- Open Access
Fixed and coupled fixed points of a new type set-valued contractive mappings in complete metric spaces
© Amini-Harandi; licensee Springer 2012
Received: 20 June 2012
Accepted: 7 November 2012
Published: 26 November 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.
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.
for every , where .
Theorem 1.1 (Nadler )
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 )
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 .
Since , letting in (4), we obtain . Then there exists such that for . Consequently, we have for all . Thus, (note that ).
Then for each . Since , then there exist and such that for . Thus, for each , and so .
where . Since , then from the above we get , and so (note that for each ).
Now, we state the main result of the paper.
Then T has a fixed point.
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.
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.
Then, by Theorem 2.4, T has a fixed point.
Then it is easy to see that , but does not satisfy the condition () of the definition of F-contraction in .
for each . Then f has a coupled fixed point , that is, and .
for each . Then from Theorem 2.4 we deduce that T has a fixed point . Then is a coupled fixed point of f. □
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.
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