Convergence of a hybrid iterative method for finite families of generalized quasi-ϕ-asymptotically nonexpansive mappings

Strong convergence theorem for finite families of generalized quasi-ϕ-asymptotically nonexpansive mappings is proved in a real uniformly convex and uniformly smooth Banach space using a new modified hybrid iterative algorithm.

MSC: 47H09, 47J25.

Let E be a real Banach space and be the dual space of E. The normalized duality mapping is defined by

for all , where denotes the duality pairing. A Banach space E is said to be uniformly convex if given , there exists such that for all with , and , we have . E is strictly convex if for all with and . The space E is said to be smooth if the limit

exists for all , where . It is also uniformly smooth if the limit exists uniformly for . It is well known that if E is strictly convex, smooth and reflexive, then the duality map J is one-to-one, single-valued and onto. Also if E is uniformly smooth, then J is norm-to-norm uniformly continuous on bounded subsets of E.

Let C be a nonempty, closed, convex subset of E. Let be a map, a point is called a fixed point of T if and the set of all fixed points of T is denoted by . We recall that a point is called an asymptotic fixed point of T if there exists a sequence which converges weakly to p and . The mapping T is called Lipschitz if there exists such that for all , and if , then T is called nonexpansive. T is asymptotically nonexpansive if there exists a sequence such that as and for all and for all . The map T is quasi-nonexpansive if and for all , , and is called asymptotically quasi-nonexpansive if and for all , and the sequence satisfies as . The mapping T is called generalized asymptotically quasi-nonexpansive if , there exist sequences , with , as and for all , and .

The map T is said to be

(i) asymptotically regular on C if for all ,

(ii) uniformly asymptotically regular on C if holds for any bounded subset K of C.

For a positive real number L, the map T is called uniformlyL-Lipschitzian if for all and .

It is clear from these definitions that every nonexpansive mapping with a fixed point is quasi-nonexpansive and all asymptotically nonexpansive maps with fixed points are asymptotically quasi-nonexpansive. Recently, the class of generalized asymptotically quasi-nonexpansive mappings was introduced and studied by Shahzad and Zegeye [21]. They proved that every asymptotically quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive mapping and the inclusion is proper. The class of quasi-nonexpansive mappings was introduced and studied first in 1967 by Diaz and Metcalf [7]. Goebel and Kirk [8] introduced the class of asymptotically nonexpansive mappings and proved that if C is a nonempty, closed, convex and bounded subset of a uniformly convex Banach space E, then an asymptotically nonexpansive mapping has a fixed point.

Kirk [16], proved that if E is a reflexive Banach space with normal structure and C is a nonempty, closed, convex and bounded subset of E, a nonexpansive map has a fixed point in C. This result was extended to a finite family of nonexpansive maps by Bellus and Kirk [3] and then to an infinite family of nonexpansive maps by Lim [17].

Let H be a real Hilbert space, C be a nonempty closed convex subset of H. Recall that for each there exists a unique nearest point in C to x denoted by . That is, for all . is called a metric projection of H onto C.

It is well known that the metric projection is nonexpansive only in a Hilbert space. This fact actually characterizes Hilbert spaces. Alber [1], introduced a generalized projection map in a Banach space which is an analogue of the metric projection in a Hilbert space.

Let E be a real normed linear space with single-valued normalized duality map. Consider the functional defined by . We observe that in a Hilbert space, reduces to . It is clear that for , the following inequality holds . The generalized projection map is a map that assigns to an arbitrary point , the minimum point of the functional over C, that is, where . Existence and uniqueness of the map follow from the properties of the functional ϕ and the strict monotonicity of J (see, for example, [2]).

Let C be a nonempty, closed, and convex subset of E, a mapping is said to be

(i) relatively nonexpansive if and for all , where denotes the set of asymptotic fixed points of T;

(ii) ϕ-nonexpansive if for all ;

(iii) ϕ-asymptotically nonexpansive if there exists a sequence satisfying as and for all , ;

(iv) quasi-ϕ-asymptotically nonexpansive if and for all , , , where is as in (iii) above.

We shall call the map Tgeneralized quasi-ϕ-asymptotically nonexpansive in the light of [21], if and there exist sequences , with , as and for all , and .

Existence and approximations of fixed points of mappings of nonexpansive type and their generalizations were studied by numerous authors, see, for example, [3,5,7,8,10,11,14-17,19,21,27] and the references therein.

In 1986, Das and Debata [6] studied the Ishikawa-like scheme defined by ,

where and are sequences in such that . They studied the scheme for two quasi-nonexpansive mapsS and T and proved strong convergence of the sequence to a common fixed point of S and T in a real strictly convex Banach space. Takahashi and Tamura [25] proved strong and weak convergence of the sequence defined by (1.1) to a common fixed point of a pair of nonexpansive mappings T and S using a weaker condition on the maps.

Using a similar scheme, Wang [26] proved strong and weak convergence theorems for a pair of nonself asymptotically nonexpansive mappings in a uniformly convex Banach space.

Shahzad and Udomene [22] proved the necessary and sufficient conditions for the strong convergence of the scheme of type (1.1) to a common fixed point of two uniformly continuous asymptotically quasi-nonexpansive mappings in a real Banach space.

Chidume and Ali [4] introduced and proved strong convergence of the scheme defined by

to a common fixed point of a finite family of nonself asymptotically nonexpansive mappings in a uniformly convex Banach space.

Khan et al.[13] introduced and studied the following scheme:

for a common fixed point of a finite family of asymptotically quasi-nonexpansive mappings in a Banach space.

It is known that only weak convergence theorems were proved for nonexpansive maps even in Hilbert spaces using Mann and Ishikawa type schemes.

In 2000 Solodov and Svaiter [23] introduced a hybrid proximal point type iterative scheme and proved the strong convergence of the scheme to a zero of a maximal monotone operator.

In 2003 Nakajo and Takahashi [19] proposed a hybrid Mann scheme for nonexpansive mappings and nonexpansive semigroups and proved strong convergence theorems.

Kim and Xu [14] generalized the result of Nakajo and Takahashi by proving strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups. Plubtieng and Ughchittrakool [20] introduced an Ishikawa type hybrid scheme for two asymptotically nonexpansive mappings and two asymptotically nonexpansive semigroups.

Takahashi et al.[24] studied a simpler hybrid scheme for nonexpansive mappings in Hilbert spaces. Inchan and Plubtieng [10], adopted this simpler scheme of Takahashi et al. with little modification for two nonexpansive maps and two nonexpansive semigroups. They proved the following theorem:

Theorem 1.1 ([10])

LetHbe a real Hilbert space and letCbe a nonempty, closed, convex, and bounded subset ofH. Letbe two asymptotically nonexpansive mappings with sequencesandrespectively and. Let. Then the sequencegenerated by

converges strongly to, whereasand, for all.

Kimura and Takahashi [15] proved strong convergence theorem for the family of relatively nonexpansive mappings in strictly convex Banach spaces having Kadec-Klee property and Frechet differentiable norm.

Recently, Zhou et al.[28] have proved strong convergence theorem for the family , of quasi-ϕ-asymptotically nonexpansive mappings, where C is a nonempty, closed, convex and bounded subset of a uniformly smooth and uniformly convex Banach space E.

More recently, Xu et al.[27] have studied a modified hybrid scheme for fixed point of families of quasi-ϕ-asymptotically nonexpansive mappings. They proved the following theorem:

Theorem 1.2 ([27])

LetCbe a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach spaceE, and let, be a family of closed and quasi-ϕ-asymptotically nonexpansive mappings such that. Assume that every, is asymptotically regular onC. Let, andbe real sequences insuch that, . Define a sequenceinCby:

Then, converges strongly to, whereis the generalized projection fromEonto F.

Motivated by these results, we have the purpose in this paper to study a new modified hybrid iterative scheme and prove a strong convergence theorem for a finite family of generalized quasi-ϕ-asymptotically nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space. Our theorems improve and unify several recent important results.

Consider a sequence of nonempty closed and convex subsets of a reflexive Banach space E. Let denotes the set of all strong limits of sequences satisfying for all and be the set of all weak limits of sequences satisfying for all where is some subsequence of . The sequence is said to converge to in the sense of Mosco [18] if . The Mosco limit of is denoted by .

We shall make use of the following important results in the sequel.

Lemma 2.1 (Kamimura and Takahashi [12])

LetEbe a real smooth and uniformly convex Banach space and, be two sequences ofE. Ifand eitheroris bounded, then.

Lemma 2.2 (Ibaraki, Kimura and Takahashi [9])

LetCbe a nonempty closed convex subset of a real uniformly smooth and uniformly convex Banach spaceE. Letbe a sequence of nonempty closed convex subsets ofC. Ifexists and is nonempty, thenconverges strongly tofor each.

The result in [9] is more general than the one presented here, but this is sufficient for our purpose.

Lemma 2.3LetCbe a nonempty closed convex subset of a real smooth Banach space andbe a closed generalized quasi-ϕ-asymptotically nonexpansive mapping. Thenis closed and convex.

Proof By the closedness assumption on T and the definition of ϕ, the result follows immediately. □

Theorem 3.1LetEbe a real uniformly convex and uniformly smooth Banach space andCbe a nonempty, bounded, closed and convex subset ofE. Let, be a finite family of closed generalized quasi-ϕ-asymptotically nonexpansive maps with corresponding sequencesand, such thatandas. Letand let, . Assume also that the maps, are uniformly asymptotically regular. Letbe arbitrary andand let. For, letbe sequences infor some, . Letbe a sequence generated by

where. Then the sequenceconverges strongly to.

Proof We start by showing that . We do this by induction. by definition. We suppose that for some . We observe that for , using convexity of and (3.1), we have

and

Similarly,

Continuing in this way, we get for ,

So for any and . This and the induction hypothesis give that for all . Therefore, and hence for all .

Also by induction and using the fact that is continuous on E for any , it follows that is closed for each and , and consequently, is closed for each .

We now prove that is convex for all . We observe that is equivalent to and . So the convexity of for each and for each follows immediately by induction. Thus is convex for each .

We now show that the sequence converges. Since is a decreasing sequence of closed, convex subsets of E, such that , then the Mosco limit exists and . By Lemma 2.2, the sequence converges to , where .

We observe that

and from the fact that is convergent, we easily deduce that

Since , we get that for each , , and so from (3.2) and (3.3), we obtain for each . By Lemma 2.1, we get that and . So, for each ,

Since j is norm-to-norm uniformly continuous on bounded subsets of E, we get that, for each , . Using (3.1) we obtain that

and for ,

Using these and the fact that is norm-to-norm uniformly continuous on bounded subsets of , we get

Since as , we obtain that , as . By the uniform asymptotic regularity of each of the maps , , we get

and

for . These imply and as , and for . By the closedness of each of the maps , , we have that .

As F is a nonempty closed convex subset of , we obtain that . This completes the proof. □

The conditions of closedness and uniform asymptotic regularity on the maps can be replaced by the condition that each of the maps is uniformly Lipschitz. So we have the following theorem:

Theorem 3.2LetE, C, , F, , , andbe as in Theorem 3.1 with the exception thatare uniformly, , Lipschitzian instead of uniformly asymptotically regular and closed. Then the sequenceconverges strongly to.

Proof The proof that , is closed, convex for each and follows as in Theorem 3.1. Also relations (3.2), (3.3), (3.4) and (3.5) are obtainable as in Theorem 3.1. We only need to show that . Let , then using (3.4) and (3.5) we get

and for ,

So we obtain

and also for ,

Finally, using these, the fact that as , and the continuity of for each k, we obtain that and this completes the proof. □

The following corollaries follow from Theorems 3.1 and 3.2.

Corollary 3.3LetEbe a real uniformly convex and uniformly smooth Banach space andCbe a nonempty, bounded, closed and convex subset ofE. Let, be a finite family of quasi-ϕ-asymptptically nonexpansive maps with corresponding sequences, , such that, as. Letand let. Assume also that the maps, are either closed and uniformly asymptotically regular onCor uniformly Lipschitzian onC. Letbe arbitrary and. For, letbe sequences infor some, . Letbe a sequence generated by (3.1). Then the sequenceconverges strongly to.

Corollary 3.4LetEbe a real uniformly convex and uniformly smooth Banach space andCbe a nonempty, bounded, closed and convex subset ofE. Let, be a finite family ofϕ-asymptotically nonexpansive maps with corresponding sequences, , such that, as. Letand let. Assume also that the maps, are either closed and uniformly asymptotically regular onCor uniformly Lipschitzian onC. Letbe arbitrary and. For, letbe sequences infor some, . Letbe a sequence generated by (3.1). Then the sequenceconverges strongly to.

Corollary 3.5LetHbe a real Hilbert space, Cbe a nonempty, bounded, closed and convex subset ofH. Let, be a finite family of generalized asymptotically quasi-nonexpansive maps with corresponding sequencesand, such thatandas. Letand let. Assume also that the maps, are either closed and uniformly asymptotically regular onCor uniformly, Lipschitzian onC. Letbe arbitrary and. For, letbe sequences infor some, . Letbe a sequence generated by

where. Then, the sequenceconverges strongly to.

Corollary 3.6LetHbe a real Hilbert space, Cbe a nonempty,closed and convex subset ofH. Let, be a finite family of asymptotically nonexpansive maps with corresponding sequences, , such thatas. Letand let. Letbe arbitrary and. For, letbe sequences infor some, . Letbe a sequence generated by (3.10). Then the sequenceconverges to.

Remark 3.7 Theorem 3.1 and Corollary 3.5 extend and improve several important recent results. For instance, Corollary 3.5 is an improvement and generalization of Theorem 1.1 and Theorem 3.1 of [20].

Remark 3.8 It is not clear whether Theorem 3.1 and Corollary 3.5 hold without the boundedness assumption on C.

The authors declare that they have no competing interest.

All the authors contributed equally in writing this article.

This work was conducted when the first author was visiting the Abdus Salam International Center for Theoretical Physics, Trieste, Italy, as an associate. He would like to thank the center for hospitality and financial support.

1. Alber, Y: Metric and generalized projection operators in Banach spaces: properties and applications. In: Karstsatos AG (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15–50. Dekker, New York (1996)

2. Alber, Y, Guerre-Delabriere, S: On the projection methods for fixed point problems. Analysis. 21, 17–39 (2001)

3. Belluce, LP, Kirk, WA: Fixed point theorem for families of contraction mappings. Pac. J. Math.. 18, 213–217 (1966). Publisher Full Text

4. Chidume, CE, Ali, B: Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl.. 326, 960–973 (2007). Publisher Full Text

5. Chidume, CE, Ali, B: Convergence theorems for finite families of asymptotically nonexpansive mappings. J. Inequal. Appl.. 326, Article ID 68616. doi:10.1155/2007/68616 (2007)

6. Das, G, Debata, JP: Fixed points of quasi-nonexpansive mappings. Indian J. Pure Appl. Math.. 17, 1263–1269 (1986)

7. Diaz, JB, Metcalf, FB: On the structure of the set of subsequential limit points of successive approximations. Bull. Am. Math. Soc.. 73, 516–519 (1967). Publisher Full Text

8. Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc.. 35, 171–174 (1972). Publisher Full Text

9. Ibaraki, T, Kimura, Y, Takahashi, W: Convergence theorems for generalized projections and maximal monotone operators in Banach spaces. Abstr. Appl. Anal.. 2003(10), 621–629 (2003). Publisher Full Text

10. Inchan, I, Plubtieng, S: Strong convergence theorem of hybrid method for two asymptotically nonexpansive mappings in Hilbert spaces. Nonlinear Anal. Hybrid Syst.. 2, 1125–1135 (2008). Publisher Full Text

11. Ishikawa, S: Fixed point theorems for asymptotically nonexpansive mappings. Proc. Am. Math. Soc.. 44, 147–150 (1974). Publisher Full Text

12. Kamimura, S, Takahashi, W: Strong convergence of a proximal type algorithm in Banach space. SIAM J. Optim.. 13, 938–945 (2002). Publisher Full Text

13. Khan, AR, Domlo, AA, Fukhar-ud-din, H: Common fixed point Noor iteration for finite family of asymptotically quasi-nonexpansive mappings in Banach spaces. J. Math. Anal. Appl.. 341, 1–11 (2008). Publisher Full Text

14. Kim, TH, Xu, HK: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal.. 24, 1140–1152 (2006)

15. Kimura, Y, Takahashi, W: On a hybrid method for family of relatively nonexpansive mappings in a Banach space. J. Math. Anal. Appl.. 357, 356–363 (2009). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

16. Kirk, WA: A fixed point theorem for mappings which do not increase distance. Am. Math. Mon.. 72, 1004–1006 (1965). Publisher Full Text

17. Lim, TC: A fixed point theorem for families of nonexpansive mappings. Pac. J. Math.. 53, 487–493 (1974). Publisher Full Text

18. Mosco, U: Convergence of convex sets and solutions of variational inequalities. Adv. Math.. 3, 510–585 (1969). Publisher Full Text

19. Nakajo, K, Takahashi, W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl.. 279, 372–379 (2003). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

20. Plubtieng, S, Ughchittrakool, K: Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups. Nonlinear Anal.. 67, 2306–2315 (2007). Publisher Full Text

21. Shahzad, N, Zegeye, H: Strong convergence of an implicit iteration process for finite family of generalized asymptotically quasi-nonexpansive maps. Appl. Math. Comput.. 189, 1058–1065 (2007). Publisher Full Text

22. Shahzad, N, Udomene, A: Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces. Fixed Point Theory Appl.. 2006, Article ID 18909 (2006)

23. Solodov, MV, Svaiter, BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program., Ser. A. 87, 189–202 (2000)

24. Takahashi, W, Takeuchi, Y, Kubota, R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert space. J. Math. Anal. Appl.. 241, 276–289 (2007)

25. Takahashi, W, Tamura, T: Convergence theorems for pair of nonexpansive mappings. J. Convex Anal.. 5, 45–56 (1998)

26. Wang, L: Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings. J. Math. Anal. Appl.. 323, 550–557 (2006). Publisher Full Text

27. Xu, Y, Zhang, X, Khang, J, Su, Y: Modified hybrid algorithm for family of quasi-ϕ-asymptotically nonexpansive mappings. Fixed Point Theory Appl.. 2010, Article ID 170701. doi:10.1155/2010170701 (2010)

28. Zhou, H, Gao, G, Tan, B: Convergence theorems of a modified hybrid algorithm for finite family of quasi-ϕ-asymptotically nonexpansive mappings. J. Appl. Math. Comput.. 32, 453–464 (2010). Publisher Full Text