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Weak and strong convergence theorems for a finite family of non-Lipschitzian nonself mappings in Banach spaces
Fixed Point Theory and Applications volume 2013, Article number: 170 (2013)
Abstract
In this paper, several weak and strong convergence theorems are established for a new modified iteration with errors for a finite family of nonself mappings which are asymptotically nonexpansive in the intermediate sense in Banach spaces. Mann-type, Ishikawa-type and Noor-type iterations are covered by this new iteration scheme. Our convergence theorems improve, unify and generalize many important results in the current literature.
MSC:47H10, 47H09, 46B20.
1 Introduction
Fixed-point iteration processes for nonexpansive and asymptotically nonexpansive mappings including Mann-type and Ishikawa-type iterations have been studied extensively by many authors (see [1–23] and the references cited therein). In 1991, Schu [14] considered the modified Mann iteration process for an asymptotically nonexpansive map. Later, Tan and Xu [16] studied the modified Ishikawa iteration process for an asymptotically nonexpansive map. Noor, in 2000, introduced a three-step iterative scheme and studied the approximate solutions of variational inclusion in Hilbert spaces [11]. Later, Cho et al. [4], Xu and Noor [18] studied weak and strong convergence theorems for the three-step Noor iterations with errors for asymptotically nonexpansive mappings in a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable. Takahashi and Tamura [15], Shahzad [24] dealt with the iterative scheme for a pair of nonexpansive and asymptotically nonexpansive mappings in a uniformly convex Banach space. In 2006, Plubtieng et al. [12] studied a class of three-step iterative scheme, for three asymptotically nonexpansive mappings, in a uniformly convex Banach space satisfying Opial’s condition. In 2007, Fukhar-ud-dina and Khan [5] studied the scheme for three nonexpansive mappings in a uniformly convex Banach space which has Opial’s condition or which has a Fréchet differentiable norm or whose dual space has the Kadec-Klee property. Also in 2007, Chidume and Bashir Ali [1] introduced the iterative scheme for a finite family of asymptotically nonexpansive mappings and obtained the weak and strong convergence theorems in a Banach space whose dual space satisfies the Kadec-Klee property.
In most of these papers, the map T has been assumed to map C into itself. If, however, C is a proper subset of a Banach space X and T maps C into X (as is the case in many applications), then may not be well defined. One method that has been used to overcome this is to introduce a retraction . Recent results on the approximation of fixed points of nonexpansive and asymptotically nonexpansive nonself mappings can be found in [24–33] and references contained therein.
In 2003, Chidume et al. [2] introduced the following modified Mann iteration process and got the weak and strong convergence theorems for an asymptotically nonexpansive nonself mapping:
Recently, Wang [32] generalized the above iteration process as follows: ,
In 2007, Chidume and Bashir Ali [27] introduced the following iteration process for a finite family of asymptotically nonexpansive nonself mappings: and
They proved strong convergence theorems in uniformly convex Banach spaces and gave the weak convergence theorem in uniformly convex Banach spaces that satisfy Opial’s condition or have a Fréchet differentiable norm. They also gave the weak convergence theorem for nonexpansive nonself mappings in uniformly convex Banach spaces whose dual spaces have the Kadec-Klee property (see [27]).
The concept of asymptotically nonexpansive nonself mappings in the intermediate sense was introduced by Chidume et al. [28] as an important generalization of asymptotically nonexpansive self-mappings in the intermediate sense.
Definition 1.1 Let C be a nonempty subset of a Banach space X. Let be a nonexpansive retraction of X onto C. A nonself mapping is called asymptotically nonexpansive in the intermediate sense if T is continuous and the following inequality holds:
It should be noted that in [28, 31, 34], an asymptotically nonexpansive mapping in the intermediate sense is required to be uniformly continuous. In Definition 1.1, we assume the continuity of T instead of uniform continuity. Chidume et al. [28], Plubtieng and Wangkeeree [31], Kim and Kim [34] gave strong convergence theorems for a uniformly continuous mapping which is asymptotically nonexpansive in the intermediate sense in uniformly convex Banach spaces if the mapping is completely continuous. Also, Chidume et al. [28] gave the weak convergence theorem for such a mapping in a uniformly convex Banach space whose dual space has the Kadec-Klee property. However, as we know, it remains open whether the weak convergence theorem of a multi-step iteration process with errors for a finite family of continuous nonself mappings which are asymptotically nonexpansive in the intermediate sense holds in a uniformly convex Banach space which satisfies Opial’s condition or whose dual space has the Kadec-Klee property. Since the asymptotically nonexpansive mappings in the intermediate sense are non-Lipschitzian and Bruck’s lemma [35] does not extend beyond Lipschitzian mappings, new techniques are needed for this more general case. It is our purpose in this paper to study the following iteration process with errors for approximating common fixed points of a finite family of nonself mappings which are asymptotically nonexpansive in the intermediate sense:
In Section 3, using the technique established in [33], we first give some weak convergence theorems of the iterative scheme (1.1) for a finite family of nonself mappings which are asymptotically nonexpansive in the intermediate sense in a uniformly convex Banach space which satisfies Opial’s condition or whose dual space has the Kadec-Klee property. We also establish some strong convergence theorems if one member of the finite family of mappings satisfies a condition weaker than complete continuity. Our results extend and improve the recently announced ones [1, 2, 4, 5, 7, 9, 12, 15, 16, 18, 24, 27, 28, 31, 32, 34] and many others.
2 Preliminaries
Let C be a nonempty closed convex subset of a Banach space X. Recall that a Banach space X is said to be uniformly convex if, for each , the modulus of convexity of X given by
satisfies the inequality for all . We say that X has the Kadec-Klee property if, for every sequence , whenever with , it follows that . We would like to remark that a reflexive Banach space X with a Fréchet differentiable norm implies that its dual has the Kadec-Klee property, while the converse implication fails [36].
Recall that a Banach space X is said to satisfy Opial’s condition [37] if and implies that
A subset C of X is said to be a retract if there exists a continuous mapping such that for all . Every closed convex subset of a uniformly convex Banach space is a retract. A mapping is said to be a retraction if . It follows that if a map P is a retraction, then for all y in the range of P.
Lemma 2.1 [17]
Let the nonnegative number sequences and satisfy
If , then exists.
Lemma 2.2 [14]
Suppose that X is a uniformly convex Banach space and for all positive integers n, . If and are two sequences of X such that , and
hold for some . Then .
Lemma 2.3 [6]
Let X be a uniformly convex Banach space. If , and , then for all ,
Lemma 2.4 (Demiclosedness principle for a nonself-map [38])
Let C be a nonempty closed convex subset of a uniformly convex Banach space X and let be a nonself mapping which is continuous and asymptotically nonexpansive in the intermediate sense. If is a sequence in C converging weakly to x and
then , i.e., .
Lemma 2.5 [36]
Let X be a reflexive Banach space whose dual has the Kadec-Klee property. Let be a bounded sequence in X and . Suppose that
exists for all , then , where denotes the set of weak limit points of , i.e., .
3 Main results
In this section, let X be a uniformly convex Banach space and let C be a nonempty closed convex subset of X. Let be a nonexpansive retraction from X onto C. Let be a finite family of continuous nonself mappings which are asymptotically nonexpansive in the intermediate sense, then we can suppose that
Hence , and for all ,
For a given , we can define the sequence by
where , , are in with , , and are bounded sequences in X, .
We start our investigation with the following lemmas, which are preparation for the proofs of the main results of this section. In the following, we always assume that and the set of common fixed points of is nonempty, i.e.,
Lemma 3.1
exists for each .
Proof Let . Since are bounded, we can set
Then
Hence we can get
and
Thus we obtain
Set , then and
By Lemma 2.1, we have that
exists. This completes the proof. □
Lemma 3.2
Proof By Lemma 3.1, we have that exists. If , then it is obvious to see that the conclusion holds. In the following, we assume that . According to (3.2), (3.3) and (3.4), we can get
Then, for any ,
and hence the sequences are bounded. By (3.5), we can obtain
We also can see
and
It follows from Lemma 2.2 and
that
Combining it with
we obtain
Thus , according to (3.4), we have
Noting
and
by Lemma 2.2 again, we have
Similarly, we can get
Therefore, by (3.1) and (3.7),
and similarly, we can have
It follows from the inequality (3.7) and
that
Thus, for any fixed m, we have and
This implies
Therefore,
Noting (3.8) and
we can see
Thus, for any fixed m,
which means
Therefore,
By the same argument, we can get
This completes the proof. □
Define the operator by
where . Then and for all , we have
and
For any , we get
and
Set , then and for any ,
We also need the following lemma, which plays a crucial role in dealing with the case of the iteration with errors. It is easy to see if for all and all .
Lemma 3.3
Proof By (3.10), (3.11) and
we get
Then fixing n and taking the limsup for m, we obtain
Thus,
This completes the proof. □
Lemma 3.4 Let and , then
exists.
Proof It follows from Lemma 3.1 that exists. If or , then the conclusion holds. In the following, we assume that and . Then, for any , there exists () such that
where δ is the modulus of convexity of the norm. Hence there exists a positive integer such that for all ,
and
Now we claim that for all ,
Otherwise, we can suppose that
for some m. Put , and , then by (3.12), (3.14) and (3.15),
and
We also have
and
So, by using Lemma 2.3, we get
and then by (3.13), (3.15) and (3.16), we have
This contradicts (3.14). Thus we can conclude that for all , (3.17) holds. Hence, for all ,
For any fixed , we can take the limsup for m and obtain
Hence we have
Since is arbitrary, this implies that
exists. This completes the proof. □
Remark 3.1 If the mappings are asymptotically nonexpansive, we can use Bruck’s lemma [35] to prove Lemma 3.4. While Bruck’s lemma is not valid for non-Lipschitzian mappings, we must introduce some new techniques to establish a similar inequality. As we haven seen, we use mainly the technique of the modulus of convexity and our proof is straightforward.
Now we can prove the weak convergence theorem of the iterative scheme (3.1).
Theorem 3.1 Let C be a nonempty closed convex subset of a uniformly convex Banach space X which satisfies Opial’s condition or whose dual has the Kadec-Klee property. Let be a nonexpansive retraction from X onto C. Let be a finite family of nonself mappings which are asymptotically nonexpansive in the intermediate sense with and the sequences satisfying . Let be defined by
where , , are in with , , and are bounded sequences in X, . Then converges weakly to a common fixed point of .
Proof It suffices to prove that the set is a singleton. Since X is reflexive and is bounded, we obtain . Assuming that , in the following, we need to show . First, by Lemma 2.4 and Lemma 3.2, we know . Second, on the one hand, if has the Kadec-Klee property, then from Lemma 3.4 and Lemma 2.5, we can get . On the other hand, if X satisfies Opial’s condition, we assume that and two subsequences and in such that and . Hence by Opial’s condition and Lemma 3.1, we get
This contraction implies . This completes the proof. □
Remark 3.2 Theorem 3.1 generalizes and improves many recent important results. For instance, if and is a uniformly continuous mapping which is asymptotically nonexpansive in the intermediate sense, then we can get Theorem 3.13 in [28]. If are asymptotically nonexpansive nonself mappings and (1) , then we can obtain Theorem 3.6 in [27]; (2) , , then we can get Theorem 3.4 in [1]; (3) and , then we can get Theorem 3.10 in [2], Theorem 2.1 in [4], Theorems 3.1-3.2 in [7], Theorem 1 in [9], Theorem 2.9 in [12], Theorem 3.3 in [15], Theorems 3.1-3.2 in [16], Theorem 3.5 in [32] and many others.
If is a family of nonexpansive mappings, we can have the following theorem, which is an extension of Theorem 4.1 in [5], Theorem 1 in [9], Theorem 3.2 in [15], Theorem 3.5 and Theorem 4.1 in [24], Theorem 3.9 and Theorem 4.2 in [27] and others. The proof is immediate corollaries of our lemmas and Theorem 3.1.
Theorem 3.2 Let C be a nonempty closed convex subset of a uniformly convex Banach space X which satisfies Opial’s condition or whose dual has the Kadec-Klee property. Let be a nonexpansive retraction from X onto C. Let be a finite family of nonself nonexpansive mappings with . Let be defined by
where , , are in with , , and are bounded sequences in X, . Then converges weakly to a common fixed point of .
Now we can give the strong convergence theorem of the scheme (3.1).
Theorem 3.3 Let C be a nonempty closed convex subset of a uniformly convex Banach space X and let be a nonexpansive retraction from X onto C. Let be a finite family of nonself mappings which are asymptotically nonexpansive in the intermediate sense and be as in Theorem 3.1. Then converges strongly to a common fixed point of if and only if
where F denotes the set of common fixed points of , i.e., .
Proof We only need to show the sufficiency. If , then for any , there exists a positive integer such that for all ,
Hence, for any , there exists such that . Therefore, for any , by (3.6),
which implies that is a Cauchy sequence in C and then it must converge to some point in C. Set , since and F is closed, we get . This completes the proof. □
In the following, we shall give a sufficient condition to ensure the strong convergence of the iterative sequence (3.1). We need the following notions. Recall that a finite family of nonself mappings with satisfies Condition () if there exists a nondecreasing function with , , for , such that at least one of the satisfies condition (), i.e.,
for at least one , , where .
A mapping is said to be demicompact if, for any bounded sequence in C such that converges, there exists a subsequence, say of , such that converges strongly to some point in C. T is said to be completely continuous if it is continuous and for every bounded sequence , there exists a subsequence, say of , such that the sequence converges to some element of the range of T.
It is well known that every continuous and demicompact mapping must satisfy condition () since every completely continuous mapping is continuous and demicompact so that it satisfies condition (). Therefore, the condition () is weaker than the demicompactness and complete continuity (see [27]). Next we shall give several strong convergence theorems in uniformly convex Banach spaces if one member of the finite family of asymptotically nonexpansive in the intermediate sense mappings satisfies condition ().
Theorem 3.4 Let C be a nonempty closed convex subset of a uniformly convex Banach space X and let be a nonexpansive retraction from X onto C. Let be a finite family of uniformly continuous nonself mappings which are asymptotically nonexpansive in the intermediate sense and let be as in Theorem 3.1. If the family satisfies condition (), then converges strongly to a common fixed point of .
Proof Without loss of generality, we assume that satisfies condition (), i.e.,
Hence we have
Then by (3.9), the uniform continuity of and
we derive
By (3.6), we have for all , , where . Hence
Then it follows from Lemma 2.1 that exists. Hence, by (3.18) and (3.19), we see and therefore,
By Theorem 3.3, we can get what we desired. This completes the proof. □
Remark 3.3 From Theorem 3.4, we can get Theorem 3.8 and Theorem 3.10 in [28], Theorem 3.5 in [31], Theorem 1 and Theorem 2 in [34].
For completeness, we conclude with the following strong convergence theorem for a finite family of nonexpansive and asymptotically nonexpansive nonself mappings.
Theorem 3.5 Let C be a nonempty closed convex subset of a uniformly convex Banach space X and let be a nonexpansive retraction from X onto C. Let be a finite family of asymptotically nonexpansive nonself mappings and be as in Theorem 3.1. If the family satisfies condition (), then converges strongly to a common fixed point of .
Theorem 3.6 Let C be a nonempty closed convex subset of a uniformly convex Banach space X and let be a nonexpansive retraction from X onto C. Let be a finite family of nonexpansive nonself mappings and be as in Theorem 3.2. If the family satisfies condition (), then converges strongly to a common fixed point of .
Remark 3.4 Theorem 3.5 and Theorem 3.6 generalize and improve many recent important results such as Theorem 3.5 in [1], Theorem 3.7 in [2], Theorem 2.4 in [4], Theorem 4.2 in [5], Theorem 2 in [9], Theorem 2.4 in [12], Theorems 2.1-2.3 in [18], Theorem 3.6 in [24], Theorem 3.4 and Theorem 4.1 in [27], Theorems 3.3-3.4 in [32] and others.
References
Chidume CE, Ali B: Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 2007, 330: 377–387. 10.1016/j.jmaa.2006.07.060
Chidume CE, Ofoedu EU, Zegeye H: Strong and weak convergence theorems for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 2003, 280: 364–374. 10.1016/S0022-247X(03)00061-1
Chidume CE, Chidume CO: Iterative methods for common fixed points for a countable family of nonexpansive mappings in uniformly convex spaces. Nonlinear Anal. 2009, 71: 4346–4356. 10.1016/j.na.2008.11.042
Cho YJ, Zhou HY, Guo G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 2004, 47: 707–717. 10.1016/S0898-1221(04)90058-2
Fukhar-ud-din H, Khan AR: Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces. Comput. Math. Appl. 2007, 53: 1349–1360. 10.1016/j.camwa.2007.01.008
Groetsch CW: A note on segmenting Mann iterates. J. Math. Anal. Appl. 1972, 40: 369–372. 10.1016/0022-247X(72)90056-X
Guo W, Guo W: Weak convergence theorems for asymptotically nonexpansive nonself-mappings. Appl. Math. Lett. 2011, 24: 2181–2185. 10.1016/j.aml.2011.06.022
Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5
Khan SH, Fukhar-ud-din H: Weak and strong convergence of a scheme with errors for two nonexpansive mappings. Nonlinear Anal. 2005, 61: 1295–1301. 10.1016/j.na.2005.01.081
Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 44: 506–510.
Noor MA: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 2000, 251: 217–229. 10.1006/jmaa.2000.7042
Plubtieng S, Wangkeeree R, Punpaeng R: On the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 2006, 322: 1018–1029. 10.1016/j.jmaa.2005.09.078
Plubtieng S, Ungchittrakool K, Wangkeeree R: Implicit iterations of two finite families for nonexpansive mappings in Banach spaces. Numer. Funct. Anal. Optim. 2007, 28: 737–749. 10.1080/01630560701348525
Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884
Takahashi W, Tamura T: Convergence theorems for a pair of nonexpansive mappings. J. Convex Anal. 1998, 5: 45–56.
Tan KK, Xu HK: Fixed point iteration processes for asymptotically nonexpansive mapping. Proc. Am. Math. Soc. 1994, 122: 733–739. 10.1090/S0002-9939-1994-1203993-5
Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178: 301–308. 10.1006/jmaa.1993.1309
Xu BL, Noor MA: Fixed-points iteration for asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 2002, 267: 444–453. 10.1006/jmaa.2001.7649
Xu HK: Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. Nonlinear Anal. 1991, 16: 1139–1146. 10.1016/0362-546X(91)90201-B
Yao YH, Chen RD, Yao JC: Strong convergence and certain control conditions for modified Mann iteration. Nonlinear Anal. 2008, 68: 1687–1693. 10.1016/j.na.2007.01.009
Zegeye H, Shahzad N: Approximation methods for a common fixed point of a finite family of nonexpansive mappings. Numer. Funct. Anal. Optim. 2007, 28: 1405–1419. 10.1080/01630560701749730
Zhou YY, Chang SS: Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces. Numer. Funct. Anal. Optim. 2002, 23: 911–921. 10.1081/NFA-120016276
Zhou HY, Cho YJ, Kang SM: A new iterative algorithm for approximating common fixed points for asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2007., 2007: Article ID 64974
Shahzad N: Approximating fixed points of non-self nonexpansive mappings in Banach spaces. Nonlinear Anal. 2005, 61: 1031–1039. 10.1016/j.na.2005.01.092
Chen LC, Yao JC: Strong convergence of an iterative algorithm for nonself multimaps in Banach spaces. Nonlinear Anal. 2009, 71: 4476–4485. 10.1016/j.na.2009.03.007
Chen W, Guo W: Convergence theorems for two finite families of asymptotically nonexpansive mappings. Math. Comput. Model. 2011, 54: 1311–1319. 10.1016/j.mcm.2011.04.002
Chidume CE, Ali B: Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 2007, 326: 960–973. 10.1016/j.jmaa.2006.03.045
Chidume CE, Shahzad N, Zegeye H: Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense. Numer. Funct. Anal. Optim. 2004, 25: 239–257.
Guo W, Cho YJ, Guo W: Convergence theorems for mixed type asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2012., 2012: Article ID 224
Pathak HK, Cho YJ, Kang SM: Strong and weak convergence theorems for nonself-asymptotically perturbed nonexpansive mappings. Nonlinear Anal. 2009, 70: 1929–1938. 10.1016/j.na.2008.02.092
Plubtieng S, Wangkeeree R: Strong convergence theorems for three-step iterations with errors for non-Lipschitzian nonself-mappings in Banach spaces. Comput. Math. Appl. 2006, 51: 1093–1102. 10.1016/j.camwa.2005.08.035
Wang L: Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings. J. Math. Anal. Appl. 2006, 323: 550–557. 10.1016/j.jmaa.2005.10.062
Zhu LP, Huang QL, Li G: Weak convergence theorem for the three-step iterations of non-Lipschitzian nonself mappings in Banach spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 106
Kim GE, Kim TH: Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces. Comput. Math. Appl. 2001, 42: 1565–1570. 10.1016/S0898-1221(01)00262-0
Bruck RE: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Isr. J. Math. 1979, 32: 107–116. 10.1007/BF02764907
Kaczor W: Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups. J. Math. Anal. Appl. 2002, 272: 565–574. 10.1016/S0022-247X(02)00175-0
Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0
Kaczor W: A nonstandard proof of a generalized demiclosedness principle. Ann. Univ. Mariae Curie-Skl̄odowska, Sect. A 2005, LIX: 43–50.
Acknowledgements
The authors are grateful to the referees for their constructive comments and helpful suggestions.
This research is supported by the National Science Foundation of China (11201410, 11271316 and 11101353), the Natural Science Foundation of Jiangsu Province (BK2012260) and the Natural Science Foundation of Jiangsu Education Committee (10KJB110012 and 11KJB110018).
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Huang, Q., Zhu, L. Weak and strong convergence theorems for a finite family of non-Lipschitzian nonself mappings in Banach spaces. Fixed Point Theory Appl 2013, 170 (2013). https://doi.org/10.1186/1687-1812-2013-170
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DOI: https://doi.org/10.1186/1687-1812-2013-170