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Some results on a modified Mann iterative scheme in a reflexive Banach space
Fixed Point Theory and Applications volume 2013, Article number: 227 (2013)
Abstract
The purpose of this paper is to study Mann iterative schemes. Strong convergence of a modified Mann iterative scheme is obtained in a reflexive Banach space.
1 Introduction-Preliminaries
Normal Mann iterative scheme is an important iterative scheme to study the class of nonexpansive mappings [1]. However, the normal Mann iterative scheme is only weak convergence for nonexpansive mappings; see [2–4]. In many disciplines, including economics [5] and image recovery [6], problems arise in infinite dimensional spaces. In such problems, strong convergence is often much more desirable than weak convergence, for it translates the physically tangible property. Strong convergence of iterative sequences properties has a direct impact when the process is executed directly in the underlying infinite dimensional space. Recently, with the aid of projections, many authors studied Mann iterative schemes. Theoretically, the advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without any compact assumptions. The purpose of this paper is to study a modified Mann iterative scheme. Strong convergence of the scheme is obtained in a reflexive Banach space.
Let E be a real Banach space, let C be a nonempty subset of E, and let be a mapping. In this paper, we use to denote the fixed point set of T. Recall that T is said to be asymptotically regular on C iff for any bounded subset K of C,
Recall that T is said to be closed iff for any sequence such that and , then . In this paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively.
Recall that T is said to be nonexpansive iff
T is said to be quasi-nonexpansive iff , and
T is said to be asymptotically nonexpansive iff there exists a sequence with as such that
T is said to be asymptotically quasi-nonexpansive iff and there exists a sequence with as such that
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [7] in 1972. In uniformly convex Banach spaces, they proved that if C is nonempty, bounded, closed, and convex, then every asymptotically nonexpansive self-mapping T on C has a fixed point. Further, the fixed point set of T is closed and convex. Since 1972, a host of authors have studied the weak and strong convergence of iterative schemes for such a class of mappings.
Recall that T is said to be asymptotically nonexpansive in the intermediate sense iff it is continuous and the following inequality holds:
T is said to be asymptotically quasi-nonexpansive in the intermediate sense iff and the following inequality holds:
The class of the mappings that are asymptotically nonexpansive in the intermediate sense was considered by Bruck et al. [8] and Kirk [9]. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense may not be Lipschitz continuous. However, asymptotically nonexpansive mappings are Lipschitz continuous. For the existence of the mapping, we can find the details in [9].
In [10], Nakajo and Takahashi first investigated fixed point problems of nonexpansive mappings based on hybrid projection methods in the framework of Hilbert spaces. Subsequently, many authors investigated fixed point problems of nonlinear mappings based on the methods in the framework of Hilbert spaces. The advantage of the method is that strong convergence of iterative sequences can be guaranteed without any compact assumptions.
Let E be a Banach space with the dual . We denote by J the normalized duality mapping from E to defined by
where denotes the generalized duality pairing. A Banach space E is said to be strictly convex if for all with and . It is said to be uniformly convex if for any two sequences and in E such that and . Let be the unit sphere of E. Then the Banach space E is said to be smooth provided exists for each . It is also said to be uniformly smooth if the above limit is attained uniformly for . It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. It is also well known that E is uniformly smooth if and only if is uniformly convex.
Recall that a Banach space E enjoys the Kadec-Klee property if for any sequence , and with , and , then as . For more details on the Kadec-Klee property, the readers can refer to [11] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.
As we all know, if C is a nonempty closed convex subset of a Hilbert space H and is the metric projection of H onto C, then is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [12] recently introduced a generalized projection operator in a Banach space E which is an analogue of the metric projection in Hilbert spaces.
Next, we assume that E is a smooth Banach space. Consider the functional defined by
Observe that, in a Hilbert space H, the equality is reduced to , . The generalized projection is a map that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution to the minimization problem
The existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping J; see, for example, [11–13]. In Hilbert spaces, . It is obvious from the definition of a function ϕ that
and
Remark 1.1 If E is a reflexive, strictly convex, and smooth Banach space, then for , if and only if . It is sufficient to show that if , then . From (2.1), we have . This implies that . From the definition of J, we have . Therefore, we have ; see [11] for more details.
Let C be a nonempty closed convex subset of E, and let T be a mapping from C into itself. A point p in C is said to be an asymptotic fixed point of T [14] if C contains a sequence which converges weakly to p such that . The set of asymptotic fixed points of T will be denoted by . A mapping T from C into itself is said to be relatively nonexpansive [15, 16] if and for all and . The mapping T is said to be relatively asymptotically nonexpansive [17, 18] if and there exists a sequence with as such that for all , and . The asymptotic behavior of relatively nonexpansive mappings was studied in [14–16].
The mapping T is said to be quasi-ϕ-nonexpansive [19] if and for all and . T is said to be asymptotically quasi-ϕ-nonexpansive [20–22] if and there exists a sequence with as such that for all , and .
Remark 1.2 The class of asymptotically quasi-ϕ-nonexpansive mappings is more general than the class of relatively asymptotically nonexpansive mappings, which requires the restriction: .
T is said to be asymptotically quasi-ϕ-nonexpansive in the intermediate sense [23] if and
Put
It follows that as . Then (1.3) is reduced to the following:
Remark 1.3 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasi-nonexpansive mappings in the intermediate sense in the framework of Banach spaces.
Let and . Define the following mapping by
Then T is an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense with the fixed point set . We also have the following
and
where . Hence, we have , .
Recently, Matsushita and Takahashi [24] first investigated fixed point problems of relatively nonexpansive mappings based on hybrid projection methods. A strong convergence theorem was established in a uniformly convex and uniformly smooth Banach space. To be more precise, they proved the following result.
Theorem MT Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let T be a relatively nonexpansive mapping from C into itself, and let be a sequence of real numbers such that and . Suppose that is given by
where J is the duality mapping on E. If is nonempty, then converges strongly to , where is the generalized projection from C onto .
Recently, Su and Qin [25] introduced a monotone projection method for computing fixed points of nonexpansive mappings. A strong convergence theorem was established in the framework of Hilbert spaces; for more details, see [25]. Takahashi et al. [26] further introduced the shrinking projection method for nonexpansive mappings; for more details, see [26]. Subsequently, a lot results were obtained on the two methods in Hilbert spaces or Banach spaces; see [27–38] and the references therein.
Motivated by the above results, some fixed point theorems of asymptotically quasi-ϕ-nonexpansive mappings based on hybrid projection methods were established in the framework of Banach spaces.
In this paper, motivated by Matsushita and Takahashi [24], Su and Qin [25] and Takahashi et al. [26], we investigate a fixed point problem of an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. A strong convergence theorem is established in a reflexive, strictly convex, and smooth Banach space such that both E and have the Kadec-Klee property. The results improve and extend the corresponding results in the literature.
For our main results, we need the following lemmas.
Lemma 1.4 [12]
Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty, closed, and convex subset of E, and let . Then
Lemma 1.5 [12]
Let C be a nonempty, closed, and convex subset of a smooth Banach space E, and let . Then if and only if
2 Main results
Theorem 2.1 Let E be a reflexive, strictly convex, and smooth Banach space such that both E and have the Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. Let be an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Assume that T is asymptotically regular on C and closed, and is nonempty and bounded. Let be a sequence generated in the following manner:
where
If the sequence satisfies the restriction , then the sequence converges strongly to , where is the generalized projection from C onto .
Proof The proof is split into six steps.
Step 1. Show that is closed and convex so that is well defined for any .
Let , and , where . We see that . Indeed, we see from (1.2) that
and
It follows from the definition of T that
and
Multiplying t and on both sides of (2.1) and (2.2), respectively, yields that . In light of (1.1), we arrive at
It follows that
Since is reflexive, we may, without loss of generality, assume that . In view of the reflexivity of E, we find that there exists an element such that . It follows that
Taking on both sides of the equality above, we obtain that
This implies that , that is, . It follows that . In view of the Kadec-Klee property of , we obtain from (2.4) that . Since is demicontinuous, we see that . By virtue of the Kadec-Klee property of E, we see from (2.3) that as . Hence, as . In view of the closedness of T, we can obtain that . This shows that is convex. Since T is closed, we can easily conclude that is also closed. This completes the proof that is convex and closed.
Step 2. Show that is closed and convex.
It is obvious that is closed and convex. Suppose that is closed and convex for some . We now show that is also closed and convex.
For , we see that . It follows that , where . Notice that
and
Notice that (2.5) and (2.6) are equivalent to
and
Multiplying t and on both sides of (2.7) and (2.8), respectively, yields that
That is,
This implies that is closed and convex. Then, for each , is closed and convex. This shows that is well defined.
Step 3. Show that .
is obvious. Suppose that for some . Then, , we have
This shows that . This implies that .
Step 4. Show that , where is some point in C.
In view of , we see from Lemma 1.5 that
It follows from that
It follows from Lemma 1.4 that
This implies that the sequence is bounded. It follows from (1.1) that the sequence is also bounded. Since the space is reflexive, we may assume that . Since is closed and convex, we find that . On the other hand, we see from the weak lower semicontinuity of the norm that
which implies that . Hence, we have . In view of the Kadec-Klee property of E, we find that as . This completes Step 4.
Step 5. Show that .
Since and , we find that . This shows that is nondecreasing. It follows from the boundedness that exists. In view of construction of , we arrive at
This implies that
In view of , we find that
This in turn implies from (2.10) that
In view of (1.1), we see that
It follows that
This is equivalent to
This implies that is bounded. Note that both E and are reflexive. We may assume that . In view of the reflexivity of E, we see that . This shows that there exists an element such that . It follows that
Taking on both sides of the equality above yields that
That is, , which in turn implies that . It follows that . Since enjoys the Kadec-Klee property, we obtain from (2.12) that
On the other hand, we have
It follows that
In view of
we find that
In view of the restriction , we find from (2.13) that
Notice that
This implies from (2.14) that
The demicontinuity of implies that . Note that
With the aid of (2.15), we see that . Since E has the Kadec-Klee property, we find that
Notice that
In view of the asymptotic regularity of T, we find from (2.16) that
that is, as . It follows from the closedness of T that . This completes Step 5.
Step 6. Show that .
Letting in (2.9), we arrive at
It follows from Lemma 1.5 that . This completes the proof of Theorem 2.1. □
Remark 2.2 The sets become increasingly complicated, which may render the algorithm unimplementable. One may use an inner-loop for calculating an approximation of at each iterative step. The advantage of the algorithm is that strong convergence of iterative sequences can be guaranteed in a reflexive, strictly convex, and smooth Banach space such that both E and have the Kadec-Klee property without any compact assumptions.
If T is quasi-ϕ-nonexpansive, then Theorem 2.1 is reduced to the following.
Corollary 2.3 Let E be a reflexive, strictly convex, and smooth Banach space such that both E and have the Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. Let be a quasi-ϕ-nonexpansive mapping with a nonempty fixed point set. Let be a sequence generated in the following manner:
If the sequence satisfies the restriction , then the sequence converges strongly to , where is the generalized projection from C onto .
Remark 2.4 Corollary 2.3 mainly improves the corresponding results [24] in the following aspects: (1) from the relatively nonexpansive mapping to the quasi-ϕ-nonexpansive mapping; (2) from a uniformly convex and uniformly smooth Banach space to a reflexive, strictly convex, and smooth Banach space such that both E and have the Kadec-Klee property. The algorithm is also different from the one in [24].
If E is a Hilbert space, then we have the following result.
Corollary 2.5 Let E be a Hilbert space. Let C be a nonempty, closed, and convex subset of E. Let be an asymptotically quasi-nonexpansive mapping in the intermediate sense. Assume that T is asymptotically regular on C and closed and is nonempty and bounded. Let be a sequence generated in the following manner:
where
If the sequence satisfies the restriction , then the sequence converges strongly to , where is the metric projection from C onto .
Proof In Hilbert spaces, we find that J is the identity and . We can immediately derive from Theorem 2.1 the desired conclusion. □
Remark 2.6 Corollary 2.5 can be viewed as an improvement of the corresponding result in Su and Qin [38]. The mapping is extended from asymptotically nonexpansive mappings to asymptotically quasi-nonexpansive mappings in the intermediate sense.
If T is quasi-nonexpansive, then we have the following result.
Corollary 2.7 Let E be a Hilbert space. Let C be a nonempty, closed, and convex subset of E. Let be a quasi-nonexpansive mapping with a nonempty fixed point set. Let be a sequence generated in the following manner:
If the sequence satisfies the restriction , then the sequence converges strongly to , where is the metric projection from C onto .
Proof In Hilbert spaces, we find that J is the identity and . We can immediately derive from Theorem 2.1 the desired conclusion. □
Remark 2.8 Corollary 2.7 can be viewed as an improvement of the corresponding result in Nakajo and Takahashi [10]. The mapping has been extended from a nonexpansive mapping to a quasi-nonexpansive mapping. The sets have also been relaxed.
References
Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3
Genel A, Lindenstrass J: An example concerning fixed points. Isr. J. Math. 1975, 22: 81–86. 10.1007/BF02757276
Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 1979, 67: 274–276. 10.1016/0022-247X(79)90024-6
Bauschke HH, Matouskova E, Reich S: Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 2004, 56: 715–738. 10.1016/j.na.2003.10.010
Khan MA, Yannelis NC: Equilibrium Theory in Infinite Dimensional Spaces. Springer, New York; 1991.
Combettes PL: The convex feasibility problem in image recovery. 95. In Advances in Imaging and Electron Physics. Edited by: Hawkes P. Academic Press, New York; 1996:155–270.
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3
Bruck RE, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math. 1993, 65: 169–179.
Kirk WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Isr. J. Math. 1974, 17: 339–346. 10.1007/BF02757136
Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 2003, 279: 372–379. 10.1016/S0022-247X(02)00458-4
Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.
Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996.
Alber YI, Reich S: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panam. Math. J. 1994, 4: 39–54.
Reich S: A weak convergence theorem for the alternating method with Bregman distance. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996.
Butnariu D, Reich S, Zaslavski AJ: Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. Numer. Funct. Anal. Optim. 2003, 24: 489–508. 10.1081/NFA-120023869
Censor Y, Reich S: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 1996, 37: 323–339. 10.1080/02331939608844225
Agarwal RP, Cho YJ, Qin X: Generalized projection algorithms for nonlinear operators. Numer. Funct. Anal. Optim. 2007, 28: 1197–1215. 10.1080/01630560701766627
Qin X, Su Y, Wu C, Liu K: Strong convergence theorems for nonlinear operators in Banach spaces. Commun. Appl. Nonlinear Anal. 2007, 14: 35–50.
Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011
Zhou H, Gao G, Tan B: Convergence theorems of a modified hybrid algorithm for a family of quasi- ϕ -asymptotically nonexpansive mappings. J. Appl. Math. Comput. 2010, 32: 453–464. 10.1007/s12190-009-0263-4
Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi- ϕ -nonexpansive mappings. Appl. Math. Comput. 2010, 215: 3874–3883. 10.1016/j.amc.2009.11.031
Qin X, Agarwal RP: Shrinking projection methods for a pair of asymptotically quasi- ϕ -nonexpansive mappings. Numer. Funct. Anal. Optim. 2010, 31: 1072–1089. 10.1080/01630563.2010.501643
Qin X, Wang L: On asymptotically quasi- ϕ -nonexpansive mappings in the intermediate sense. Abstr. Appl. Anal. 2012., 2012: Article ID 636217
Matsushita SY, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J. Approx. Theory 2005, 134: 257–266. 10.1016/j.jat.2005.02.007
Su Y, Qin X: Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators. Nonlinear Anal. 2008, 68: 3657–3664. 10.1016/j.na.2007.04.008
Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2008, 341: 276–286. 10.1016/j.jmaa.2007.09.062
Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008
Cho SY: Hybrid projection algorithms for treating common fixed points of a family of demicontinuous pseudocontractions. Appl. Math. Lett. 2012, 25: 854–857. 10.1016/j.aml.2011.10.031
Kang SM, Cho SY, Liu Z: Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J. Inequal. Appl. 2010., 2010: Article ID 827082
Ye J, Huang J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. J. Math. Comput. Sci. 2011, 1: 1–18.
Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618.
Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10
Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi- ϕ -nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 2010, 234: 750–760. 10.1016/j.cam.2010.01.015
Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.
Qin X, Cho SY, Kang SM: Iterative algorithms for variational inequality and equilibrium problems with applications. J. Glob. Optim. 2010, 48: 423–445. 10.1007/s10898-009-9498-8
Chen Z: A strong convergence theorem of common elements in Hilbert spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 59
Hao Y, Cho SY: Fixed point iterations of a pair of hemirelatively nonexpansive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 270150
Su Y, Qin X: Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups. Fixed Point Theory Appl. 2006., 2006: Article ID 96215
Acknowledgements
The study was supported by the Natural Science Foundation of Zhejiang Province (Y6110270). The author is grateful to the editor and the three anonymous reviewers’ suggestions which improved the contents of the article.
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Hao, Y. Some results on a modified Mann iterative scheme in a reflexive Banach space. Fixed Point Theory Appl 2013, 227 (2013). https://doi.org/10.1186/1687-1812-2013-227
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DOI: https://doi.org/10.1186/1687-1812-2013-227