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On fixed point generalizations of Suzuki’s method. (English) Zbl 05883254
Summary: In order to generalize the well-known Banach contraction theorem, many authors have introduced various types of contraction inequalities. In 2008, Suzuki introduced a new method (Suzuki (2008) [4]) and then his method was extended by some authors (see for example, Dhompongsa and Yingtaweesittikul (2009), Kikkawa and Suzuki (2008) and Mot and Petrusel (2009) [7], [10], [5] and [6]). Kikkawa and Suzuki extended the method in (Kikkawa and Suzuki (2008) [5]) and then Mot and Petrusel further generalized it in (Mot and Petrusel (2009) [6]). 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:
 46 Functional analysis
##### References:
 [1] Petrusel, A.: Operatorial inclusions, (2002) [2] Kannan, R.: Some results on fixed points II, American mathematical monthly 76, 405-408 (1969) · Zbl 0179.28203 · doi:10.2307/2316437 [3] Subrahmanyam, P. V.: Completeness and fixed points, Monatshefte für Mathematik 74, No. 4, 325-330 (1969) [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 · doi:10.1090/S0002-9939-07-09055-7 [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 · doi:10.1016/j.na.2007.08.064 [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 · doi:10.1016/j.na.2008.05.005 [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 · doi:10.1155/2009/972395 [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, No. 1, 65-73 (1994) · Zbl 0813.60058 · doi:10.1080/07362999408809338 [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 · doi:10.1155/2008/649749