Some generalizations of Suzuki and Edelstein type theorems
© Popescu; licensee Springer. 2013
Received: 29 July 2013
Accepted: 28 October 2013
Published: 25 November 2013
We prove some generalizations of Suzuki’s fixed point theorem and Edelstein’s theorem.
KeywordsBanach principle contraction Suzuki’s theorem Edelstein’s theorem
Introduction and preliminaries
for all .
The following famous theorem is referred to as the Banach contraction principle.
Theorem 1 (Banach )
Let be a complete metric space, and let T be a contraction on X. Then T has a unique fixed point.
In 2008, Suzuki  introduced a new type of mapping and presented a generalization of the Banach contraction principle in which the completeness can also be characterized by the existence of a fixed point of these mappings.
Theorem 2 
Assume that there exists such that implies for all . Then there exists a unique fixed point z of T. Moreover, for all .
Its further outcomes by Altun and Erduran , Karapinar [22, 23], Kikkawa and Suzuki [24, 25], Moţ and Petruşel , Dhompongsa and Yingtaweesittikul , Popescu [28, 29], Singh and Mishra [30–32] are important contributions to metric fixed point theory.
Popescu  introduced a new type of contractive operator and proved the following theorem.
Theorem 3 
Then T has a fixed point. Moreover, if , then T has a unique fixed point.
As a direct consequence of Theorem 3, we obtain the following result.
for all . Then there exists a fixed point z of T. Further, if , then there exists a unique fixed point of T.
The following theorem is a well-known result in fixed point theory.
Theorem 5 (Edelstein )
Let be a compact metric space, and let T be a mapping on X. Assume for all with . Then T has a unique fixed point.
Theorem 6 
Let be a compact metric space, and let T be a mapping on X. Assume that implies for all . Then T has a unique fixed point.
In this paper, we prove generalizations of Theorem 2, Theorem 4, Theorem 5 and extend Theorem 6. The direction of our extension is new, very simple and inspired by Theorem 3.
We start this section by proving the following theorem.
for all . Then there exists a unique fixed point z of T. Moreover, for all .
for all , . Since , , there exists a positive integer ν such that for all . By hypothesis, we get . Letting n tend to ∞, we obtain . That is, we have shown (4).
This is a contradiction.
This is also a contradiction.
for all . By hypothesis, we get for all . Letting k tend to ∞, we get , that is, . This is a contradiction.
Thus there exists an integer such that . By (3) we get , so , that is, .
so, by hypothesis, . Hence . This is a contradiction. □
hence our condition implies Suzuki’s condition. We also note that if we take , for , we get Suzuki’s condition. Therefore, our theorem generalizes, extends and complements Suzuki’s theorem.
Example 1 Define a complete metric space X by and a mapping T on X by if and . Then T satisfies our condition from Theorem 7 for every , but T does not satisfy Suzuki’s condition from Theorem 2.
Proof Since for every , and , T does not satisfy Suzuki’s condition. If , we have , so taking , we get . Hence and . Now it is obvious that T satisfies our condition. If , we take . We have two cases: and . In the first case we put and in the second . We have in both cases, so T satisfies our condition. If for , , it is obvious that T satisfies our condition. □
The following theorem is a generalization of Theorem 4.
for all . Then T has a unique fixed point. Moreover, if , then T has a unique fixed point.
Hence is a Cauchy sequence. Since X is complete, converges to some point .
for all . By hypothesis, we get . Letting , we obtain , that is, .
If , we assume that y is another fixed point of T. Then , so, by hypothesis, . Since , this is a contradiction. □
The following theorem extends Theorem 6 and generalizes Theorem 5.
for , where , , . Then T has a unique fixed point.
which contradicts the definition of β. Therefore we obtain . We have , so . Thus, .
for all . Thus, by (10), we obtain , which is a contradiction. Therefore there exists such that . Fix with . Then since , we have and hence y is not a fixed point of T. Therefore, the fixed point of T is unique. □
Remark 2 The proof of Theorem 9 is available for , . In this case we obtained Theorem 6. We do not know if Theorem 9 is still correct for , , or, more generally, for . This is an open question.
Example 2 Define a complete metric space X by such that , , , and a mapping T on X by , , , , . Then T satisfies our condition from Theorem 9 for , , but T does not satisfy Suzuki’s condition from Theorem 6.
Proof We have and , so T does not satisfy Suzuki’s condition from Theorem 6. Moreover, we have . It is now obvious that T satisfies our condition from Theorem 9. □
The author is highly indebted to the referees for their careful reading of the manuscript and valuable suggestions.
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