# zbMATH — the first resource for mathematics

On fixed point generalizations of Suzuki’s method. (English) Zbl 1290.54024
Summary: In order to generalize the well-known Banach contraction theorem, many authors have introduced various types of contraction inequalities. In [Proc. Am. Math. Soc. 136, No. 5, 1861–1869 (2008; Zbl 1145.54026)], T. Suzuki introduced a new method which was then extended by some authors (see, for example, [S. Dhompongsa and H. Yingtaweesittikul, Fixed Point Theory Appl. 2009, Article ID 972395, 15 p. (2009; Zbl 1179.54055)], [M. Kikkawa and T. Suzuki, Fixed Point Theory Appl. 2008, Article ID 649749, 8 p. (2008; Zbl 1162.54019)] and [G. Moţ and A. Petruşel, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 9, 3371–3377 (2009; Zbl 1213.54068)]). M. Kikkawa and T. Suzuki [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 9, 2942–2949 (2008; Zbl 1152.54358)] extended the method and then G. Moţ and A. Petruşel further generalized it in [loc. cit.].
In this paper, we shall provide a new condition for $$T$$ which guarantees the existence of its fixed point. Our results generalize some old results.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text:
##### References:
 [1] Petrusel, A., Operatorial inclusions, (2002), House of the Book of Science Cluj-Napoca · Zbl 1057.47004 [2] Kannan, R, Some results on fixed points II, American mathematical monthly, 76, 405-408, (1969) · Zbl 0179.28203 [3] Subrahmanyam, P.V., Completeness and fixed points, Monatshefte für Mathematik, 74, 4, 325-330, (1969) · Zbl 0312.54048 [4] Suzuki, T., A generalized Banach contraction principle that characterized metric completeness, Proceedings of the American mathematical society, 136, 1861-1869, (2008) · Zbl 1145.54026 [5] Kikkawa, M.; Suzuki, T, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear analysis, 69, 2942-2949, (2008) · Zbl 1152.54358 [6] Mot, G.; Petrusel, A., Fixed point theory for a new type of contractive multivalued operators, Nonlinear analysis, 70, 3371-3377, (2009) · Zbl 1213.54068 [7] S. Dhompongsa, H. Yingtaweesittikul, Fixed point for multivalued mappings and the metric completeness, Fixed Point Theory and Applications, 2009, 15 pages, Article ID 972395. doi:10.1155/2009/972395. · Zbl 1179.54055 [8] Reich, S., Kannan’s fixed point theorem, Bollettino Della unione matematica italiana. serie 9, 4, 1-11, (1971) · Zbl 0219.54042 [9] Constantin, A., A random fixed point theorem for multifunctions, Stochastic analysis and applications, 12, 1, 65-73, (1994) · Zbl 0813.60058 [10] M. Kikkawa, T. Suzuki, Some similarity between contractions and Kannan mappings, Fixed Point Theory and Applications, 2008, 8 pages, Article ID 649749. doi:10.1155/2008/649749. · Zbl 1162.54019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.