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Fixed point theorems for N-generalized hybrid mappings in uniformly convex metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 188 (2013)
Abstract
In this paper, we prove some fixed point theorems for N-generalized hybrid mappings in both uniformly convex metric spaces and spaces. We also introduce a new iteration method for approximating a fixed point of N-generalized hybrid mappings in spaces and obtain Δ-convergence to a fixed point of N-generalized hybrid mappings in such spaces. Our results improve and extend the corresponding results existing in the literature.
MSC:47H09, 47H10.
1 Introduction and preliminaries
Let C be a nonempty closed subset of a metric space and let T be a mapping of C into itself. The set of all fixed points of T is denoted by . In 1970, Takahashi [1] introduced the concept of convex metric spaces by using the convex structure as follows.
Definition 1.1 Let be a metric space. A mapping is said to be a convex structure on X if for each and ,
for all . A metric space together with a convex structure W is called a convex metric space which will be denoted by .
A nonempty subset C of X is said to be convex if for all and . Clearly, a normed space and each of its convex subsets are convex metric spaces, but the converse does not hold. For each and , it is known that a convex metric space has the following properties [1, 2]:
-
(i)
, and ;
-
(ii)
and .
In 1996, Shimizu and Takahashi [3] introduced the concept of uniform convexity in convex metric spaces and studied some properties of these spaces. A convex metric space is said to be uniformly convex if for any , there exists such that for all and with , and imply that . Obviously, uniformly convex Banach spaces are uniformly convex metric spaces.
Let C be a nonempty closed and convex subset of a convex metric space and let be a bounded sequence in X. For , we define a mapping by
Clearly, is a continuous and convex function. The asymptotic radius of relative to C is given by
and the asymptotic center of relative to C is the set
It is clear that the asymptotic center is always closed and convex. It may either be empty or consist of one or many points. The asymptotic center is singleton for uniformly convex Banach spaces [4, 5] or spaces [6]. The following lemma obtained by Phuengrattana and Suantai [7] is useful for our results.
Lemma 1.2 Let C be a nonempty closed and convex subset of a complete uniformly convex metric space and let be a bounded sequence in X. Then is a singleton set.
One of the special spaces of uniformly convex metric spaces is a space; see [8]. It was noted in [9] that any space () is uniformly convex in a certain sense but it is not a space. Fixed point theory in spaces was first studied by Kirk [9, 10]. 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 fixed point theory for single-valued and multivalued mappings in spaces has been rapidly developed, and many papers have appeared (e.g., see [11–27]).
Let be a metric space. A geodesic path joining to (or, more briefly, a geodesic from x to y) is a map c from a closed interval to X such that , and for all . In particular, c is an isometry and . The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by . Write for . The space is said to be a geodesic metric space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each . A subset Y of X is said to be convex if Y includes every geodesic segment joining any two of its points.
A geodesic triangle in a geodesic metric space consists of three points , , in X (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparison triangle for the geodesic triangle in is a triangle in the Euclidean plane such that for .
A geodesic metric space is said to be a space if all geodesic triangles satisfy the following comparison axiom: Let △ be a geodesic triangle in X and let be a comparison triangle for △. Then △ is said to satisfy the inequality if for all and all comparison points ,
It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a space. Other examples include pre-Hilbert spaces [8], ℝ-trees [16], the complex Hilbert ball with a hyperbolic metric [5], and many others.
If z, x, y are points in a space and if is the midpoint of the segment , then the inequality implies
This is the (CN) inequality of Bruhat and Tits [28], which is equivalent to
for any . The (CN*) inequality has appeared in [29]. Moreover, if X is a space and , then for any , there exists a unique point such that
for any . It follows that spaces have a convex structure .
Remark 1.3
-
(i)
By using the (CN) inequality, it is easy to see that spaces are uniformly convex.
-
(ii)
A geodesic metric space is a space if and only if it satisfies the (CN) inequality; see [8].
In 2012, Dhompongsa et al. [12] introduced the following notation in spaces: Let be points in a space X and with , we write
The definition of ⨁ is an ordered one in the sense that it depends on the order of points . Under (1.1) we obtain that
In 1976, Lim [30] introduced the concept of Δ-convergence in a general metric space. Later in 2008, Kirk and Panyanak [15] extended the concept of Lim to a space.
Definition 1.4 [15]
A sequence in a space X is said to Δ-converge to if x is the unique asymptotic center of for every subsequence of . In this case, we write and call x the Δ-limit of .
Lemma 1.5 [15]
Every bounded sequence in a complete space has a Δ-convergent subsequence.
For any nonempty subset C of a space X, let be the nearest point projection mapping from X to a subset C of X. In [8], it is known that if C is closed and convex, the mapping π is well defined, nonexpansive, and the following inequality holds:
for all and . By using the same argument as in [[31], Lemma 3.2], we can prove the following result for nearest point projection mappings in spaces.
Lemma 1.6 Let C be a nonempty closed and convex subset of a complete space X, let be the nearest point projection mapping, and let be a sequence in X. If for all and , then converges strongly to some element in C.
Proof Let . By the (CN) inequality and the property of π, it follows that
This implies that
Then exists. Letting in (1.2), we have that is a Cauchy sequence in a closed subset C of a complete space X, hence it converges to some element in C. □
Let C be a nonempty closed and convex subset of a Hilbert space H. A mapping is called generalized hybrid if there exist such that
for all . We note that the generalized hybrid mappings generalize several well-known mappings. For example, a generalized hybrid mapping is nonexpansive for and , nonspreading for and , and hybrid for and . In 2010, Kocourek et al. [32] proved the fixed point theorems for generalized hybrid mappings in Hilbert spaces. Later in 2011, Takahashi and Yao [33] extended the results of Kocourek et al. to uniformly convex Banach spaces.
Recently, Maruyama et al. [34] introduced a new nonlinear mapping in a Hilbert space as follows. Let . A mapping is called N-generalized hybrid if there are such that
for all . They obtained the existence and weak convergence theorems for N-generalized hybrid mappings in Hilbert spaces. Hojo et al. [35] also studied the fixed point theorems for N-generalized hybrid mappings in Hilbert spaces and provided an example of N-generalized hybrid mappings which are not generalized hybrid mappings as follows.
Example 1.7 Let H be a Hilbert space, and define a mapping as follows:
We observe that the N-generalized hybrid mappings generalize several well-known mappings, for instance, nonexpansive mappings, nonspreading mappings, hybrid mappings, λ-hybrid mappings, generalized hybrid mappings, and 2-generalized hybrid mappings. Many researchers have studied the fixed point theorems of those mappings in both Hilbert spaces and Banach spaces (e.g., see [32, 33, 36–38]). However, no researcher has studied the fixed point theorems for N-generalized hybrid mappings in more general spaces. So, in this paper, we are interested in studying and extending those mappings to both uniformly convex metric spaces and spaces.
2 Fixed point theorems in uniformly convex metric spaces
We first define N-generalized hybrid mappings in convex metric spaces. Let C be a nonempty subset of a convex metric space . Let . A mapping is called N-generalized hybrid if there are such that
for all . Now, we prove a fixed point theorem for N-generalized hybrid mappings in complete uniformly convex metric spaces.
Theorem 2.1 Let C be a nonempty closed and convex subset of a complete uniformly convex metric space and let be an N-generalized hybrid mapping with and . Then T has a fixed point if and only if there exists an such that is bounded.
Proof The necessity is obvious. Conversely, we assume that there exists an such that is bounded. We will show that is nonempty. From Lemma 1.2, is a singleton set. Let . Since T is N-generalized hybrid, there are such that
If and , then (2.1) becomes
This implies that
If and , then (2.1) becomes
This implies again that
Therefore, we have
Since and , it implies that . Hence, is nonempty. □
As a direct consequence of Theorem 2.1, we obtain a fixed point theorem for N-generalized hybrid mappings in uniformly convex metric spaces as follows.
Theorem 2.2 Let C be a nonempty bounded closed and convex subset of a complete uniformly convex metric space and let be an N-generalized hybrid mapping with and . Then T has a fixed point.
We can show that if T is an N-generalized hybrid mapping and , then for any , we get
and hence . This means that an N-generalized hybrid mapping with a fixed point is quasi-nonexpansive. Then, using the methods of the proof of Theorem 1.3 in [13], we can prove the following.
Corollary 2.3 Let C be a nonempty convex subset of a complete uniformly convex metric space . Suppose that is an N-generalized hybrid mapping and has a fixed point. Then is closed and convex.
Remark 2.4
3 Fixed point theorems in spaces
In this section, we study the existence and Δ-convergence theorems for N-generalized hybrid mappings in complete spaces.
We first recall the definition of a Banach limit. Let μ be a continuous linear functional on , the Banach space of bounded real sequences, and . We write instead of . We call μ a Banach limit if μ satisfies and for each . For a Banach limit μ, we know that for all . So if with , then ; see [39] for more details.
Now, we obtain the following lemma in spaces.
Lemma 3.1 Let C be a nonempty closed and convex subset of a complete space X, let be a bounded sequence in X, and let μ be a Banach limit. If a function is defined by
then there exists a unique such that
Proof It is easy to show that f is continuous. By (CN) inequality, we obtain that
This implies by Proposition 1.7 in [40] that there exists a unique such that . □
By using Lemma 3.1, we can prove the following fixed point theorem for N-generalized hybrid mappings in spaces without the assumptions that and .
Theorem 3.2 Let C be a nonempty closed and convex subset of a complete space X and let be an N-generalized hybrid mapping. Then T has a fixed point if and only if there exists an such that is bounded.
Proof The necessity is obvious. Conversely, we assume that there exists an such that is bounded. Let μ be a Banach limit. Since T is N-generalized hybrid, there are such that
for any and . Since is bounded, we have
This implies that
for all . It follows by Lemma 3.1 that . Hence, is nonempty. □
As a direct consequence of Theorem 3.2, we obtain a fixed point theorem for N-generalized hybrid mappings in spaces as follows.
Theorem 3.3 Let C be a nonempty bounded closed and convex subset of a complete space X and let be an N-generalized hybrid mapping. Then T has a fixed point.
Remark 3.4 Theorems 3.2 and 3.3 extend and generalize the corresponding results in [17, 32–34, 36–38] to N-generalized hybrid mappings on spaces.
Next, we study the Δ-convergence theorem for N-generalized hybrid mappings in spaces. Before proving the theorem, we need the following lemma.
Lemma 3.5 Let C be a nonempty closed and convex subset of a complete space X and let be an N-generalized hybrid mapping with and . If is a bounded sequence in C with and for all , then .
Proof Since T is an N-generalized hybrid mapping, there are such that
Case 1: and . It follows by (3.1) that
This implies that
Since is bounded and for all , we have that are bounded. So, we have
where .
Case 2: and . In the same way as Case 1, we can show that
By Case 1, Case 2, and the assumption for all , we obtain
Since , it follows by the uniqueness of asymptotic centers that . Hence, . □
Fixed point iteration methods are very useful for approximating a fixed point of various nonlinear mappings such as Mann iteration, Ishikawa iteration, Noor iteration and so on. We now introduce a new iteration method for approximating a fixed point of mappings in a space X as follows: Let C be a nonempty closed and convex subset of X, let be a mapping and . For , the sequence generated by
where is a sequence in for all with .
Remark 3.6 If we put
then by (1.1) we get
Indeed, we put for . Thus
Therefore, (3.3) is justified.
Using Lemma 3.5, we can prove the Δ-convergence theorem for N-generalized hybrid mappings in complete spaces as follows.
Theorem 3.7 Let C be a nonempty closed and convex subset of a complete space X and let be an N-generalized hybrid mapping with and and . Let be the nearest point projection mapping. Suppose that is a sequence in C defined by (3.2) with for all . Then Δ-converges to a fixed point u of T, where .
Proof Since T is N-generalized hybrid and , we get T is quasi-nonexpansive. Then, for , we have
Therefore, exists and hence is bounded.
For each , we obtain, by (3.2), (3.3), and the (CN*) inequality, that
This implies that
Since exists and for all , we get that
For , we have
This implies by (3.4) that
We now let , where the union is taken over all subsequences of . We claim that . Let . Then there exists a subsequence of such that . By Lemma 1.5, there exists a subsequence of such that . It implies by (3.5) and Lemma 3.5 that . Then, exists. Suppose that . By the uniqueness of asymptotic centers, we get
This is a contradiction, hence . This shows that .
Next, we show that consists of exactly one point. Let be a subsequence of with and let . Since , it follows that exists. We will show that . To show this, suppose not. By the uniqueness of asymptotic centers, we get
which is a contradiction, and so . Hence, Δ-converges to a fixed point u of T. Since is a closed convex subset of X and for all and , we obtain by Lemma 1.6 that converges strongly to some element in , say q. Thus, by the property of π, we obtain that
This implies, by the uniqueness of asymptotic centers, that . This means . □
Taking in Theorem 3.7, we obtain the following Δ-convergence theorem of a 2-generalized hybrid mapping in spaces.
Theorem 3.8 Let C be a nonempty closed and convex subset of a complete space X. Let be a 2-generalized hybrid mapping, i.e., there are such that
for all . Assume that and and . Let be the nearest point projection mapping. For , let be a sequence defined by
where is a sequence in with for all and . Then Δ-converges to a fixed point u of T, where .
Taking in Theorem 3.7, we obtain the following Δ-convergence theorem of a generalized hybrid mapping in spaces.
Theorem 3.9 Let C be a nonempty closed and convex subset of a complete space X. Let be a generalized hybrid mapping, i.e., there are such that
for all . Assume that and and . Let be the nearest point projection mapping. For , let be a sequence defined by
where and are sequences in with and . Then Δ-converges to a fixed point u of T, where .
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Phuengrattana, W. Fixed point theorems for N-generalized hybrid mappings in uniformly convex metric spaces. Fixed Point Theory Appl 2013, 188 (2013). https://doi.org/10.1186/1687-1812-2013-188
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DOI: https://doi.org/10.1186/1687-1812-2013-188