Abstract
It is our purpose in this paper first to introduce the class of total asymptotically nonexpansive nonself mappings and to prove the demiclosed principle for such mappings in spaces. Then, a new mixed Agarwal-O’Regan-Sahu type iterative scheme for approximating a common fixed point of two total asymptotically nonexpansive mappings and two total asymptotically nonexpansive nonself mappings is constructed. Under suitable conditions, some strong convergence theorems and Δ-convergence theorems are proved in a space. Our results improve and extend the corresponding results of Agarwal, O’Regan and Sahu (J. Nonlinear Convex Anal. 8(1):61-79, 2007), Guo et al. (Fixed Point Theory Appl. 2012:224, 2012. doi:10.1186/1687-1812-2012-224), Sahin et al. (Fixed Point Theory Appl. 2013:12, 2013. doi:10.1186/1687-1812-2013-12), Chang et al. (Appl. Math. Comput. 219:2611-2617, 2012), Khan and Abbas (Comput. Math. Appl. 61:109-116, 2011), Khan et al. (Nonlinear Anal. 74:783-791, 2011), Xu (Nonlinear Anal., Theory Methods Appl. 16(12):1139-1146, 1991), Chidume et al. (J. Math. Anal. Appl. 280:364-374, 2003) and others.
MSC:47J05, 47H09, 49J25.
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1 Introduction and preliminaries
Let be a metric space and with . A geodesic path from x to y is an isometry such that and . The image of a geodesic path is called a geodesic segment. A metric space X is a (uniquely) geodesic space if every two points of X are joined by only one geodesic segment. A geodesic triangle in a geodesic space X consists of three points of X and three geodesic segments joining each pair of vertices. A comparison triangle of a geodesic triangle is the triangle in the Euclidean space such that
A geodesic space X is a space if for each geodesic triangle in X and its comparison triangle in , the inequality
is satisfied for all and .
The initials of the term ‘CAT’ are in honor of Cartan, Alexanderov and Toponogov. A space is a generalization of the Hadamard manifold, which is a simply connected, complete Riemannian manifold such that the sectional curvature is nonpositive. A thorough discussion of these spaces and their important role in various branches of mathematics are given in [1].
In this paper, we write for the unique point z in the geodesic segment joining from x to y such that
We also denote by the geodesic segment joining from x to y, that is, .
A subset C of a space is convex if for all . For elementary facts about spaces, we refer the readers to [1] or [2].
The following lemma plays an important role in our paper.
Lemma 1.1 [2]
A geodesic space X is a space if and only if the following inequality holds:
for all and all . In particular, if x, y, z are points in a space and , then
Let be a metric space, and let C be a nonempty subset of X. Recall that C is said to be a retract of X if there exists a continuous map such that , . A map is said to be a retraction if . If P is a retraction, then for all y in the range of P.
A mapping is said to be nonexpansive if
is said to be asymptotically nonexpansive if there is a sequence with such that
is said to be an asymptotically nonexpansive nonself mapping if there is a sequence with such that
where P is a nonexpansive retraction of X onto C.
is said to be uniformly L-Lipschitzian if there exists a constant such that
Definition 1.2 A self-mapping is said to be -total asymptotically nonexpansive if there exist nonnegative sequences , with , and a strictly increasing continuous function with such that
Definition 1.3 is said to be a -total asymptotically nonexpansive nonself mapping if there exist nonnegative sequences , with , and a strictly increasing continuous function with such that
where P is a nonexpansive retraction of X onto C.
Definition 1.4 A nonself mapping is said to be uniformly L-Lipschitzian if there exists a constant such that
where P is a nonexpansive retraction of X onto C.
Remark 1.5 From the definitions, it is to know that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence , and each asymptotically nonexpansive mapping is a -total asymptotically nonexpansive mapping with , , and , .
In 1976, Lim [3] introduced the concept of Δ-convergence in a general metric space. In 2008, Kirk and Panyanak [4] specialized Lim’s concept to spaces and proved that it is very similar to the weak convergence in a Banach space setting.
Fixed point theory in a space was first studied by Kirk (see [5, 6]). He showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete space always has a fixed point. Since then the existence problem of fixed point and the Δ-convergence problem of iterative sequences to a fixed point for nonexpansive mappings, asymptotically nonexpansive mappings in a space have been rapidly developed and many papers have appeared (see, e.g., [7–26]).
The purpose of this paper is first to introduce the class of total asymptotically nonexpansive nonself mappings and to prove the demiclosed principle for such mappings in spaces. Then, a new mixed Agarwal-O’Regan-Sahu type iterative scheme [27] for approximating a common fixed point of two total asymptotically nonexpansive mappings and two total asymptotically nonexpansive nonself mappings is constructed. Under suitable conditions, some strong convergence theorems and Δ-convergence theorems are proved in a space. Our results extend and improve the corresponding results of Agarwal, O’Regan and Sahu [27], Guo et al. [28], Sahin [26], Chang et al. [24], Khan and Abbas [22], Khan et al. [23], Chidume et al. [29], Xu [30], Chang et al. [31] and many other recent results.
2 Demiclosed principle for total asymptotically nonexpansive nonself mappings
Let be a bounded sequence in a space X. For , we set
The asymptotic radius of is given by
The asymptotic radius of with respect to is given by
The asymptotic center of is the set
And the asymptotic center of with respect to is the set
Proposition 2.1 [7]
Let X be a complete space, let be a bounded sequence in X and let C be a closed convex subset of X. Then
-
(1)
there exists a unique point such that
-
(2)
and both are singleton.
Let X be a space. A sequence in X is said to Δ-converge to if p is the unique asymptotic center of for each subsequence of . In this case, we write and call p the Δ-limit of .
Lemma 2.3
-
(1)
Let X be a complete space, let C be a closed convex subset of X. If is a bounded sequence in C, then the asymptotic center of is in C [8];
-
(2)
Every bounded sequence in a complete space always has a Δ-convergent subsequence [4].
Remark 2.4 Let X be a space and let C be a closed convex subset of X. Let be a bounded sequence in C. In what follows, we denote it by
where .
Now we give a connection between the ‘⇀’ convergence and Δ-convergence.
Proposition 2.5 Let X be a space, let C be a closed convex subset of X and let be a bounded sequence in C. Then implies that .
Proof In fact, if , then it follows from Lemma 2.3 that . Since , we have . This implies that , i.e., . The desired conclusion is obtained. □
It is well known that one of the fundamental and celebrated results in the theory of nonexpansive mappings is Browder’s demiclosed principle [32] which states that if X is a uniformly convex Banach space, C is a nonempty closed convex subset of X, and is a nonexpansive mapping, then is demiclosed at 0, i.e., for any sequence in C if weakly and , then .
Later, Xu [30] and Chang et al. [31] proved the demiclosed principle for asymptotically nonexpansive mappings in a uniformly convex Banach space. In 2003, Chidume et al. [29] proved the demiclosed principle for asymptotically nonexpansive nonself mappings in uniformly convex Banach spaces.
In this section, by using the convergence ‘⇀’ defined by (2.5), we prove the demiclosed principle for total asymptotically nonexpansive nonself mappings in spaces, which extends the results of Xu [30], Chang et al. [31] and Chidume et al. [29] to spaces.
Theorem 2.6 (Demiclosed principle for total asymptotically nonexpansive nonself mappings in spaces)
Let C be a nonempty closed and convex subset of a complete space X, and let be a uniformly L-Lipschitzian and -total asymptotically nonexpansive nonself mapping. Let be a bounded sequence in C such that defined by (2.5) and . Then .
Proof By the definition and Proposition 2.1, if and only if . By Lemma 2.3, we have .
Since , by induction we can prove that
In fact, it is obvious that the conclusion is true for . Suppose the conclusion holds for , now we prove that the conclusion is also true for .
Indeed, since , we have . In addition, since T is uniformly L-Lipschitzian, we have
Equation (2.6) is proved. Hence for each and , we have
In (2.7), taking , , we have
Letting and taking superior limit on both sides, we get that
Furthermore, for any , it follows from inequality (1.3) with that
Letting and taking superior limit on both sides of the above inequality, for any , we get
Since , for any , we have
This implies that
From (2.8) and (2.12), we have . Hence we have
i.e., as desired. □
The following theorem can be obtained from Theorem 2.6 immediately which is a generalization of Kirk et al. [[4], Proposition 3.7], Xu [30], Chang et al. [31] and Chidume et al. [[29], Theorem 3.4].
Theorem 2.7 Let C be a closed and convex subset of a complete space X. Let T be a mapping satisfying one of the following conditions:
-
(1)
is an asymptotically nonexpansive mapping with a sequence , ;
-
(2)
is an asymptotically nonexpansive nonself mapping with a sequence , ;
-
(3)
is a -total asymptotically nonexpansive mapping.
Let be a bounded sequence in C such that and . Then .
3 Δ-convergence theorems for total asymptotically nonexpansive mappings in spaces
In this section we prove some Δ-convergence theorems for the mixed Agarwal-O’Regan-Sahu type iterative scheme [27]
where C is a nonempty bounded closed and convex subset of a complete space X, P is a nonexpansive retraction of X onto C, , , is a uniformly -Lipschitzian and -total asymptotically nonexpansive nonself mapping (defined by (1.7)), and , , is a uniformly -Lipschitzian and total asymptotically nonexpansive mapping (defined by (1.6)) such that the following conditions are satisfied:
-
(1)
, , , , ;
-
(2)
There exists a constant such that , , , .
Remark 3.1 Without loss of generality, in the sequel, we can assume that and , , both are uniformly L-Lipschitzian and -total asymptotically nonexpansive mappings satisfying the conditions (1) and (2). In fact, letting , , and , then and , , are the mappings satisfying the required conditions.
The following lemmas will be used to prove our main results.
Lemma 3.2 (Chang et al. [24])
Let X be a space, be a given point and be a sequence in with and . Let and be any sequences in X such that
for some . Then
Lemma 3.3 Let , and be the sequences of nonnegative numbers such that
If and , then exists. If there exists a subsequence such that , then .
Lemma 3.4 [2]
Let X be a complete space, be a bounded sequence in X with , and be a subsequence of with and the sequence converges, then .
Now we are in a position to give the main results of this paper.
Theorem 3.5 Let C be a bounded closed and convex subset of a complete X. Let , , be a uniformly L-Lipschitzian and -total asymptotically nonexpansive nonself mapping, and let , , be a uniformly L-Lipschitzian and -total asymptotically nonexpansive mapping. If and the following conditions are satisfied:
-
(i)
; ;
-
(ii)
there exist constants with such that ;
-
(iii)
there exists a constant such that , ;
-
(iv)
for all and ,
then the sequence defined by (3.1) Δ-converges to some point (a common fixed point of and , ).
Proof (I) First we prove that the following limits exist
In fact, since , . In addition, since and , , are total asymptotically nonexpansive mappings, by the condition (iii), we have
and
Substituting (3.4) into (3.5) and simplifying it, we have
and so
where , . By virtue of the condition (i),
By Lemma 3.3 the limits and exist for each .
(II) Next we prove that
In fact, it follows from (3.3) that for each given , exists. Without loss of generality, we can assume that
From (3.4) we have
Since
and
then we have
and
In addition, it follows from (3.6) that
This implies that
From (3.12)-(3.14) and Lemma 3.2, one gets that
By the same method, we can also prove that
By virtue of the condition (iv), it follows from (3.15) and (3.16) that
and
Since , . By (3.1) and (3.16) we have
Observe that
From (3.18) and (3.19) we get
This together with (3.17) implies that
On the other hand, by the condition (iv), . Hence from (3.17) and (3.20), we have
By the condition (iv), . Hence from (3.22) we have that
This together with (3.17) shows that
Hence from (3.18), (3.21) and (3.23), for each , we have
By virtue of the condition (iv), . It follows from (3.18), (3.21) and (3.22) that
Equation (3.9) is proved.
(III) Now we prove that
and consists of exactly one point.
In fact, let , then there exists a subsequence of such that . By Lemma 2.3, there exists a subsequence of such that . In view of (3.9), , , . It follows from Theorem 2.7 that . So, by (3.3), the limit exists. By Lemma 3.4 . This implies that .
Next we prove that consists of exactly one point. Let be a subsequence of with and let . Since , from (3.3) the limit exists. In view of Lemma 3.4, . The conclusion is proved.
(IV) Finally we prove Δ-converges to a point in ℱ.
In fact, it follows from (3.3) that is convergent for each . By (3.9) and (3.26), , , and consists of exactly one point. This shows that Δ-converges to a point of ℱ.
The conclusion of Theorem 3.5 is proved. □
Remark 3.6 (1) Now we give an example which satisfies the condition (iv) in Theorem 3.5.
Let be a subset in ℛ. Define two mappings by
and
It is proved in Guo [28] that both S and T are asymptotically nonexpansive mappings (therefore they are total asymptotically nonexpansive mappings) with and satisfy the condition (iv).
(2) Theorem 3.5 contains the main results of Sahin [26], Khan Abbas [22], Khan et al. [23] and Chang et al. [24] as its special cases. Theorem 3.5 also extends the main result of Guo et al. [28] from a Banach space to a space.
The following results can be obtained from Theorem 3.5 immediately.
Theorem 3.7 Let C, X and , be the same as in Theorem 3.5. If and the following conditions are satisfied:
-
(i)
; ;
-
(ii)
there exist constants with such that .
-
(iii)
there exists a constant such that , ;
then the sequence defined by
Δ-converges to a common fixed point of and .
Proof Take (the identity mapping on C) in Theorem 3.5 and note that in this case the condition (iv) in Theorem 3.5 is satisfied automatically. Hence the conclusion of Theorem 3.7 can be obtained from Theorem 3.5 immediately. □
Theorem 3.8 Let C and X be the same as in Theorem 3.5. Let and , , be uniformly L-Lipschitzian and -total asymptotically nonexpansive mappings. If and the (i)-(iv) in Theorem 3.5 are satisfied, then the sequence defined by
Δ-converges to a common fixed point of and , .
Proof Since , , is a self-mapping from C to C, take (the identity mapping on C), then . The conclusion of Theorem 3.8 is obtained from Theorem 3.5. □
Remark 3.9 Theorem 3.8 improves and extends the main results of Agawal O’Regan Sahu [27] from a Banach space to a space. As well as it also extends and improves the main results in Sahin [26].
4 Strong convergence theorems for total asymptotically nonexpansive mappings in spaces
Recall that a mapping is said to be demi-compact if for any sequence in C such that (as ), there exists a subsequence such that converges strongly (i.e., in metric topology) to some point .
Theorem 4.1 Under the assumptions of Theorem 3.5, if one of , , and is demi-compact, then the sequence defined by (3.1) converges strongly (i.e., in metric topology) to a common fixed point .
Proof By virtue of (3.9): , , and one of , , and is demi-compact, there exists a subsequence such that converges strongly to some point . Moreover, by the continuity of , , and , for each , we have
This implies that . Again by (3.3) the limit exists. Hence we have . This completes the proof of Theorem 4.1. □
Theorem 4.2 Under the assumptions of Theorem 3.5, if there exists a nondecreasing function with , , such that
then the sequence defined by (3.1) converges strongly (i.e., in metric topology) to a common fixed point .
Proof It follows from (3.9) that
Therefore we have . Since f is a nondecreasing function with and , , we have . Next we prove that is a Cauchy sequence in C. In fact, it follows from (3.6) that for any
where and . Hence for any positive integers n, m, we have
Since for each , , we have
where . By (3.3) . Therefore we have
This shows that is a Cauchy sequence in C. Since C is a closed subset in a complete space X, it is complete. Without loss of generality, we can assume that converges strongly (i.e., in metric topology in X) to some point . It is easy to prove that and , are closed subsets in C, so is ℱ. Since , . This completes the proof of Theorem 4.2. □
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The authors would like to express their thanks to the referees for their helpful comments and suggestions. This work was supported by the Natural Science Foundation of Yunnan Province, Grant No. 2011FB074.
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Chang, Ss., Wang, L., Joseph Lee, H.W. et al. Strong and Δ-convergence for mixed type total asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl 2013, 122 (2013). https://doi.org/10.1186/1687-1812-2013-122
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DOI: https://doi.org/10.1186/1687-1812-2013-122