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.
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.
- Banach S: Sur les opérationes dans les ensembles abstraits et leur application aux équation intégrales. Fundam. Math. 1922, 3: 133–181.Google Scholar
- Caristi J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 1976, 215: 241–251.MathSciNetView ArticleGoogle Scholar
- Caristi J, Kirk WA: Geometric fixed point theory and inwardness conditions. Lect. Notes Math. 1975, 490: 74–83. 10.1007/BFb0081133MathSciNetView ArticleGoogle Scholar
- Ćirić LB: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 1974, 45: 267–273.Google Scholar
- Ćirić LB: A new fixed-point theorem for contractive mappings. Publ. Inst. Math. (Belgr.) 1981, 30: 25–27.Google Scholar
- Chauhan S, Kadelburg Z, Dalal S: A common fixed point theorem in metric space under general contractive condition. J. Appl. Math. 2013., 2013: Article ID 510691Google Scholar
- Ekeland I: On the variational principle. J. Math. Anal. Appl. 1974, 47: 324–353. 10.1016/0022-247X(74)90025-0MathSciNetView ArticleGoogle Scholar
- Imdad M, Chauhan S: Employing common limit range property to prove unified metrical common fixed point theorems. Int. J. Anal. 2013., 2013: Article ID 763261Google Scholar
- Imdad M, Chauhan S, Kadelburg Z:Fixed point theorems for mappings with common limit range property satisfying generalized -weak contractive conditions. Math. Sci. 2013, 2013: 7–16.MathSciNetGoogle Scholar
- Kannan R: Some results on fixed points II. Am. Math. Mon. 1969, 76: 405–408. 10.2307/2316437View ArticleGoogle Scholar
- Kirk WA: Contractions mappings and extensions. In Handbook of Metric Fixed Point Theory. Edited by: Kirk WA, Sims B. Kluwer Academic, Dordrecht; 2001:1–34.View ArticleGoogle Scholar
- Kirk WA: Fixed point of asymptotic contractions. J. Math. Anal. Appl. 2003, 277: 645–650. 10.1016/S0022-247X(02)00612-1MathSciNetView ArticleGoogle Scholar
- Meir A, Keeler E: A theorem on contraction mappings. J. Math. Anal. Appl. 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6MathSciNetView ArticleGoogle Scholar
- Nadler SB Jr.: Multi-valued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475MathSciNetView ArticleGoogle Scholar
- Reich S: Kannan’s fixed point theorem. Boll. Unione Mat. Ital. 1971, 4: 1–11.Google Scholar
- Subrahmanyam PV: Remarks on some fixed point theorems related to Banach’s contraction principle. J. Math. Phys. Sci. 1974, 8: 445–457.MathSciNetGoogle Scholar
- Sehgal VM, Bharucha-Reid AT: Fixed points of contraction mappings on probabilistic metric spaces. Math. Syst. Theory 1972, 6: 97–102. 10.1007/BF01706080MathSciNetView ArticleGoogle Scholar
- Khan MA, Sumitra , Kumar R: Subcompatible and subsequential continuous maps in non Archimedean Menger PM-spaces. Jordan J. Math. Stat. 2012, 5: 137–150.Google Scholar
- Khan MA, Sumitra , Kumar R: Semi-compatible maps and common fixed point theorems in non-Archimedean Menger PM-spaces. Jordan J. Math. Stat. 2012, 5: 185–199.Google Scholar
- Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 2008, 136: 1861–1869.View ArticleGoogle Scholar
- Altun I, Erduran A: A Suzuki type fixed-point theorem. Int. J. Math. Math. Sci. 2011., 2011: Article ID 736063Google Scholar
- Karapinar E, Tas K: Generalized (C)-conditions and related fixed point theorems. Comput. Math. Appl. 2011, 61(11):3370–3380. 10.1016/j.camwa.2011.04.035MathSciNetView ArticleGoogle Scholar
- Karapinar E: Remarks on Suzuki (C)-condition. Dynamical Systems and Methods 2012, 227–243.View ArticleGoogle Scholar
- Kikkawa M, Suzuki T: Some similarity between contractions and Kannan mappings. Fixed Point Theory Appl. 2009., 2009: Article ID 192872Google Scholar
- Kikkawa M, Suzuki T: Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Anal. TMA 2008, 69: 2942–2949. 10.1016/j.na.2007.08.064MathSciNetView ArticleGoogle Scholar
- Moţ G, Petruşel A: Fixed point theory for a new type of contractive multivalued operators. Nonlinear Anal. TMA 2009, 70: 3371–3377. 10.1016/j.na.2008.05.005View ArticleGoogle Scholar
- Dhompongsa S, Yingtaweesittikul H: Fixed points for multivalued mappings and the metric completeness. Fixed Point Theory Appl. 2009., 2009: Article ID 972395Google Scholar
- Popescu O: Two fixed point theorems for generalized contractions with constants in complete metric spaces. Cent. Eur. J. Math. 2009, 7: 529–538. 10.2478/s11533-009-0019-2MathSciNetView ArticleGoogle Scholar
- Popescu O: A new type of multivalued contractive operators. Bull. Sci. Math. 2013, 137: 30–44. 10.1016/j.bulsci.2012.07.001MathSciNetView ArticleGoogle Scholar
- Singh SL, Mishra SN: Remarks on recent fixed point theorems. Fixed Point Theory Appl. 2010., 2010: Article ID 452905Google Scholar
- Singh SL, Pathak HK, Mishra SN: On a Suzuki type general fixed point theorem with applications. Fixed Point Theory Appl. 2010., 2010: Article ID 234717Google Scholar
- Singh SL, Mishra SN, Chugh R, Kamal R: General common fixed point theorems and applications. Fixed Point Theory Appl. 2012., 2012: Article ID 902312Google Scholar
- Edelstein M: On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 1962, 37: 74–79.MathSciNetView ArticleGoogle Scholar
- Suzuki T: A new type of fixed point theorem in metric spaces. Nonlinear Anal. 2009, 71: 5313–5317. 10.1016/j.na.2009.04.017MathSciNetView ArticleGoogle Scholar
- Dorić D, Lazović R: Some Suzuki type fixed point theorem for generalized multivalued mappings and applications. Fixed Point Theory Appl. 2011., 2011: Article ID 40Google Scholar
- Dorić D, Kadelburg Z, Radenović S: Edelstein-Suzuki-type fixed point results in metric and abstract metric spaces. Nonlinear Anal. TMA 2012, 75(4):1927–1932. 10.1016/j.na.2011.09.046View ArticleGoogle Scholar
- Karapinar E: Edelstein type fixed point theorems. Fixed Point Theory Appl. 2012., 2012: Article ID 107Google Scholar
- Karapinar E, Salimi P: Suzuki-Edelstein type contractions via auxiliary functions. Math. Probl. Eng. 2013., 2013: Article ID 648528Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.