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The fixed point theory for some generalized nonexpansive mappings. (English) Zbl 1215.47042

Summary: We study some aspects of the fixed point theory for a class of generalized nonexpansive mappings, which among others contain the class of generalized nonexpansive mappings recently defined by T. Suzuki [J. Math. Anal. Appl. 340, No. 2, 1088–1095 (2008; Zbl 1140.47041)].

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems

Citations:

Zbl 1140.47041
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References:

[1] M. A. Smyth, Aspects of the fixed point theory for some metrically defined maps, Ph.D. dissertation, University of Newcastle, 1994.
[2] J. Y. Park and J. U. Jeong, “Weak convergence to a fixed point of the sequence of Mann type iterates,” Journal of Mathematical Analysis and Applications, vol. 184, no. 1, pp. 75-81, 1994. · Zbl 0811.47067 · doi:10.1006/jmaa.1994.1184
[3] K. Goebel, W. A. Kirk, and T. N. Shimi, “A fixed point theorem in uniformly convex spaces,” Bollettino della Unione Matematica Italiana, vol. 7, no. 4, pp. 67-75, 1973. · Zbl 0265.47045
[4] A. A. Gillespie and B. B. Williams, “Some theorems on fixed points in Lipschitz- and Kannan-type mappings,” Journal of Mathematical Analysis and Applications, vol. 74, no. 2, pp. 382-387, 1980. · Zbl 0448.47039 · doi:10.1016/0022-247X(80)90135-3
[5] J. Bogin, “A generalization of a fixed point theorem of Goebel, Kirk and Shimi,” Canadian Mathematical Bulletin, vol. 19, no. 1, pp. 7-12, 1976. · Zbl 0329.47021 · doi:10.4153/CMB-1976-002-7
[6] T. Suzuki, “Fixed point theorems and convergence theorems for some generalized nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 340, no. 2, pp. 1088-1095, 2008. · Zbl 1140.47041 · doi:10.1016/j.jmaa.2007.09.023
[7] S. Dhompongsa, W. Inthakon, and A. Kaewkhao, “Edelstein’s method and fixed point theorems for some generalized nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 350, no. 1, pp. 12-17, 2009. · Zbl 1153.47046 · doi:10.1016/j.jmaa.2008.08.045
[8] J. García-Falset, E. Llorens- Fuster, and T. Suzuki, “Some generalized nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 375, no. 2011, pp. 185-195, 2010. · Zbl 1214.47047
[9] J. B. Díaz and F. T. Metcalf, “On the set of subsequential limit points of successive approximations,” Transactions of the American Mathematical Society, vol. 135, pp. 459-485, 1969. · Zbl 0174.25904 · doi:10.2307/1995027
[10] W. G. Dotson Jr., “Fixed points of quasi-nonexpansive mappings,” Australian Mathematical Society Journal, vol. 13, pp. 167-170, 1972. · Zbl 0227.47047 · doi:10.1017/S144678870001123X
[11] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0708.47031 · doi:10.1017/CBO9780511526152
[12] B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257-290, 1977. · Zbl 0365.54023 · doi:10.2307/1997954
[13] M. Gregu\vs Jr., “A fixed point theorem in Banach space,” Bollettino della Unione Matematica Italiana, vol. 17, no. 1, pp. 193-198, 1980. · Zbl 0538.47035
[14] C. S. Wong, “On Kannan maps,” Proceedings of the American Mathematical Society, vol. 47, pp. 105-111, 1975. · Zbl 0265.47039 · doi:10.2307/2040215
[15] A. A. Gillespie and B. B. Williams, “Fixed point theorem for nonexpansive mappings on Banach spaces with uniformly normal structure,” Applicable Analysis, vol. 9, no. 2, pp. 121-124, 1979. · Zbl 0424.47035 · doi:10.1080/00036817908839259
[16] P.-K. Lin, “A uniformly asymptotically regular mapping without fixed points,” Canadian Mathematical Bulletin, vol. 30, no. 4, pp. 481-483, 1987. · Zbl 0645.47050 · doi:10.4153/CMB-1987-071-6
[17] K. Goebel and W. A. Kirk, “Classical theory of nonexpansive mappings,” in Handbook of Metric Fixed Point Theory, W. A. Kirk and B. Sims, Eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. · Zbl 1035.47033
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