Results on n-tupled fixed points in metric spaces with uniform normal structure
© Soliman; licensee Springer. 2014
Received: 24 January 2014
Accepted: 4 July 2014
Published: 18 August 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).
Keywordscoupled fixed point tripled fixed point n-fixed point asymptotically regular semigroup uniform normal structure convexity structure
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 
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 
If there exist such that and , then there exist such that and .
Definition 2.3 
Definition 2.4 
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,
Proposition 2.1 
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.
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 
Lemma 2.1 
If , then .
Definition 2.10 
3 Main results
, , …, ;
, the self-mappings , , …, from G into X are continuous when G has the relative topology of .
If ℑ is asymptotically regular at each , then ℑ is called an asymptotically regular semigroup on .
, where .
Definition 3.3 A mapping has an r-tupled fixed point if , , , …, .
the sequences , …, are bounded;
- (b)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 (∗).
from which (II) follows.
where , …, .
Therefore (I) holds. □
We are ready to prove our main theorem for this paper.
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 .
Then we have , …, , i.e., , , …, for each . □
The following corollary is related to the simplest uniformly Lipschitzian semigroup defined in Definition 3.2.
Then there exist some such that , …, and for all .
From Remark 2.1 and Theorem 3.1, we immediately obtain the following corollary.
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 
Then there exist some such that , and for all .
Corollary 3.4 
Then there exist some such that , and for all .
for all and .
The author is grateful to anonymous referees for their fruitful comments.
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