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Convergence theorems for fixed points of multivalued strictly pseudocontractive mappings in Hilbert spaces. (English) Zbl 1273.47109

Summary: Let \(K\) be a nonempty, closed, and convex subset of a real Hilbert space \(H\). Suppose that \(T : K \to 2^K\) is a multivalued strictly pseudocontractive mapping such that \(F(T) \neq \emptyset\). A Krasnoselskii-type iteration sequence \(\{x_n\}\) is constructed and shown to be an approximate fixed point sequence of \(T\); that is, \(\lim_{n \to \infty}d(x_n, Tx_n) = 0\) holds. Convergence theorems are also proved under appropriate additional conditions.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H04 Set-valued operators
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[1] Brouwer, L. E. J., Über Abbildung von Mannigfaltigkeiten, Mathematische Annalen, 71, 4, 598 (1912) · JFM 42.0419.01 · doi:10.1007/BF01456812
[2] Kakutani, S., A generalization of Brouwer’s fixed point theorem, Duke Mathematical Journal, 8, 457-459 (1941) · Zbl 0061.40304 · doi:10.1215/S0012-7094-41-00838-4
[3] Nash, J. F., Non-cooperative games, Annals of Mathematics. Second Series, 54, 286-295 (1951) · Zbl 0045.08202 · doi:10.2307/1969529
[4] Nash,, J. F., Equilibrium points in \(n\)-person games, Proceedings of the National Academy of Sciences of the United States of America, 36, 1, 48-49 (1950) · Zbl 0036.01104 · doi:10.1073/pnas.36.1.48
[5] Geanakoplos, J., Nash and Walras equilibrium via Brouwer, Economic Theory, 21, 2-3, 585-603 (2003) · Zbl 1049.91105 · doi:10.1007/s001990000076
[6] Nadler, S. B., Multi-valued contraction mappings, Pacific Journal of Mathematics, 30, 475-488 (1969) · Zbl 0187.45002 · doi:10.2140/pjm.1969.30.475
[7] Downing, D.; Kirk, W. A., Fixed point theorems for set-valued mappings in metric and Banach spaces, Mathematica Japonica, 22, 1, 99-112 (1977) · Zbl 0372.47030
[8] Filippov, A. F., Diffrential equations with discontinuous right hand side, Matematicheskii Sbornik, 51, 99-128 (1960) · Zbl 0138.32204
[9] Filippov, A. F., Diffrential equations with discontinuous right hand side, Transactions of the American Mathematical Society, 42, 199-232 (1964) · Zbl 0148.33002
[10] Chang, K. C., The obstacle problem and partial differential equations with discontinuous nonlinearities, Communications on Pure and Applied Mathematics, 33, 2, 117-146 (1980) · Zbl 0405.35074 · doi:10.1002/cpa.3160330203
[11] Erbe, L.; Krawcewicz, W., Existence of solutions to boundary value problems for impulsive second order differential inclusions, The Rocky Mountain Journal of Mathematics, 22, 2, 519-539 (1992) · Zbl 0784.34012 · doi:10.1216/rmjm/1181072746
[12] Frigon, M.; Granas, A.; Guennoun, Z., A note on the Cauchy problem for differential inclusions, Topological Methods in Nonlinear Analysis, 1, 2, 315-321 (1993) · Zbl 0783.34009
[13] Deimling, K., Multivalued Differential Equations, 1 (1992), Berlin, Germany: Walter de Gruyter & Co., Berlin, Germany · Zbl 0760.34002 · doi:10.1515/9783110874228
[14] Rockafellar, R. T., On the maximality of sums of nonlinear monotone operators, Transactions of the American Mathematical Society, 149, 75-88 (1970) · Zbl 0222.47017 · doi:10.1090/S0002-9947-1970-0282272-5
[15] Minty, G. J., Monotone (nonlinear) operators in Hilbert space, Duke Mathematical Journal, 29, 341-346 (1962) · Zbl 0111.31202 · doi:10.1215/S0012-7094-62-02933-2
[16] Martinet, B., Régularisation d’inéquations variationnelles par approximations successives, Revue Francaise d’informatique et de Recherche operationelle, 4, 154-159 (1970) · Zbl 0215.21103
[17] Rockafellar, R. T., Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization, 14, 5, 877-898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056
[18] Bruck, R. E.; Sine, R. C., Asymptotic behavior of nonexpansive mappings, Contemporary Mathematics. Contemporary Mathematics, Fixed Points and Nonexpansive Mappings, 18 (1980), Providence, RI, England: AMS, Providence, RI, England · Zbl 0528.47039
[19] Browder, F. E.; Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, Journal of Mathematical Analysis and Applications, 20, 197-228 (1967) · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[20] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20, 1, 103-120 (2004) · Zbl 1051.65067 · doi:10.1088/0266-5611/20/1/006
[21] Sastry, K. P. R.; Babu, G. V. R., Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point, Czechoslovak Mathematical Journal, 55, 4, 817-826 (2005) · Zbl 1081.47069 · doi:10.1007/s10587-005-0068-z
[22] Panyanak, B., Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Computers & Mathematics with Applications, 54, 6, 872-877 (2007) · Zbl 1130.47050 · doi:10.1016/j.camwa.2007.03.012
[23] Song, Y.; Wang, H., 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, 55, 2999-3002 (2008) · Zbl 1142.47344
[24] Khan, S. H.; Yildirim, I., Fixed points of multivalued nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications, 2012 (2012) · Zbl 1460.47041 · doi:10.1186/1687-1812-2012-73
[25] Khan, S. H.; Yildirim, I.; Rhoades, B. E., A one-step iterative process for two multivalued nonexpansive mappings in Banach spaces, Computers & Mathematics with Applications, 61, 10, 3172-3178 (2011) · Zbl 1223.47081 · doi:10.1016/j.camwa.2011.04.011
[26] Abbas, M.; Khan, S. H.; Khan, A. R.; Agarwal, R. P., Common fixed points of two multivalued nonexpansive mappings by one-step iterative scheme, Applied Mathematics Letters, 24, 2, 97-102 (2011) · Zbl 1223.47068 · doi:10.1016/j.aml.2010.08.025
[27] García-Falset, J.; Lorens-Fuster, E.; Suzuki, T., Fixed point theory for a class of generalized nonexpansive mappings, Journal of Mathematical Analysis and Applications, 375, 1, 185-195 (2011) · Zbl 1214.47047 · doi:10.1016/j.jmaa.2010.08.069
[28] Daffer, P. Z.; Kaneko, H., Fixed points of generalized contractive multi-valued mappings, Journal of Mathematical Analysis and Applications, 192, 2, 655-666 (1995) · Zbl 0835.54028 · doi:10.1006/jmaa.1995.1194
[29] Shahzad, N.; Zegeye, H., On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Analysis. Theory, Methods & Applications, 71, 3-4, 838-844 (2009) · Zbl 1218.47118 · doi:10.1016/j.na.2008.10.112
[30] Krasnosel’skiĭ, M. A., Two remarks on the method of successive approximations, Uspekhi Matematicheskikh Nauk, 10, 1(63), 123-127 (1955) · Zbl 0064.12002
[31] Mann, W. R., Mean value methods in iteration, Proceedings of the American Mathematical Society, 4, 506-510 (1953) · Zbl 0050.11603 · doi:10.1090/S0002-9939-1953-0054846-3
[32] Ishikawa, S., Fixed points by a new iteration method, Proceedings of the American Mathematical Society, 44, 147-150 (1974) · Zbl 0286.47036 · doi:10.1090/S0002-9939-1974-0336469-5
[33] Song, Y.; Cho, Y. J., Some notes on Ishikawa iteration for multi-valued mappings, Bulletin of the Korean Mathematical Society, 48, 3, 575-584 (2011) · Zbl 1218.47120 · doi:10.4134/BKMS.2011.48.3.575
[34] Husain, T.; Latif, A., Fixed points of multivalued nonexpansive maps, Mathematica Japonica, 33, 3, 385-391 (1988) · Zbl 0667.47028
[35] Xu, H. K., On weakly nonexpansive and \(x 2 a;\)-nonexpansive multivalued mappings, Mathematica Japonica, 36, 3, 441-445 (1991) · Zbl 0733.54010
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