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A tripled fixed point theorem for semigroups of Lipschitzian mappings on metric spaces with uniform normal structure
Fixed Point Theory and Applications volume 2013, Article number: 346 (2013)
Abstract
In this work, we establish a tripled fixed point theorem for an asymptotically regular one-parameter semigroup ℑ = {, where G is an unbounded subset of } of Lipschitzian self-mappings on in a complete bounded metric space X with uniform normal structure.
1 Introduction
The classical Banach contraction principle proved in complete metric spaces continues to be an indispensable and effective tool in theory as well as 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 refer to [1–5].
In 2006, Bhaskar and Lakshmikantham [6] initiated the idea of a coupled fixed point in partially ordered metric spaces and proved some interesting coupled fixed point theorems for a mapping satisfying the mixed monotone property. Many authors obtained important coupled fixed point (see [6–9]). In this continuation, Lakshmikantham and Ciric [8] 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 [6]. Recently, Samet and Vetro [10] introduced the concept of fixed point of N-order 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 [6].
On the other hand, in 1989, Khamsi [11] 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 the result of Maluta [12] to metric spaces. In 1995, Lim and Hong-Kun Xu [13] proved a fixed point theorem for uniformly Lipschitzian mappings in metric spaces with both property (P) and uniform normal structure, which extends the result of Khamsi [11]. This is the metric space version of Casini and Maluta’s theorem [2]. In 2007, Jen-Chih Yao and Lu-Chuan Zeng [14] established a fixed point theorem for an asymptotically regular semigroup of uniformly Lipschitzian mappings with property (∗) in a complete bounded metric space with uniform normal structure which extends the results of Lim and Hong-Kun Xu [13]. Recently, Imdad and Soliman [15] introduced fixed point theorems for an asymptotically regular semigroup of uniformly generalized Lipschitzian mappings which generalize the results due to Jen-Chih Yao and Lu-Chuan Zeng [14].
In the present paper, we prove that asymptotically regular one parameter semigroups of Lipschitzian self-mappings on , has a tripled fixed point, where X denotes a complete bounded metric space with uniform normal structure. Also, some corollaries of our results are presented.
2 Preliminaries
Definition 2.1 [6]
An element is called a coupled fixed point of the mapping if
Theorem 2.1 [6]
Let be a partially ordered set and suppose 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 2.2 [10]
An element is called a tripled fixed point of the mapping if
Definition 2.3 A mapping is said to be a Lipschitzian mapping if for each integer , there exists a constant such that
where .
If , then F is called uniformly Lipschitzian and if , then F is called nonexpansive.
Definition 2.4 A mapping is called asymptotically regular if
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:
-
(i)
, and ;
-
(ii)
and ;
-
(iii)
, the self-mappings , and from G into X are continuous when G has the relative topology of .
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 .
Definition 2.5 A semigroup on is called a uniformly Lipschitzian semigroup if
where
In a metric space , let μ denote a nonempty family of subsets of X. Following Khamsi [11], we say that μ defines a convexity structure on X if μ is stable under intersection. We say that μ has property (R) if any decreasing sequence of nonempty bounded closed subsets of X with has a nonempty intersection. Recall that a subset of X is said to be admissible [5] if it is an intersection of closed balls. We denote by the family of all admissible subsets of X. It is obvious 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 [13], we shall adopt the following notations:
is the closed ball centered at x with radius r,
for ,
,
.
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 [13]
For a point and a bounded subset A of X, we have
Definition 2.6 [11]
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 [11]
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 2.7 [14]
Let be a metric space and be a semigroup on . Let us write the set
Lemma 2.1 [14]
If , then .
Definition 2.8 [13]
A metric space is said to have property (P) if given any two bounded sequences and in X, one can find some such that
Definition 2.9 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:
-
(a)
the sequences , and are bounded;
-
(b)
for any sequence in , there exists some such that
for any sequence in , there exists some such that
for any sequence in , there exists some such that
Remark 2.1 If X is a complete bounded metric space with property (P), then each semigroup on has property (∗).
3 Main results
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 , and satisfying the properties:
-
(I)
where
-
(II)
Proof For each integer , let , and . Then , and are decreasing sequences of admissible subsets of X, hence , and by Proposition 2.1. From Proposition 2.1, it is not difficult to see that , and . Indeed, observe that
Similarly, one can obtain
On the other hand, for any and any , we have
Therefore,
Also, one can deduce that for any , and any , we have
from which (ii) follows.
We now claim that for each , there exist , and such that
Indeed, if , then , we conclude that (3) holds. Without loss of generality, we may assume that . Then, for , we choose so small satisfying the following:
By the definition of , one can find such that
This shows that (3) holds. Obviously, it follows from (3) that for each ,
which implies
where . Noticing
we know that property (∗) yields a point such that
Since , and satisfies
similarly one can obtain that
where and .
Therefore (i) holds. □
Theorem 3.1 Let be a complete bounded metric space with uniform normal structure, and let be an asymptotically regular semigroup on with property (∗) and satisfying
Then there exist some such that , and for all .
Proof First, we write and . Choose a constant such that and . We can select a sequence such that and , where .
Observe that
for each and .
Now fix . Then, by Lemma 3.1, we can inductively construct sequences such that
(III)
where
(IV)
Let
Observe that for each , using (IV) we have
By the asymptotic regularity of on , we see that
which implies
Similarly, one can show that
Then it follows from (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 ,
which implies
It follows from (14) that
Similarly, one can deduce that
Thus, we have , and . Consequently, , and are Cauchy and hence convergent as X is complete. Let , and . Then 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., , and . Hence, for each , by the continuity of , we deduce
Similarly, we get that
Then we have , and , i.e., , and for each . □
From Remark 2.1 and Theorem 3.1, we immediately obtain the following results.
Corollary 3.1 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 3.1 It will be interesting to establish Theorem 3.1 for representative on of a left amenable semigroup S as a complete bounded metric space with uniform normal structure as in Holmes and Lau [16], Lau and Takahashi [17] and Lau [18].
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Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Khaled university, under project No. (KKU_S028_33). Also, the author is grateful to an anonymous referee for his fruitful comments.
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Soliman, A.H. A tripled fixed point theorem for semigroups of Lipschitzian mappings on metric spaces with uniform normal structure. Fixed Point Theory Appl 2013, 346 (2013). https://doi.org/10.1186/1687-1812-2013-346
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DOI: https://doi.org/10.1186/1687-1812-2013-346
Keywords
- coupled fixed point
- tripled fixed point
- asymptotically regular semigroup
- uniform normal structure
- convexity structure