satisfies (C2) and one of (C4) P 1 an1 n¼1 jan anþ1 j < 1 and (C5) limn!1 an ¼ 1. If Kmin ˙ F(T) 5 ;, then as n ! 1, {xn} converges strongly to some fixed point p of T. Proof. It follows from Theorem 1.1 that lim kxnþ1 xn k ¼ 0:
n!1
The remainder of the proof follows from Theorem 3.1.
h
Remark 1. It is noted that the assumption of the space with the fixed point property for non-expansive selfmappings is necessary in almost all known results about the convergence of Halpern type iteration such that uniformly convex Banach space, uniformly smooth Banach space, Banach space with Opial’s condition (see [3,5,7–11,14,13,15–19,23] and the literature cited there), whereas the proof of our results does not depend necessarily on them. On the other hand, our proof also shakes off the dependence on limn!1 kxn Txnk = 0 and the convergence of the implicit anchor-like continuous path zt, defined by zt = tu+(1t)Tzt, and only need Kmin ˙ F(T) 5 ;. Next, we give several examples satisfy our condition Kmin ˙ F(T) 5 ; which is weaker than the condition (C1) limn!1 an = 0 since (C1) means lim kxnþ1 Txn k ¼ lim an ku Txn k ¼ 0:
n!1
n!1
Example 3.3. Let K be a non-empty closed convex subset of a real reflexive Banach space E with fixed point property for non-expansive self-mappings, and T : K ! K be a non-expansive mapping with a fixed point, and {xn} be defined by (3.1) with limn!1 kxn+1 Txnk = 0. Then Kmin ˙ F(T) 5 ;. Proof. The reflexivity of E assures Kmin 5 ;. Then Kmin is a non-empty bounded and closed convex subset of E by the definition of Kmin. Since limn!1 kxn+1 Txnk = 0, then It follows from the property of Banach limit ln and the definition of the function u that for all z 2 Kmin, we have 2
2
2
2
uðTzÞ ¼ ln kxn Tzk ¼ ln kxnþ1 Tzk 6 ln ðkxnþ1 Txn k þ kTxn TzkÞ 6 ln kxn zk ¼ uðzÞ: Therefore, Tz 2 Kmin. That is, T(Kmin) Kmin. Thus, there exists x* 2 Kmin such that x* = Tx* since E has the fixed point property for non-expansive self-mappings. That is x* 2 Kmin \ F(T). h Example 3.4. Let K be a non-empty closed convex subset of a reflexive and strictly convex Banach space E, and T : K ! K be a non-expansive mapping with a fixed point, and {xn} be defined by (3.1) with limn!1 kxn+1 Txnk = 0. Then Kmin ˙ F(T) 5 ;. Proof. It follows from the same argumentation of Example 3.3 that T(Kmin) Kmin. Since F(T)5;, then let y be one of those. From [6, Corollary 5.1.19] and the non-empty closed convexity of Kmin, we get that there exists unique p 2 Kmin such that kp yk ¼ inf x2K min ky xk: By y = Ty and Tp 2 Kmin, we have ky Tpk ¼ kTy Tpk 6 ky pk: Hence p = Tp by the uniqueness of p 2 Kmin. Thus, p 2 Kmin \ F(T). Hence, the following corollary obtained obviously.
h
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Corollary 3.5. Let E be a real reflexive Banach space with fixed point property for non-expansive self-mappings and with a uniformly Gaˆteaux differentiable norm, K, T, {xn} be as Theorem 3.1. Suppose the sequence {an} satisfies (C1) and (C2) and one of (C4) and (C5). Then as n ! 1, {xn} converges strongly to p 2 Kmin \ F(T). Proof. It follows from Theorem 1.1 that limn!1 kxn+1 xnk = 0. Example 3.3 and (C1) assure Kmin \ F(T) 5 ;. Proceeding as in Theorem 3.1, we reach the conclusion. h Using the similar argumentation as Corollary 3.5 (the application of Example 3.3 is replaced by Example 3.4), we obtain the following corollary. Corollary 3.6. Let E be a real reflexive and strictly convex Banach space with a uniformly Gaˆteaux differentiable norm, K, T, {xn}, {an} be as Corollary 3.5. Then as n ! 1, {xn} converges strongly to p 2 Kmin \ F(T). Remark 2. (1) There are many Banach space which satisfy the fixed point property for non-expansive selfmappings in the known Banach spaces. For example, uniformly convex Banach space, uniformly smooth Banach space, reflexive Banach space with normal structure and so on.(2) We remark that Corollary 3.6 is independent of Corollary 3.5. On the one hand, it is easy to find examples of spaces which satisfy the fixed point property for non-expansive self-mappings, which are not strictly convex. On the other hand, it appears to be unknown whether a reflexive and strictly convex Banach space satisfies the fixed point property for nonexpansive self-mappings. 4. Some applications In this section, we shall investigate the modified Mann’s iteration (4.1) for finding a fixed point of nonexpansive mapping T. xnþ1 ¼ an u þ ð1 an Þðbn Txn þ ð1 bn Þxn Þ:
ð4:1Þ
If F(T) 5 ;, then {xn} is bounded. In fact, take p 2 F(T), we have kxnþ1 pk 6 an ku pk þ ð1 an Þðbn kTxn pk þ ð1 bn Þkxn pkÞ 6 an ku pk þ ð1 an Þkxn pk 6 maxfku pk; kxn pkg .. . 6 maxfku pk; kx0 pkg: Thus, we have the following results which is a new result even in a Hilbert space since its proof does not require limn!1 kxn Txnk = 0. Theorem 4.1. Let K be a non-empty closed convex subset of a real Banach space E with a uniformly Gaˆteaux differentiable norm, and T : K ! K be a non-expansive mapping with a fixed point. Suppose that Kmin ˙ F(T) 5 ;, and {xn} is given by (4.1). Let {an} and {bn} are sequences in (0, 1) satisfying the conditions (C1) limn!1 an = 0 and (B1) limn!1 bn = 0. Then as n ! 1, {xn} converges strongly to some fixed point p of T. Proof. Take p 2 Kmin ˙ F(T). The boundedness of {xn} assures that so is {Txn} since kTxn pk 6 kxn pk. Therefore, hereinafter we may assume for all nP1, max{kTxn xnk, kxn uk, kxn pk} 6 M. Since kxnþ1 xn k 6 an ku xn k þ ð1 an Þbn kxn Txn k 6 ðan þ bn ÞM; then limn!1 kxn+1 xnk = 0 by the conditions (C1) and (B1). As p 2 Kmin ˙ F(T), then ln kxnþ1 pk2 ¼ inf ln kxn yk2 y2K
and p ¼ Tp:
Using the same argumentation of Theorem 3.1, we have limsuphu p; J ðxnþ1 pÞi 6 0: n!1
ð4:2Þ
Y. Song / Applied Mathematics and Computation 198 (2008) 721–728
727
Using Eq. (4.1), we make the following estimates: 2
kxnþ1 pk ¼ ð1 an Þhbn ðTxn pÞ þ ð1 bn Þðxn pÞ; J ðxnþ1 pÞi þ an hu p; J ðxnþ1 pÞi 6 ð1 an Þkxn pkkxnþ1 pk þ an hu p; J ðxnþ1 pÞi 2
6 ð1 an Þ
2
kxn pk þ kxnþ1 pk þ an hu p; J ðxnþ1 pÞi: 2
Thus, 2
2
kxnþ1 pk 6 ð1 an Þkxn pk þ 2an hu p; J ðxnþ1 pÞi: Hence, noting Eq. (4.2), an application of Lemma 2.3 onto Eq. (4.3), the conclusion desired is reached.
ð4:3Þ h
Clearly, we can obtain the following results which develops and complements the corresponding ones of Chidume and Chidume [1], Kim and Xu [4], Song and Chen [20] and many other existing literatures. Corollary 4.2. Let E be a real reflexive Banach space with the fixed point property and a uniformly Gaˆteaux differentiable norm or be a real reflexive and strictly convex Banach space E with a uniformly Gaˆteaux differentiable norm, and K, T,{xn},{an},{bn} be as Theorem 4.1. Suppose that Kmin ˙ F(T) 5 ;, then as n ! 1, {xn} converges strongly to some fixed point p of T. Corollary 4.3. Let E, K, T,{xn} be as Corollary 4.2. Let {an} and {bP n} be a sequence of positive P1 numbers in [0, 1] 1 satisfying the following conditions: (i) a ! 0, b ! 1; (ii) a ¼ 1; (iii) n n n n¼1 n¼1 janþ1 an j < þ1; P1 jb b j < þ1. Then {x } converges strongly to p 2 K ˙ F(T) as n ! 1. n min n n¼1 nþ1 Proof. It follows from Kim and Xu [4, Theorem 1] that limn!1 kxn+1 xnk = 0 and limn!1 kxn+1-Txnk = 0. Example 3.3 or 3.4 assures Kmin \ F(T) 5 ;. Proceeding as in Theorem 4.1, the conclusion is obtained. h Corollary 4.4. Let E, K, T, {xn} be as Corollary 4.2.Suppose that {an} and {bn} are two sequences in (0, 1) satisfying the conditions (C1), (C2) and (B2) 0 < lim inf n!1 bn 6 lim inf n!1 bn < 1. Then {xn} converges strongly to p 2 Kmin ˙ F(T) as n ! 1. Proof. It follows from Song and Chen [20, Theorem 3.2] that limn!1kxn+1 xnk = 0 and limn!1kxn+1 Txnk = 0. Example 3.3 or 3.4 assures Kmin \ F(T) 5 ;. Following as in Theorem 4.1, the conclusion is obtained. h References [1] C.E. Chidume, C.O. Chidume, Iterative approximation of fixed points of non-expansive mappings, J. Math. Anal. Appl. 318 (2006) 288–295. [2] K. Deimling, Non-linear Functional Analysis, Springer-Verlag, New Tork, 1988. [3] B. Halpern, Fixed points of non-expansive maps, Bull. Am. Math. Soc. 73 (1967) 957–961. [4] T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Non-linear Anal. 61 (2005) 51–60. [5] P.L. Lions, Approximation de points fixes de contraction, C.R. Acad. Sci. Se´r A-B, Pairs, 284, 1977, pp. 1357–1359. [6] R.E. Megginson, An Introduction to Banach Space Theory, Springer-Verlag, New Tork, 1998. [7] S. Reich, Asymptotic behavior of contractions in Banach spaces, J. Math. Anal. Appl. 44 (1973) 57–70. [8] S. Reich, Some fixed point problems, Atti. Accad. Maz. Lincei. 57 (1974) 194–198. [9] S. Reich, Approximating zeros of accretive operators, Proc. Am. Math. Soc. 51 (1975) 381–384. [10] S. Reich, Constructive techniques for accretive and monotone operators, in: Applied Non-linear Analysis, Academic Press, New York, 1979, pp. 335–345. [11] S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal Appl. 75 (1980) 287– 292. [12] S. Reich, Some problems and results in fixed point theory, Contempor. Math. 21 (1983) 179–187. [13] S. Reich, Approximating fixed points of non-expansive mappings, Panam. Math. J. 4 (2) (1994) 23–28. [14] R. Wittmann, Approximation of fixed points of non-expansive mappings, Arch. Math. 59 (1992) 486–491.
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[15] N. Shioji, W. Takahashi, Strong convergence of approximated sequences for non-expansive mappings in Banach spaces, Proc. Am. Math. Soc. 125 (1997) 3641–3645. [16] Yisheng Song, Rudong Chen, Strong convergence theorems on an iterative method for a family of finite non-expansive mappings, Appl. Math. Comput. 180 (2006) 275–287. [17] Yisheng Song, Rudong Chen, Viscosity approximation methods for non-expansive nonself-mappings, J. Math. Anal. Appl. 321 (2006) 316–326. [18] Yisheng Song, Rudong Chen, Iterative approximation to common fixed points of non-expansive mapping sequences in reflexive Banach spaces, Non-linear Anal. 66 (2007) 591–603. [19] Yisheng Song, Rudong Chen, Haiyun Zhou, Viscosity approximation methods for non-expansive mapping sequences in Banach spaces, Non-linear Anal. 66 (2007) 1016–1024. [20] Yisheng Song, Rudong Chen, Strong convergence of an iterative method for non-expansive mappings, Math Nachr, accepted for publication. [21] W. Takahashi, Non-linear Functional Analysis – Fixed Point Theory and its Applications, Yokohama Publishers Inc, Yokohama, 2000 (in Japanese). [22] W. Takahashi, Y. Ueda, On Reich’s strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984) 546–553. [23] H.K. Xu, Iterative algorithms for non-linear operators, J. Lond. Math. Soc. 66 (2002) 240–256.
Applied Mathematics and Computation 198 (2008) 721–728 www.elsevier.com/locate/amc
A new sufficient condition for the strong convergence of Halpern type iterations Yisheng Song College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, PR China
Abstract The aim of this work is to give a new sufficient condition of the strong convergence of the Halpern type iteration for a non-expansive self-mapping defined on a Banach space with a uniformly Gaˆteaux differentiable norm. Several examples satisfying our condition are presented. Our results not only remove the restriction of the space with the fixed point property for non-expansive self-mappings, but also get rid of the dependence on the convergence of the implicit anchor-like continuous path zt in the proof. 2007 Elsevier Inc. All rights reserved. Keywords: Non-expansive mappings; Halpern type iteration; Uniformly Gaˆteaux differentiable norm
1. Introduction Let E be a real Banach space with dual E* and K a non-empty closed convex subset of E. Let J denote the normalized duality mapping from E into 2E given by J(x) = {f 2 E*, hx, fi = kxkkfk, kxk = kfk} "x 2 E, where hÆ, Æi denotes the generalized duality pairing. We say F(T) = {x 2 E; Tx = x}, the set of all fixed point for a mapping T and N, the set of all positive integer. We write xn N x (respectively xn * x) to indicate that the sequence xn weakly (respectively weak*) converges to x; as usual xn ! x will symbolize strong convergence. Let T be a non-expansive mappings from K into itself (a mapping T : K ! K is said to be non-expansive if kTx Tyk 6 kx yk "x, y 2 K). In 1967, Halpern [3] was the first who introduced the following Halpern type iteration in a Hilbert spaces: for any initialization x0 2 K and any anchor u 2 K, an 2 [0, 1], xnþ1 ¼ an u þ ð1 an ÞTxn ;
n P 0: P1
ð1:1Þ
He pointed out that the control conditions ðC1Þlimn!1 an ¼ 0 and ðC2Þ n¼1 an ¼ 1 are necessary for the convergence of the Halpern type iteration (1.1) to a fixed point of T. At the same time, he put forth the following open problem, which was put forward by Reich in Refs. [8,12] also. Question 1. Are sufficient the control conditions (C1) and (C2) for the convergence of the Halpern type iteration to a fixed point of T?
E-mail addresses: [email protected], [email protected] 0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.09.010
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Subsequently, many mathematic workers carefully researched the problem. For example, Lions [5] investigated the problem and established a strong convergence theorem under the control conditions (C1), (C2) and nþ1 (C3) limn!1 anaa ¼ 0. In 1980, Reich [11] studied the iteration scheme (1.1) with a specific sequence of param2 nþ1
eters {an} = {na}, where a 2 (0, 1). However, both Lions’ and Reich’s control conditions excluded the natural choice of an = 1/(n + 1). In 1992, Wittmann [14] studied the iterative scheme (1.1) in a Hilbert space, and obtained the strong convergence of the iteration. In particular,P he proved a strong convergence theorem [14, Theorem 2] under the control conditions (C1), (C2) and (C4) 1 n¼1 jan anþ1 j < 1. The main advantage of Wittmann’s convergence theorem lies in the fact that it allows 1/(n + 1) to be a candidate of an. An analogy of Wittmann’s result was first proved outside Hilbert space by Reich in [13] for spaces with a weakly sequentially continuous duality mapping in 1994. In 1997, Shioji and Takahashi [15] also extended Wittmann’s result to a real Banach space with a uniformly Gaˆteaux differentiable norm. In 2002, Xu [23] showed Wittmann’s result in a uniformly smooth Banach space using the conditions (C1), (C2) and (C5) limn!1 aan1 ¼ 1. We observe n from above results that the aim of the additive conditions (C4) (or (C5)) is to guarantee limn!1 kxn xn+1k = 0, and that limn!1 kxn Txnk = 0. In nature, they proved the following result. Theorem 1.1. Let K be a non-empty closed convex subset of a Banach space E, and T : K ! K be a non-expansive mapping. Assumed that {xn} is given by (1.1) with {an} (0, 1). If F(T) 5 ;, then (i) {xn} is bounded. If, in addition, {an} satisfies (C2) together with one of (C4) and (C5), then (ii) limn!1 kxn xn+1k = 0. Suppose that, {an} fulfils (C1), then (iii) limn!1 kxn+1 Txnk = 0. By the means of Theorem 1.1 along with the fact that the implicit anchor-like continuous path zt, defined by zt = tu + (1t)Tzt, strongly converges to a fixed point of T, the strong convergence of {xn} is obtained. That is, the strong convergence of {xn} depends upon the convergence of {zt} even if the conditions (C1) and (C2) together with one of the conditions (C4) and (C5) are satisfied. Thus, it will be a very interesting to establish a sufficient condition to assure the strong convergence of the explicit iteration {xn} which is independent of the strong convergence of zt. In this paper, we will study and explore the condition reached the above desire. Let {xn} be a bounded sequence defined by (1.1) and ln be any Banach limit (see Preliminaries). Set 2
uðyÞ ¼ ln kxn yk ; then u(y) is convex and continuous, and u(y) ! 1 as kyk ! 1. If E is reflexive, there exists z 2 K such that u(z) = infy2Ku(y) (see [21, Theorem 1.3.11]). So the set K min ¼ z 2 K; uðzÞ ¼ inf uðyÞ 6¼ ;: y2K
Clearly, Kmin is closed convex by the convexity and continuity of u(y). It is easily seen that if xn ! x* 2 K (n ! 1), then x* 2 Kmin since u(x*) = 0. In addition, suppose that the condition (C1) is satisfied, then x* 2 F(T) as kxnþ1 Tx k 6 an ku Tx k þ ð1 an Þkxn x k ! 0: Namely, x* 2 Kmin \ F(T) is necessary for the convergence of the Halpern type iteration (1.1) to x*. Then we spontaneously put forth the following problem. Question 2. Are sufficient the control conditions Kmin \ F(T) 5 ; and (C2) for the convergence of the Halpern type iteration to a fixed point of T? Our main aim in this paper is to not only give a sufficient condition of the strong convergence of the Halpern type iteration {xn} for a non-expansive self-mapping defined on a Banach space with a uniformly Gaˆteaux differentiable norm, but also make the strong convergence of the Halpern type iteration {xn} to shake off the restriction and dependence on the implicit anchor-like continuous path zt. Several examples satisfying our condition Kmin \ F(T) 5 ; are presented. As application, we also investigate the modified Mann’s iteration (1.2) for finding a fixed point of non-expansive mapping T xnþ1 ¼ an u þ an ðð1 bn ÞTxn þ bn xn Þ:
ð1:2Þ
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2. Preliminaries Let S(E): = {x 2 E; kxk = 1} denote the unit sphere of a Banach space E. The space E is said to have (i) a Gaˆteaux differentiable norm (we also say that E is smooth), if the limit ðÞ
lim t!0
kx þ tyk kxk t
exists for each x, y 2 S(E); (ii) a uniformly Gaˆteaux differentiable norm, if for each y in S(E), the limit (*) is uniformly attained for x 2 S(E); (iii) a Fre´chet differentiable norm, if for each x 2 S(E), the limit (*) is attained uniformly for y 2 S(E); (iv) a uniformly Fre´chet differentiable norm (we also say that E is uniformly smooth), if the limit (*) is attained uniformly for (x, y) 2 S(E) · S(E). (v) fixed point property for non-expansive self-mappings, if each non-expansive self-mapping defined on any bounded closed convex subset K of E has at least a fixed point. A Banach space E is said to be (vi) strictly convex if kxk ¼ kyk ¼ 1; x 6¼ y implies
kx þ yk < 1; 2
(vii) uniformly convex if for all e 2 [0, 2], $de > 0 such that kxk ¼ kyk ¼ 1 implies
kx þ yk < 1 de whenever kx yk P e: 2
The following results is well known which are found in Refs. [2,21,6]: the normalized duality mapping J in a Banach space E with a uniformly Gaˆteaux differentiable norm is single-valued and strong-weak* uniformly continuous on any bounded subset of E; each uniformly convex Banach space E is reflexive and strictly convex and has fixed point property for non-expansive self-mappings; every uniformly smooth Banach space E is a reflexive Banach space with a uniformly Gaˆteaux differentiable norm and has fixed point property for nonexpansive self-mappings. In the proof of our main theorems, we also need the following definitions and results. Let l be a continuous linear functional on l1 satisfying klk = 1 = l(1). Then we know that l is a mean on N if and only if inffan ; n 2 N g 6 lðaÞ 6 supfan ; n 2 N g for every a = (a1, a2, . . .)2 l1. According to time and circumstances, we ln(an) instead of l(a). A mean l on N is called a Banach limit if ln ðan Þ ¼ ln ðanþ1 Þ for every a = (a1, a2, . . .) 2 l1. Lemma 2.1 [22, Lemma 1]. Let C be a nonempty closed convex subset of a Banach space E with a uniformly Gaˆteaux differentiable norm. Let {xn} be a bounded sequence of E and let ln be a Banach limit and z 2 C. Then 2
ln kxn zk ¼ min ln kxn yk
2
y2C
if and only if ln hy z; J ðxn zÞi 6 0;
8y 2 C:
Lemma 2.2 [15, Proposition 2]. Let a is a real number and (x0, x1, . . .) 2 l1 such that lnxn 6 a for all Banach Limits. If lim supn!1 (xn+1 xn) 6 0, then lim supn!1 xn 6 a. In the sequel, we shall make use of the following lemma which can be found in the existing literature [23]. Furthermore, a variant of Lemma 2.3 has already been used by Reich in [10, Theorem 1].
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Lemma 2.3 [23, Lemma 2.5]. Let {an} be a sequence of non-negative real numbers satisfying the property anþ1 6 ð1 cn Þan þ cn bn ; n P 0: Suppose that {cn} (0, 1) and {bn} are real number sequence such that (i) Then {an} converges to zero, as n ! 1.
P1
n¼0 cn
¼ 1; (ii) lim supn!1 bn 6 0.
3. Strong convergence of Halpern type iteration In this section, let K be a non-empty closed convex subset of a real Banach space E, and T : K ! K be a non-expansive mapping with a fixed point. For arbitrary initial value x0 2 K and fixed anchor u 2 K, define iteratively a sequence {xn} as follows: xnþ1 ¼ an u þ ð1 an ÞTxn ;
ð3:1Þ
where {an} is a sequence in (0, 1). Clearly, following Theorem 1.1, {xn} is bounded. Then there exists r > 0 sufficiently large such that xn 2 Br ¼ fx; kxk 6 rg8n 2 N: Furthermore, the set K \ Br is a bounded closed and convex non-empty subset of E. If we define a map u : E ! R by 2
uðyÞ ¼ ln kxn yk ; then u(y) is convex and continuous, and u(y) ! 1 as kyk ! 1. If E is reflexive, there exists z 2 K such that u(z) = infy2Ku(y) (see [21, Theorem 1.3.11]). So the set K min ¼ z 2 K; uðzÞ ¼ inf uðyÞ 6¼ ;: y2K
Clearly, Kmin is closed convex by the convexity and continuity of u(y). Theorem 3.1. Let K be a non-empty closed convex subset of a real Banach space E with a uniformly Gaˆteaux differentiable norm, and T : K ! K be a non-expansive mapping with a fixed point. Suppose that P Kmin ˙ F(T) 5 ;, and {an} is a sequence in (0, 1) satisfying the conditions ðC2Þ þ1 n¼1 an ¼ þ1 and limn!1 kxn+1 xnk = 0. Then as n ! 1, {xn} given by (3.1) converges strongly to some fixed point p of T. Proof. Take p 2 Kmin ˙ F(T). Then ln kxn pk2 ¼ inf ln kxn yk2
and
y2K
p ¼ Tp:
It follows from Lemma 2.1 that for u 2 K, ln hu p; J ðxn pÞi 6 0: Since limn!1 kxn+1 xnk = 0, then it follows from the norm-weak* uniformly continuity of the duality mapping J that lim ðhu p; J ðxnþ1 pÞi hu p; J ðxn pÞiÞ ¼ 0:
n!1
Hence, the sequence {hu p, J(xn p)i} satisfies the conditions of Lemma 2.2. As a result, we must have lim suphu p; J ðxnþ1 pÞi 6 0:
ð3:2Þ
n!1
Using Eq. (3.1), we make the following estimates: 2
kxnþ1 pk ¼ ð1 an ÞhTxn p; J ðxnþ1 pÞi þ an hu p; J ðxnþ1 pÞi 6 ð1 an ÞkTxn pkkxnþ1 pk þ an hu p; J ðxnþ1 pÞi 2
6 ð1 an Þ
2
kxn pk þ kxnþ1 pk þ an hu p; J ðxnþ1 pÞi: 2
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Thus, 2
2
kxnþ1 pk 6 ð1 an Þkxn pk þ 2an hu p; J ðxnþ1 pÞi:
ð3:3Þ
Hence, noting Eq. (3.2), an application of Lemma 2.3 onto Eq. (3.3), the conclusion desired is reached.
h
Using Theorem 1.1, we can obtain the following result. Theorem 3.2. Let E, K, T, {xn} are as Theorem 3.1. Suppose that {an} satisfies (C2) and one of (C4) P 1 an1 n¼1 jan anþ1 j < 1 and (C5) limn!1 an ¼ 1. If Kmin ˙ F(T) 5 ;, then as n ! 1, {xn} converges strongly to some fixed point p of T. Proof. It follows from Theorem 1.1 that lim kxnþ1 xn k ¼ 0:
n!1
The remainder of the proof follows from Theorem 3.1.
h
Remark 1. It is noted that the assumption of the space with the fixed point property for non-expansive selfmappings is necessary in almost all known results about the convergence of Halpern type iteration such that uniformly convex Banach space, uniformly smooth Banach space, Banach space with Opial’s condition (see [3,5,7–11,14,13,15–19,23] and the literature cited there), whereas the proof of our results does not depend necessarily on them. On the other hand, our proof also shakes off the dependence on limn!1 kxn Txnk = 0 and the convergence of the implicit anchor-like continuous path zt, defined by zt = tu+(1t)Tzt, and only need Kmin ˙ F(T) 5 ;. Next, we give several examples satisfy our condition Kmin ˙ F(T) 5 ; which is weaker than the condition (C1) limn!1 an = 0 since (C1) means lim kxnþ1 Txn k ¼ lim an ku Txn k ¼ 0:
n!1
n!1
Example 3.3. Let K be a non-empty closed convex subset of a real reflexive Banach space E with fixed point property for non-expansive self-mappings, and T : K ! K be a non-expansive mapping with a fixed point, and {xn} be defined by (3.1) with limn!1 kxn+1 Txnk = 0. Then Kmin ˙ F(T) 5 ;. Proof. The reflexivity of E assures Kmin 5 ;. Then Kmin is a non-empty bounded and closed convex subset of E by the definition of Kmin. Since limn!1 kxn+1 Txnk = 0, then It follows from the property of Banach limit ln and the definition of the function u that for all z 2 Kmin, we have 2
2
2
2
uðTzÞ ¼ ln kxn Tzk ¼ ln kxnþ1 Tzk 6 ln ðkxnþ1 Txn k þ kTxn TzkÞ 6 ln kxn zk ¼ uðzÞ: Therefore, Tz 2 Kmin. That is, T(Kmin) Kmin. Thus, there exists x* 2 Kmin such that x* = Tx* since E has the fixed point property for non-expansive self-mappings. That is x* 2 Kmin \ F(T). h Example 3.4. Let K be a non-empty closed convex subset of a reflexive and strictly convex Banach space E, and T : K ! K be a non-expansive mapping with a fixed point, and {xn} be defined by (3.1) with limn!1 kxn+1 Txnk = 0. Then Kmin ˙ F(T) 5 ;. Proof. It follows from the same argumentation of Example 3.3 that T(Kmin) Kmin. Since F(T)5;, then let y be one of those. From [6, Corollary 5.1.19] and the non-empty closed convexity of Kmin, we get that there exists unique p 2 Kmin such that kp yk ¼ inf x2K min ky xk: By y = Ty and Tp 2 Kmin, we have ky Tpk ¼ kTy Tpk 6 ky pk: Hence p = Tp by the uniqueness of p 2 Kmin. Thus, p 2 Kmin \ F(T). Hence, the following corollary obtained obviously.
h
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Corollary 3.5. Let E be a real reflexive Banach space with fixed point property for non-expansive self-mappings and with a uniformly Gaˆteaux differentiable norm, K, T, {xn} be as Theorem 3.1. Suppose the sequence {an} satisfies (C1) and (C2) and one of (C4) and (C5). Then as n ! 1, {xn} converges strongly to p 2 Kmin \ F(T). Proof. It follows from Theorem 1.1 that limn!1 kxn+1 xnk = 0. Example 3.3 and (C1) assure Kmin \ F(T) 5 ;. Proceeding as in Theorem 3.1, we reach the conclusion. h Using the similar argumentation as Corollary 3.5 (the application of Example 3.3 is replaced by Example 3.4), we obtain the following corollary. Corollary 3.6. Let E be a real reflexive and strictly convex Banach space with a uniformly Gaˆteaux differentiable norm, K, T, {xn}, {an} be as Corollary 3.5. Then as n ! 1, {xn} converges strongly to p 2 Kmin \ F(T). Remark 2. (1) There are many Banach space which satisfy the fixed point property for non-expansive selfmappings in the known Banach spaces. For example, uniformly convex Banach space, uniformly smooth Banach space, reflexive Banach space with normal structure and so on.(2) We remark that Corollary 3.6 is independent of Corollary 3.5. On the one hand, it is easy to find examples of spaces which satisfy the fixed point property for non-expansive self-mappings, which are not strictly convex. On the other hand, it appears to be unknown whether a reflexive and strictly convex Banach space satisfies the fixed point property for nonexpansive self-mappings. 4. Some applications In this section, we shall investigate the modified Mann’s iteration (4.1) for finding a fixed point of nonexpansive mapping T. xnþ1 ¼ an u þ ð1 an Þðbn Txn þ ð1 bn Þxn Þ:
ð4:1Þ
If F(T) 5 ;, then {xn} is bounded. In fact, take p 2 F(T), we have kxnþ1 pk 6 an ku pk þ ð1 an Þðbn kTxn pk þ ð1 bn Þkxn pkÞ 6 an ku pk þ ð1 an Þkxn pk 6 maxfku pk; kxn pkg .. . 6 maxfku pk; kx0 pkg: Thus, we have the following results which is a new result even in a Hilbert space since its proof does not require limn!1 kxn Txnk = 0. Theorem 4.1. Let K be a non-empty closed convex subset of a real Banach space E with a uniformly Gaˆteaux differentiable norm, and T : K ! K be a non-expansive mapping with a fixed point. Suppose that Kmin ˙ F(T) 5 ;, and {xn} is given by (4.1). Let {an} and {bn} are sequences in (0, 1) satisfying the conditions (C1) limn!1 an = 0 and (B1) limn!1 bn = 0. Then as n ! 1, {xn} converges strongly to some fixed point p of T. Proof. Take p 2 Kmin ˙ F(T). The boundedness of {xn} assures that so is {Txn} since kTxn pk 6 kxn pk. Therefore, hereinafter we may assume for all nP1, max{kTxn xnk, kxn uk, kxn pk} 6 M. Since kxnþ1 xn k 6 an ku xn k þ ð1 an Þbn kxn Txn k 6 ðan þ bn ÞM; then limn!1 kxn+1 xnk = 0 by the conditions (C1) and (B1). As p 2 Kmin ˙ F(T), then ln kxnþ1 pk2 ¼ inf ln kxn yk2 y2K
and p ¼ Tp:
Using the same argumentation of Theorem 3.1, we have limsuphu p; J ðxnþ1 pÞi 6 0: n!1
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Using Eq. (4.1), we make the following estimates: 2
kxnþ1 pk ¼ ð1 an Þhbn ðTxn pÞ þ ð1 bn Þðxn pÞ; J ðxnþ1 pÞi þ an hu p; J ðxnþ1 pÞi 6 ð1 an Þkxn pkkxnþ1 pk þ an hu p; J ðxnþ1 pÞi 2
6 ð1 an Þ
2
kxn pk þ kxnþ1 pk þ an hu p; J ðxnþ1 pÞi: 2
Thus, 2
2
kxnþ1 pk 6 ð1 an Þkxn pk þ 2an hu p; J ðxnþ1 pÞi: Hence, noting Eq. (4.2), an application of Lemma 2.3 onto Eq. (4.3), the conclusion desired is reached.
ð4:3Þ h
Clearly, we can obtain the following results which develops and complements the corresponding ones of Chidume and Chidume [1], Kim and Xu [4], Song and Chen [20] and many other existing literatures. Corollary 4.2. Let E be a real reflexive Banach space with the fixed point property and a uniformly Gaˆteaux differentiable norm or be a real reflexive and strictly convex Banach space E with a uniformly Gaˆteaux differentiable norm, and K, T,{xn},{an},{bn} be as Theorem 4.1. Suppose that Kmin ˙ F(T) 5 ;, then as n ! 1, {xn} converges strongly to some fixed point p of T. Corollary 4.3. Let E, K, T,{xn} be as Corollary 4.2. Let {an} and {bP n} be a sequence of positive P1 numbers in [0, 1] 1 satisfying the following conditions: (i) a ! 0, b ! 1; (ii) a ¼ 1; (iii) n n n n¼1 n¼1 janþ1 an j < þ1; P1 jb b j < þ1. Then {x } converges strongly to p 2 K ˙ F(T) as n ! 1. n min n n¼1 nþ1 Proof. It follows from Kim and Xu [4, Theorem 1] that limn!1 kxn+1 xnk = 0 and limn!1 kxn+1-Txnk = 0. Example 3.3 or 3.4 assures Kmin \ F(T) 5 ;. Proceeding as in Theorem 4.1, the conclusion is obtained. h Corollary 4.4. Let E, K, T, {xn} be as Corollary 4.2.Suppose that {an} and {bn} are two sequences in (0, 1) satisfying the conditions (C1), (C2) and (B2) 0 < lim inf n!1 bn 6 lim inf n!1 bn < 1. Then {xn} converges strongly to p 2 Kmin ˙ F(T) as n ! 1. Proof. It follows from Song and Chen [20, Theorem 3.2] that limn!1kxn+1 xnk = 0 and limn!1kxn+1 Txnk = 0. Example 3.3 or 3.4 assures Kmin \ F(T) 5 ;. Following as in Theorem 4.1, the conclusion is obtained. h References [1] C.E. Chidume, C.O. Chidume, Iterative approximation of fixed points of non-expansive mappings, J. Math. Anal. Appl. 318 (2006) 288–295. [2] K. Deimling, Non-linear Functional Analysis, Springer-Verlag, New Tork, 1988. [3] B. Halpern, Fixed points of non-expansive maps, Bull. Am. Math. Soc. 73 (1967) 957–961. [4] T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Non-linear Anal. 61 (2005) 51–60. [5] P.L. Lions, Approximation de points fixes de contraction, C.R. Acad. Sci. Se´r A-B, Pairs, 284, 1977, pp. 1357–1359. [6] R.E. Megginson, An Introduction to Banach Space Theory, Springer-Verlag, New Tork, 1998. [7] S. Reich, Asymptotic behavior of contractions in Banach spaces, J. Math. Anal. Appl. 44 (1973) 57–70. [8] S. Reich, Some fixed point problems, Atti. Accad. Maz. Lincei. 57 (1974) 194–198. [9] S. Reich, Approximating zeros of accretive operators, Proc. Am. Math. Soc. 51 (1975) 381–384. [10] S. Reich, Constructive techniques for accretive and monotone operators, in: Applied Non-linear Analysis, Academic Press, New York, 1979, pp. 335–345. [11] S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal Appl. 75 (1980) 287– 292. [12] S. Reich, Some problems and results in fixed point theory, Contempor. Math. 21 (1983) 179–187. [13] S. Reich, Approximating fixed points of non-expansive mappings, Panam. Math. J. 4 (2) (1994) 23–28. [14] R. Wittmann, Approximation of fixed points of non-expansive mappings, Arch. Math. 59 (1992) 486–491.
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[15] N. Shioji, W. Takahashi, Strong convergence of approximated sequences for non-expansive mappings in Banach spaces, Proc. Am. Math. Soc. 125 (1997) 3641–3645. [16] Yisheng Song, Rudong Chen, Strong convergence theorems on an iterative method for a family of finite non-expansive mappings, Appl. Math. Comput. 180 (2006) 275–287. [17] Yisheng Song, Rudong Chen, Viscosity approximation methods for non-expansive nonself-mappings, J. Math. Anal. Appl. 321 (2006) 316–326. [18] Yisheng Song, Rudong Chen, Iterative approximation to common fixed points of non-expansive mapping sequences in reflexive Banach spaces, Non-linear Anal. 66 (2007) 591–603. [19] Yisheng Song, Rudong Chen, Haiyun Zhou, Viscosity approximation methods for non-expansive mapping sequences in Banach spaces, Non-linear Anal. 66 (2007) 1016–1024. [20] Yisheng Song, Rudong Chen, Strong convergence of an iterative method for non-expansive mappings, Math Nachr, accepted for publication. [21] W. Takahashi, Non-linear Functional Analysis – Fixed Point Theory and its Applications, Yokohama Publishers Inc, Yokohama, 2000 (in Japanese). [22] W. Takahashi, Y. Ueda, On Reich’s strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984) 546–553. [23] H.K. Xu, Iterative algorithms for non-linear operators, J. Lond. Math. Soc. 66 (2002) 240–256.