Convergence of a hybrid iterative method for finite families of generalized quasi-ϕ-asymptotically nonexpansive mappings
© Ali and Minjibir; licensee Springer 2012
Received: 10 January 2012
Accepted: 4 July 2012
Published: 23 July 2012
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.
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 .
asymptotically regular on C if for all ,
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 uniformly L-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 . 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 . Goebel and Kirk  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 , 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  and then to an infinite family of nonexpansive maps by Lim .
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 , 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, ).
relatively nonexpansive if and for all , where denotes the set of asymptotic fixed points of T;
ϕ-nonexpansive if for all ;
ϕ-asymptotically nonexpansive if there exists a sequence satisfying as and for all , ;
quasi-ϕ-asymptotically nonexpansive if and for all , , , where is as in (iii) above.
We shall call the map T generalized quasi-ϕ-asymptotically nonexpansive in the light of , 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.
where and are sequences in such that . They studied the scheme for two quasi-nonexpansive maps S 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  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  proved strong and weak convergence theorems for a pair of nonself asymptotically nonexpansive mappings in a uniformly convex Banach space.
Shahzad and Udomene  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.
to a common fixed point of a finite family of nonself asymptotically nonexpansive mappings in a uniformly convex Banach space.
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  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  proposed a hybrid Mann scheme for nonexpansive mappings and nonexpansive semigroups and proved strong convergence theorems.
Kim and Xu  generalized the result of Nakajo and Takahashi by proving strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups. Plubtieng and Ughchittrakool  introduced an Ishikawa type hybrid scheme for two asymptotically nonexpansive mappings and two asymptotically nonexpansive semigroups.
Takahashi et al.  studied a simpler hybrid scheme for nonexpansive mappings in Hilbert spaces. Inchan and Plubtieng , 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 ()
converges strongly to , where as and , for all .
Kimura and Takahashi  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.  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.  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 ()
Then, converges strongly to , where is the generalized projection from E onto 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  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 )
Let E be a real smooth and uniformly convex Banach space and , be two sequences of E. If and either or is bounded, then .
Lemma 2.2 (Ibaraki, Kimura and Takahashi )
Let C be a nonempty closed convex subset of a real uniformly smooth and uniformly convex Banach space E. Let be a sequence of nonempty closed convex subsets of C. If exists and is nonempty, then converges strongly to for each .
The result in  is more general than the one presented here, but this is sufficient for our purpose.
Lemma 2.3 Let C be a nonempty closed convex subset of a real smooth Banach space and be a closed generalized quasi-ϕ-asymptotically nonexpansive mapping. Then is closed and convex.
Proof By the closedness assumption on T and the definition of ϕ, the result follows immediately. □
3 Main results
where . Then the sequence converges strongly to .
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 .
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 3.2 Let E, C, , F, , , and be as in Theorem 3.1 with the exception that are uniformly , , Lipschitzian instead of uniformly asymptotically regular and closed. Then the sequence converges strongly to .
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 3.3 Let E be a real uniformly convex and uniformly smooth Banach space and C be a nonempty, bounded, closed and convex subset of E. Let , be a finite family of quasi-ϕ-asymptptically nonexpansive maps with corresponding sequences , , such that , as . Let and let . Assume also that the maps , are either closed and uniformly asymptotically regular on C or uniformly Lipschitzian on C. Let be arbitrary and . For , let be sequences in for some , . Let be a sequence generated by (3.1). Then the sequence converges strongly to .
Corollary 3.4 Let E be a real uniformly convex and uniformly smooth Banach space and C be a nonempty, bounded, closed and convex subset of E. Let , be a finite family of ϕ-asymptotically nonexpansive maps with corresponding sequences , , such that , as . Let and let . Assume also that the maps , are either closed and uniformly asymptotically regular on C or uniformly Lipschitzian on C. Let be arbitrary and . For , let be sequences in for some , . Let be a sequence generated by (3.1). Then the sequence converges strongly to .
where . Then, the sequence converges strongly to .
Corollary 3.6 Let H be a real Hilbert space, C be a nonempty,closed and convex subset of H. Let , be a finite family of asymptotically nonexpansive maps with corresponding sequences , , such that as . Let and let . Let be arbitrary and . For , let be sequences in for some , . Let be a sequence generated by (3.10). Then the sequence converges 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 .
Remark 3.8 It is not clear whether Theorem 3.1 and Corollary 3.5 hold without the boundedness assumption on C.
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.
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