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Results on n-tupled fixed points in metric spaces with uniform normal structure
Fixed Point Theory and Applications volume 2014, Article number: 168 (2014)
In this paper, we introduce the concept of new notions related to n-tupled fixed point and prove some related results for an asymptotically regular one-parameter semigroup of Lipschitzian self-mappings on in the case when is a complete bounded metric space with uniform normal structure. Our results extend the results due to Yao and Zeng (J. Nonlinear Convex Anal. 8(1):153-163, 2007) and Soliman (Fixed Point Theory Appl. 2013:346, 2013; J. Adv. Math. Stud. 7(2):2-14, 2014).
The Banach contraction principle is the most natural and significant result of fixed point theory. In complete metric spaces it continues to be an indispensable and effective tool in theory and applications, which guarantees the existence and uniqueness of fixed points of contraction self-mappings besides offering a constructive procedure to compute the fixed point of the underlying mapping. There already exists an extensive literature on this topic. Keeping in view the relevance of this paper, we merely refer to [1–5]. In 1987, the idea of coupled fixed point was initiated by Guo and Lakshmikantham ; it was also followed by Bhaskar and Lakshmikantham  wherein authors proved some interesting coupled fixed point theorems for mappings satisfying the mixed monotone property. Many authors obtained important coupled, tripled and n-tupled fixed point theorems (see [7–16]). In this continuation, Lakshmikantham and Ćirić  introduced coupled common fixed point theorems for nonlinear ϕ-contraction mappings in partially ordered complete metric spaces which indeed generalize the corresponding fixed point theorems contained in Bhaskar and Lakshmikantham . In 2010, Samet and Vetro  introduced the concept of fixed point of n-tupled fixed point (where ) for nonlinear mappings in complete metric spaces. They obtained the existence and uniqueness theorems for contractive type mappings. Their results generalized and extended coupled fixed point theorems established by Bhaskar and Lakshmikantham . Recently, Imdad et al.  introduced a generalization of n-tupled fixed point and n-tupled coincidence point by considering n even besides using the idea of mixed g-monotone property on and proved an n-tupled (where n is even) coincidence point theorem for nonlinear ϕ-contraction mappings satisfying the mixed g-monotone property. For more information about n-tupled fixed points, see [10, 17–19].
On the other hand, normal structure is one of the most important aspects of metric fixed point theory. It was introduced by Brodskii and Milman in . They found the first application of normal structure to fixed point theory. In 1965, Kirk  introduced the following theorem: Every nonexpansive self-mapping on a weakly compact convex subset of a Banach space with normal structure has a fixed point. In 1969, Kijima and Takahashi  established the metric space version of Kirk’s theorem . Subsequently, many authors successfully generalized certain fixed point theorems and structure properties from Banach spaces to metric spaces. For example, Khamsi  defined normal and uniform normal structure for metric spaces and proved that if is a complete bounded metric space with uniform normal structure, then it has the fixed point property for nonexpansive mappings and a kind of intersection property which extends a result of Maluta  to metric spaces. In 1995, Lim and Xu  proved a fixed point theorem for uniformly Lipschitzian mappings in metric spaces with both property (P) and uniform normal structure, which extended the result of Khamsi . This is the metric space version of Casini and Maluta’s theorem . In 2007, Yao and Zeng  established a fixed point theorem for an asymptotically regular one-parameter semigroup of uniformly k-Lipschitzian mappings with property (∗) in a complete bounded metric space with uniform normal structure, which extended the results of Lim and Xu . Recently, the idea of coupled and tripled fixed point results in a complete bounded metric space X with uniform normal structure was initiated by Soliman [27, 28]. He proved that every asymptotically regular one-parameter semigroup of Lipschitzian mappings on has a coupled fixed point and on has a tripled fixed point.
In the present paper, we prove an n-tupled fixed point theorem for asymptotically regular Lipschitzian one-parameter semigroups on , where X is a complete bounded metric space with uniform normal structure. Also, some corollaries of our main theorem are presented.
Definition 2.1 
An element is called a coupled fixed point of the mapping if
Definition 2.2 
Let be a partially ordered set and . We say that F has the mixed monotone property if is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any , , and , .
Theorem 2.1 
Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a continuous mapping having the mixed monotone property on X. Assume that there exists a constant with
If there exist such that and , then there exist such that and .
Definition 2.3 
An element is called a tripled fixed point of the mapping if
Definition 2.4 
Let X be a nonempty set. An element is called an r-tupled fixed point of the mapping if
Definition 2.5 
Suppose that is a metric space, and let μ denote a nonempty family of subsets of X. Then μ defines a convexity structure on X if it is stable under intersection.
Definition 2.6 
Let μ be a convexity structure on a metric . Then μ has property (R) if any decreasing sequence of nonempty bounded closed subsets of X with has a nonempty intersection.
Definition 2.7 
A subset of X is said to be admissible if it is an intersection of closed balls.
Remark 2.1 Let be a family of all admissible subsets of X. Then we note that defines a convexity structure on X.
In this paper any other convexity structure μ on X is always assumed to contain .
Let M be a bounded subset of X. Following Lim and Xu , we shall adopt the following notations:
is the closed ball centered at x with radius r,
For a bounded subset A of X, we define the admissible hull of A, denoted by , as the intersection of all those admissible subsets of X which contain A, i.e.,
Proposition 2.1 
For a point and a bounded subset A of X, we have
Definition 2.8 
A metric space is said to have normal (resp. uniform normal) structure if there exists a convexity structure μ on X such that (resp. for some constant ) for all which is bounded and consists of more than one point. In this case μ is said to be normal (resp. uniformly normal) in X.
We define the normal structure coefficient of X (with respect to a given convexity structure μ) as the number
where the supremum is taken over all bounded with . X then has uniform normal structure if and only if .
Khamsi proved the following result that will be very useful in the proof of our main theorem.
Proposition 2.2 
Let X be a complete bounded metric space and μ be a convexity structure of X with uniform normal structure. Then μ has property (R).
Definition 2.9 
Let be a metric space and be a semigroup on . Let us write the set
Lemma 2.1 
If , then .
Definition 2.10 
A metric space is said to have property (P) if given any two bounded sequences and in X, one can find some such that
3 Main results
Let G be a subsemigroup of with addition ‘+’ such that
This condition is satisfied if or , the set of nonnegative integers. Let be a family of self-mappings on . Then ℑ is called a (one-parameter) semigroup on if the following conditions are satisfied:
, , …, ;
, the self-mappings , , …, from G into X are continuous when G has the relative topology of .
Definition 3.1 A semigroup on is said to be asymptotically regular at a point if
If ℑ is asymptotically regular at each , then ℑ is called an asymptotically regular semigroup on .
A semigroup on is called simplest asymptotically regular at a point if
Definition 3.2 A semigroup on is called a uniformly Lipschitzian semigroup if
The simplest uniformly Lipschitzian semigroup is a semigroup of iterates of a mapping with
, where .
Definition 3.3 A mapping has an r-tupled fixed point if , , , …, .
Definition 3.4 Let be a complete bounded metric space and be a semigroup on . Then ℑ has property (∗) if for each and each , the following conditions are satisfied:
the sequences , …, are bounded;
for any sequence in , there exists some such that
for any sequence in , there exists some such that
Remark 3.1 If X is a complete bounded metric space with property (P), then each semigroup on has property (∗).
Lemma 3.1 Let be a complete bounded metric space with uniform normal structure, and let be a semigroup on with property (∗). Then, for each , each and for any constant , the normal structure coefficient with respect to the given convexity structure μ, there exist some , …, satisfying the following properties:
Proof For each integer , let , , …, . Then are decreasing sequences of admissible subsets of X hence , , …, by Proposition 2.2. From Proposition 2.1, it is not difficult to see that , , …, . Indeed, observe that
Similarly, one can obtain
On the other hand, for any and any , we have
Also, one can deduce that for any , …, and any , we have
from which (II) follows.
We now suppose that for each , there exist , , …, such that
Indeed, if , then , we conclude that (3) holds. Without loss of generality, we may assume that . Then, for , we choose so small that it satisfies the following:
From the definition of , one can find such that
which implies that (3) holds. Obviously, it follows from (3) that for each ,
where . Noticing
we know that property (∗) yields a point such that
Since , and satisfies
similarly one can obtain that
where , …, .
Therefore (I) holds. □
We are ready to prove our main theorem for this paper.
Theorem 3.1 Let be a complete bounded metric space with uniform normal structure, and let be an asymptotically regular uniformly Lipschitzian semigroup of self-mappings on with property (∗) and satisfying
where and .
Then there exist some such that , …, and for all .
Proof First, we choose a constant c such that and . We can select a sequence , from Lemma 2.1 we find that and .
Now fix . Then, by Lemma 3.1, we can inductively construct sequences , , …, such that , , …, for each integer ,
Observe that for each , using (IV) we have
By the asymptotic regularity of on , we see that
Similarly, one can show that
Then it follows from (9), (10), (11) and (12) that for each ,
which implies that for each ,
Hence, by using (III) and (9), we have
Hence, by the asymptotic regularity of ℑ on , we have, for each integer ,
It follows from (14) that
Similarly, one can deduce that
Thus, we have , …, . Consequently, are Cauchy and hence convergent as X is complete. Let , …, , then, for each , by the continuity of we have
On the other hand, from (15) we have actually proven the following inequalities:
Since , it follows that
Similarly, one can obtain that
i.e., , …, . Hence, for each , by the continuity of , we deduce
Similarly, we get that
Then we have , …, , i.e., , , …, for each . □
The following corollary is related to the simplest uniformly Lipschitzian semigroup defined in Definition 3.2.
Corollary 3.1 Let be a complete bounded metric space with uniform normal structure, and let be the simplest asymptotically regular uniformly Lipschitzian semigroup of self-mappings on with property (P) and satisfying
Then there exist some such that , …, and for all .
From Remark 2.1 and Theorem 3.1, we immediately obtain the following corollary.
Corollary 3.2 Let be a complete bounded metric space with property (P) and uniform normal structure, and let be an asymptotically regular semigroup on satisfying
Then there exist some such that , , …, for all .
For in Theorem 3.1, we get the following two corollaries which are due to Soliman .
Corollary 3.3 
Let be a complete bounded metric space with property (∗) and uniform normal structure, and let be an asymptotically regular semigroup on satisfying
Then there exist some such that , and for all .
Corollary 3.4 
Let be a complete bounded metric space with property (P) and uniform normal structure, and let be an asymptotically regular semigroup on satisfying
Then there exist some such that , and for all .
Remark 3.2 It is well known that the Lipschitzian mapping is uniformly continuous. It is natural to ask if there is a contractive mapping definition which does not force it to be continuous. It was answered affirmatively by Kannan. It is clear that Lipschitzian mappings are always continuous and Kannan type mappings are not necessarily continuous. It will be interesting to establish Theorem 3.1 for representative on satisfying the following condition:
for all and .
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The author is grateful to anonymous referees for their fruitful comments.
The author declares that they have no competing interests.
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Cite this article
Soliman, A.H. Results on n-tupled fixed points in metric spaces with uniform normal structure. Fixed Point Theory Appl 2014, 168 (2014). https://doi.org/10.1186/1687-1812-2014-168
- coupled fixed point
- tripled fixed point
- n-fixed point
- asymptotically regular semigroup
- uniform normal structure
- convexity structure