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Some generalizations of Suzuki and Edelstein type theorems
Fixed Point Theory and Applications volume 2013, Article number: 319 (2013)
Abstract
We prove some generalizations of Suzuki’s fixed point theorem and Edelstein’s theorem.
MSC:54H25.
Introduction and preliminaries
Let be a complete metric space and T be a selfmap of X. Then T is called a contraction if there exists such that
for all .
The following famous theorem is referred to as the Banach contraction principle.
Theorem 1 (Banach [1])
Let be a complete metric space, and let T be a contraction on X. Then T has a unique fixed point.
This theorem is a very forceful and simple, and it has become a classical tool in nonlinear analysis. It has many generalizations, see [2–19].
In 2008, Suzuki [20] 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 [20]
Let be a complete metric space, and let T be a mapping on X. Define a nonincreasing function θ from onto by
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 [21], Karapinar [22, 23], Kikkawa and Suzuki [24, 25], Moţ and Petruşel [26], Dhompongsa and Yingtaweesittikul [27], Popescu [28, 29], Singh and Mishra [30–32] are important contributions to metric fixed point theory.
Popescu [28] introduced a new type of contractive operator and proved the following theorem.
Theorem 3 [28]
Let be a complete metric space and be a -contractive single-valued operator:
where , and
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.
Theorem 4 Let be a complete metric space, and let T be a mapping on X. Assume that there exist and such that
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 [33])
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.
Inspired by Theorem 2, Suzuki [34] proved a generalization of Edelstein’s fixed point theorem (see also [35–38]).
Theorem 6 [34]
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.
Main results
We start this section by proving the following theorem.
Theorem 7 Let be a complete metric space, and let T be a mapping on X. Assume that there exist , , , if , if such that
for all . Then there exists a unique fixed point z of T. Moreover, for all .
Proof Since holds for every , by hypothesis, we get
for all . We now fix and define a sequence by . Then (3) yields , so . Hence is a Cauchy sequence. Since X is complete, converges to some point . We next show that
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).
Now we assume that for every integer . Then (4) yields
for every integer . We consider the following three cases:
-
(a)
,
-
(b)
,
-
(c)
.
In the case (a) we note that . Then, by (3) and (5), we have
This is a contradiction.
In the case (b), we note that . If we assume , then we have, in view of (3) and (5),
This is a contradiction. Hence . By hypothesis and (5), we have
This is also a contradiction.
In the case (c), we assume there exists an integer such that
for all . Then
Continuing this process, we get
for all , . Letting p tend to ∞, we obtain
for all . Thus,
for all , so
for all . This is a contradiction. Hence there exists a subsequence of such that
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, .
Now we suppose that y is another fixed point of T, that is, . Then
so, by hypothesis, . Hence . This is a contradiction. □
Remark 1 For , taking , , we obtain Suzuki’s condition from Theorem 2. Moreover, from our condition and the triangle inequality, we get
that is,
If , we have
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.
Theorem 8 Let be a complete metric space, and let T be a mapping on X. Assume that there exist , such that
for all . Then T has a unique fixed point. Moreover, if , then T has a unique fixed point.
Proof Let and the sequence be defined by . Since
we get from hypothesis for all . Therefore, for all . Thus
Hence is a Cauchy sequence. Since X is complete, converges to some point .
Now, we will show that there exists a subsequence of such that
for all . Arguing by contradiction, we suppose that there exists a positive integer ν such that
for all . Then we have
By induction, we get for all , that
Then we have
Hence
On the other hand,
Letting , we get for all that . Thus
By (6) and (7) we have for all , that
so
Taking the limit as , we obtain that for all . Then we have
and
This implies for all . Thus,
This is a contradiction. Therefore there exists a subsequence of such that
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. □
Edelstein’s theorem
The following theorem extends Theorem 6 and generalizes Theorem 5.
Theorem 9 Let be a compact metric space, and let T be a mapping on X. Assume that
for , where , , . Then T has a unique fixed point.
Proof We put
and choose a sequence in X such that . Since X is compact, without loss of generality, we may assume that and converge to some elements , respectively. We have
We shall show . Arguing by contradiction, we assume . Since
we can choose a positive integer ν such that
for all . By hypothesis, holds for . This implies
From the definition of β, we obtain . Since , we have
which contradicts the definition of β. Therefore we obtain . We have , so . Thus, .
We next show that T has a fixed point. Arguing by contradiction, we assume that T does not have a fixed point. Since for all , we get , so . By induction, we obtain that and for all integers . If there exist an integer and a subsequence of such that
for all , by hypothesis we get . Taking the limit as , we obtain , that is, , which is a contradiction. Hence, we can assume that for every , there exists an integer such that
for all . Since
and
we can choose p satisfying
We put . Then by (9) we have
for all . Since
we get
so
for all . Similarly, we can obtain
for all . Using (11), we get
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. □
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Popescu, O. Some generalizations of Suzuki and Edelstein type theorems. Fixed Point Theory Appl 2013, 319 (2013). https://doi.org/10.1186/1687-1812-2013-319
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DOI: https://doi.org/10.1186/1687-1812-2013-319