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Fixed and coupled fixed points of a new type set-valued contractive mappings in complete metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 215 (2012)
In this paper, motivated by the recent work of Wardowski (Fixed Point Theory Appl. 2012:94, 2012), we introduce a new concept of set-valued contraction and prove a fixed point theorem which generalizes some well-known results in the literature. As an application, we derive a new coupled fixed point theorem. Some examples are also given to support our main results.
In the literature, there are plenty of extensions of the famous Banach contraction principle , which states that every self-mapping T defined on a complete metric space satisfying
where , has a unique fixed point, and for every , the sequence is convergent to the fixed point. Some of the extensions weaken the right side of the inequality in the condition (1) by replacing k with a mapping; see, e.g., [2–4]. In other results, the underlying space is more general; see, e.g., [5–8]. In 1969, Nadler  extended the Banach contraction principle to set-valued mappings. For other extensions of the Banach contraction principle, see [10–21] and the references therein.
Recently, Wardowski  introduced a new concept of contraction and proved a fixed point theorem which generalizes the Banach contraction principle in a different way than in the known results from the literature. In this paper, we present an improvement and generalization of the main result of Wardowski . To set up our results, in the next section, we introduce some definitions and facts.
Let be a metric space and let denote the class of all nonempty bounded closed subsets of X. Let H be the Hausdorff metric with respect to d, that is,
for every , where .
Theorem 1.1 (Nadler )
Let be a complete metric space and let be a set-valued map. Assume that there exists such that
Then T has a fixed point.
In 1989 Mizoguchi and Takahashi  proved the following generalization of Theorem 1.1.
Theorem 1.2 (Mizoguchi and Takahashi )
Let be a complete metric space and let be a set-valued map satisfying
where satisfies for each . Then T has a fixed point.
2 Main results
Let and be two mappings. Throughout the paper, let Δ be the set of all pairs satisfying the following:
() for each strictly decreasing sequence ;
() F is strictly increasing;
() For each sequence of positive numbers, if and only if ;
() If and for each , then .
Example 2.1 Let for each , where is a constant, and let be a mapping satisfying for some where is strictly increasing. Then the proof of the main result in  shows that . We give the details for completeness. Using (), the following holds for every :
By (3), the following holds for every :
Since , letting in (4), we obtain . Then there exists such that for . Consequently, we have for all . Thus, (note that ).
Example 2.2 Let and let for each , where satisfying
Now, we show that . It is easy to see that and satisfy ()-(). To show (), assume that and
Then for each . Since , then there exist and such that for . Thus, for each , and so .
Example 2.3 Let and let for each , where is a constant. Now, we show that . We only show (). Suppose that and
where . Since , then from the above we get , and so (note that for each ).
Now, we state the main result of the paper.
Theorem 2.4 Let be a complete metric spaces, let be a set-valued mapping and let . Assume that either T is compact valued or F is continuous from the right. Furthermore, assume that
Then T has a fixed point.
Proof Let and . If , then and is a fixed point of T. So, we may assume that . Since either T is compact valued or F is continuous from the right, and then there exists such that (note that F is increasing)
From (5) and (6), we have
We may also assume that (otherwise, ). Proceeding this manner, we can define a sequence in X satisfying
where . Since then from (8), we have for each . Since F is strictly increasing, then we deduce that is a nonnegative strictly decreasing sequence and so is convergent to some , . Now we show that . On the contrary, assume that . From (8), we get
Since is strictly decreasing, then from () we get . Thus, , and then from (9), we have . Then by (), , a contradiction. Hence,
From (8), (10) and (), we have
Then, by the triangle inequality, is a Cauchy sequence. From the completeness of X, there exists such that . Now, we prove that x is a fixed point of T. To prove the claim, we may assume that for sufficiently large . On the contrary, assume that for each . Since Tx is closed, and , then , and we are finished.
From (5), we have (note that and for )
Since , then (11) together with () imply that
and so . Hence, (note that Tx is closed). □
Remark 2.5 By Example 2.1, Theorem 2.4 is an extension and improvement of Theorem 2.1 of Wardowski . From Example 2.2, we infer that Theorem 2.4 is a generalization of the above mentioned Theorem 1.2 of Mizoguchi and Takahashi.
Now, we illustrate our main result by the following example.
Example 2.6 Consider the complete metric space , where d is defined as
Let be defined as
Let be given by
Now, we show that T satisfies (5), where for each and for each . To show the claim, notice first that for each . Now let with . Since , then we have
Thus, from the above, we have
Therefore, (note that )
Then, by Theorem 2.4, T has a fixed point.
Now, we show that T does not satisfy the condition of Nadler’s theorem. On the contrary, assume that there exists a function such that
for all . Then
Then, for each and , we have
Example 2.7 For each , let and let
Then it is easy to see that , but does not satisfy the condition () of the definition of F-contraction in .
Corollary 2.8 Let be a complete metric space and let . Let be a mapping satisfying
for each . Then f has a coupled fixed point , that is, and .
Proof Let and let d be the metric on M which is defined by
Then it is straightforward to show that is a complete metric space. Let be defined by . From (12), we get
for each . Then from Theorem 2.4 we deduce that T has a fixed point . Then is a coupled fixed point of f. □
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The author is grateful to the referees for their helpful comments leading to improvement of the presentation of the work. This work was supported by the University of Shahrekord and by the Center of Excellence for Mathematics, University of Shahrekord, Iran. This research was also in part supported by a grant from IPM (No. 91470412). This research is partially carried out in the IPM-Isfahan Branch.
The author declares that they have no competing interests.
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Amini-Harandi, A. Fixed and coupled fixed points of a new type set-valued contractive mappings in complete metric spaces. Fixed Point Theory Appl 2012, 215 (2012). https://doi.org/10.1186/1687-1812-2012-215
- fixed point
- coupled fixed point
- set-valued contractions