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Random fixed point theorems under mild continuity assumptions. (English) Zbl 1408.47014

Summary: In this paper, we study the existence of the random fixed points under mild continuity assumptions. The main theorems consider the almost lower semicontinuous operators defined on Banach spaces and also operators having properties weaker than lower semicontinuity. Our results either extend or improve corresponding ones present in literature.

MSC:

47H10 Fixed-point theorems
47H40 Random nonlinear operators
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References:

[1] Fierro, R.; Martinez, C.; Morales, CH, Fixed point theorems for random lower semi-continuous mappings, No. 2009 (2009) · Zbl 1187.47044
[2] Fierro R, Martinez C, Morales CH: Random coincidence theorems and applications.J. Math. Anal. Appl. 2011, 378:213-219. · Zbl 1223.47066 · doi:10.1016/j.jmaa.2010.12.008
[3] Fierro R, Martinez C, Orellana E: Weak conditions for existence of random fixed points.Fixed Point Theory 2011,12(1):83-90. · Zbl 1273.47081
[4] Lazar TA, Petrusel A, Shahzad N: Fixed points for non-self operators and domain invariance theorems.Nonlinear Anal. 2009, 70:117-125. · Zbl 1183.47052 · doi:10.1016/j.na.2007.11.037
[5] Patriche M: A new fixed-point theorem and its applications in the equilibrium theory.Fixed Point Theory 2009, 1:159-171. · Zbl 1185.91107
[6] Patriche M: Fixed point theorems for nonconvex valued correspondences and applications in game theory.Fixed Point Theory 2013, 14:435-446. · Zbl 1305.47039
[7] Patriche, M: Fixed point theorems and applications in theory of games. Fixed Point Theory (forthcoming) · Zbl 1305.91178
[8] Patriche M: Equilibrium in Games and Competitive Economies. The Publishing House of the Romanian Academy, Bucharest; 2011.
[9] Petrusel A: Multivalued operators and fixed points.Pure Math. Appl. 2000, 11:361-368. · Zbl 0980.47047
[10] Petrusel A, Rus IA: Fixed point theory of multivalued operators on a set with two metrics.Fixed Point Theory 2007, 8:97-104. · Zbl 1133.47036
[11] Shahzad N: Random fixed points of set-valued maps.Nonlinear Anal., Theory Methods Appl. 2001, 45:689-692. · Zbl 0994.47057 · doi:10.1016/S0362-546X(99)00412-5
[12] Shahzad N: Random fixed points ofK-set- and pseudo-contractive random maps.Nonlinear Anal. 2004, 57:173-181. · Zbl 1083.47046 · doi:10.1016/j.na.2004.02.006
[13] Shahzad N, Hussain N: Deterministic and random coincidence point results forf-nonexpansive maps.J. Math. Anal. Appl. 2006, 323:1038-1046. · Zbl 1107.47042 · doi:10.1016/j.jmaa.2005.10.057
[14] Shahzad N: Some general random coincidence point theorems.N.Z. J. Math. 2004, 33:95-103. · Zbl 1083.47047
[15] Shahzad N: Random fixed points of discontinuous random maps.Math. Comput. Model. 2005, 41:1431-1436. · Zbl 1171.54338 · doi:10.1016/j.mcm.2004.02.036
[16] Ionescu Tulcea C: On the approximation of upper semicontinuous correspondences and the equilibria of generalized games.J. Math. Anal. Appl. 1968, 136:267-289. · Zbl 0685.90100 · doi:10.1016/0022-247X(88)90130-8
[17] Agarwal RP, Dshalalow JH, O’Regan D: Fixed point theory for Mönch-type maps defined on closed subsets of Fréchet spaces: the projective limit approach.Int. J. Math. Math. Sci. 2005, 2005:2775-2782. · Zbl 1102.47039 · doi:10.1155/IJMMS.2005.2775
[18] Agarwal RP, O’Regan D:Random fixed point theorems and Leray-Schauder alternatives for[InlineEquation not available: see fulltext.]maps.Commun. Korean Math. Soc. 2005, 20:299-310. · Zbl 1097.47051 · doi:10.4134/CKMS.2005.20.2.299
[19] Agarwal, RP; Frigon, M.; O’Regan, D., A survey of recent fixed point theory in Fréchet spaces (2003) · Zbl 1078.47002 · doi:10.1007/978-94-010-0035-2
[20] O’Regan D: An essential map approach for multimaps defined on closed subsets of Fréchet spaces.Appl. Anal. 2006, 85:503-513. · Zbl 1106.47042 · doi:10.1080/00036810500474861
[21] O’Regan D, Agarwal RP: Fixed point theory for admissible multimaps defined on closed subsets of Fréchet spaces.J. Math. Anal. Appl. 2003, 277:438-445. · Zbl 1022.47037 · doi:10.1016/S0022-247X(02)00517-6
[22] Shahzad N: Random fixed points of multivalued maps in Fréchet spaces.Arch. Math. 2002, 38:95-100. · Zbl 1068.47075
[23] Debreu, G., Integration of correspondences, 351-372 (1966), Berkeley
[24] Kuratowski K, Ryll-Nardzewski C: A general theorem on selectors.Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 1965, 13:397-403. · Zbl 0152.21403
[25] Deutsch F, Kenderov P: Continuous selections and approximate selection for set-valued mappings and applications to metric projections.SIAM J. Math. Anal. 1983, 14:185-194. · Zbl 0518.41031 · doi:10.1137/0514015
[26] Wu X, Shen S: A further generalisation of Yannelis-Prabhakar’s continuous selection theorem and its applications.J. Math. Anal. Appl. 1996, 197:61-74. · Zbl 0852.54019 · doi:10.1006/jmaa.1996.0007
[27] Zheng X: Approximate selection theorems and their applications.J. Math. Anal. Appl. 1997, 212:88-97. · Zbl 0918.54018 · doi:10.1006/jmaa.1997.5466
[28] Ansari QH, Yao J-C: A fixed point theorem and its applications to a system of variational inequalities.Bull. Aust. Math. Soc. 1999, 59:433-442. · Zbl 0944.47037 · doi:10.1017/S0004972700033116
[29] Petryshyn WV: Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces.Trans. Am. Math. Soc. 1973, 182:323-352. · Zbl 0277.47033
[30] Michael E: Continuous selection.Ann. Math. 1956, 63:361-382. · Zbl 0071.15902 · doi:10.2307/1969615
[31] Patriche, M: Random fixed point theorems for lower semicontinuous condensing random operators (submitted) · Zbl 1398.91394
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