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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 629468, 10 pages
Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces
1Mathematics Institute, African University of Science and Technology, PMB 681, Garki, Abuja, Nigeria
2Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA
3Université Gaston Berger, 234 Saint Louis, Senegal
4Department of Mathematical Sciences, Bayero University, PMB 3011, Kano, Nigeria
Received 10 September 2012; Accepted 15 April 2013
Academic Editor: Josef Diblík
Copyright © 2013 C. E. Chidume et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- L. E. J. Brouwer, “Über Abbildung von Mannigfaltigkeiten,” Mathematische Annalen, vol. 71, no. 4, p. 598, 1912.
- S. Kakutani, “A generalization of Brouwer's fixed point theorem,” Duke Mathematical Journal, vol. 8, pp. 457–459, 1941.
- J. F. Nash, “Non-cooperative games,” Annals of Mathematics. Second Series, vol. 54, pp. 286–295, 1951.
- J. F. Nash, Jr., “Equilibrium points in -person games,” Proceedings of the National Academy of Sciences of the United States of America, vol. 36, no. 1, pp. 48–49, 1950.
- J. Geanakoplos, “Nash and Walras equilibrium via Brouwer,” Economic Theory, vol. 21, no. 2-3, pp. 585–603, 2003.
- S. B. Nadler Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969.
- D. Downing and W. A. Kirk, “Fixed point theorems for set-valued mappings in metric and Banach spaces,” Mathematica Japonica, vol. 22, no. 1, pp. 99–112, 1977.
- A. F. Filippov, “Diffrential equations with discontinuous right hand side,” Matematicheskii Sbornik, vol. 51, pp. 99–128, 1960.
- A. F. Filippov, “Diffrential equations with discontinuous right hand side,” Transactions of the American Mathematical Society, vol. 42, pp. 199–232, 1964.
- K. C. Chang, “The obstacle problem and partial differential equations with discontinuous nonlinearities,” Communications on Pure and Applied Mathematics, vol. 33, no. 2, pp. 117–146, 1980.
- L. Erbe and W. Krawcewicz, “Existence of solutions to boundary value problems for impulsive second order differential inclusions,” The Rocky Mountain Journal of Mathematics, vol. 22, no. 2, pp. 519–539, 1992.
- M. Frigon, A. Granas, and Z. Guennoun, “A note on the Cauchy problem for differential inclusions,” Topological Methods in Nonlinear Analysis, vol. 1, no. 2, pp. 315–321, 1993.
- K. Deimling, Multivalued Differential Equations, vol. 1, Walter de Gruyter & Co., Berlin, Germany, 1992.
- R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970.
- G. J. Minty, “Monotone (nonlinear) operators in Hilbert space,” Duke Mathematical Journal, vol. 29, pp. 341–346, 1962.
- B. Martinet, “Régularisation d'inéquations variationnelles par approximations successives,” Revue Francaise d'informatique et de Recherche operationelle, vol. 4, pp. 154–159, 1970.
- R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976.
- R. E. Bruck, “Asymptotic behavior of nonexpansive mappings,” in Contemporary Mathematics, R. C. Sine, Ed., vol. 18 of Fixed Points and Nonexpansive Mappings, AMS, Providence, RI, England, 1980.
- F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.
- C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
- K. P. R. Sastry and G. V. R. Babu, “Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point,” Czechoslovak Mathematical Journal, vol. 55, no. 4, pp. 817–826, 2005.
- B. Panyanak, “Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 54, no. 6, pp. 872–877, 2007.
- Y. Song and H. Wang, “Erratum to, “Mann and Ishikawa iterative processes for multi-valued mappings in Banach Spaces” [Comput. Math. Appl.54 (2007),872–877],” Computers & Mathematics With Applications, vol. 55, pp. 2999–3002, 2008.
- S. H. Khan and I. Yildirim, “Fixed points of multivalued nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2012, article 73, 2012.
- S. H. Khan, I. Yildirim, and B. E. Rhoades, “A one-step iterative process for two multivalued nonexpansive mappings in Banach spaces,” Computers & Mathematics with Applications, vol. 61, no. 10, pp. 3172–3178, 2011.
- M. Abbas, S. H. Khan, A. R. Khan, and R. P. Agarwal, “Common fixed points of two multivalued nonexpansive mappings by one-step iterative scheme,” Applied Mathematics Letters, vol. 24, no. 2, pp. 97–102, 2011.
- J. García-Falset, E. Lorens-Fuster, and T. Suzuki, “Fixed point theory for a class of generalized nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 375, no. 1, pp. 185–195, 2011.
- P. Z. Daffer and H. Kaneko, “Fixed points of generalized contractive multi-valued mappings,” Journal of Mathematical Analysis and Applications, vol. 192, no. 2, pp. 655–666, 1995.
- N. Shahzad and H. Zegeye, “On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 3-4, pp. 838–844, 2009.
- M. A. Krasnosel'skiĭ, “Two remarks on the method of successive approximations,” Uspekhi Matematicheskikh Nauk, vol. 10, no. 1(63), pp. 123–127, 1955.
- W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.
- S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974.
- Y. Song and Y. J. Cho, “Some notes on Ishikawa iteration for multi-valued mappings,” Bulletin of the Korean Mathematical Society, vol. 48, no. 3, pp. 575–584, 2011.
- T. Husain and A. Latif, “Fixed points of multivalued nonexpansive maps,” Mathematica Japonica, vol. 33, no. 3, pp. 385–391, 1988.
- H. K. Xu, “On weakly nonexpansive and -nonexpansive multivalued mappings,” Mathematica Japonica, vol. 36, no. 3, pp. 441–445, 1991.