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Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. (English) Zbl 1176.54030

Summary: Let \((X,\leq)\) be a partially ordered set and suppose there is a metric \(d\) on \(X\) such that \((X,d)\) is a complete separable metric space and \((\Omega,\Sigma)\) be a measurable space. In this article, a pair of random mappings \(F:\Omega\times (X\times X)\rightarrow X\) and \(g:\Omega\times X\rightarrow X\), where \(F\) has a mixed \(g\)-monotone property on \(X\), and \(F\) and \(g\) satisfy a certain nonlinear contractive condition, are introduced and investigated. Two coupled random coincidence and coupled random fixed point theorems are proved. These results are random versions and extensions of recent results of the authors [V.Lakshmikantham and {Lj.Ćirić}, Nonlinear Anal., Theory Methods Appl.70, No.12 (A), 4341–4349 (2009; Zbl 1176.54032)] and include several recent developments.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
47H40 Random nonlinear operators
34B15 Nonlinear boundary value problems for ordinary differential equations

Citations:

Zbl 1176.54032
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References:

[1] Agarwal R.P., Appl. Anal. 87 pp 109– (2008) · Zbl 1140.47042
[2] Agarwal R.P., Appl. Anal. 83 pp 711– (2004) · Zbl 1088.47044
[3] Agarwal R.P., Appl. Anal. 87 pp 1– (2008) · Zbl 1154.35313
[4] Beg I., J. Appl. Math. Stochastic Anal. 7 pp 569– (1994) · Zbl 0823.47055
[5] Bhaskar T.G., Nonlinear Anal.–Theor. 65 pp 1379– (2006) · Zbl 1106.47047
[6] Bhaskar T.G., Nonlinear Anal. Theor. 66 pp 2237– (2007) · Zbl 1121.34065
[7] Ćirić Lj.B., Fixed Point Theory Appl. 2008 pp 11– (2008)
[8] Ćirić Lj.B., J. Inequal. Appl. 2006 pp 1– (2006) · Zbl 1129.47312
[9] Ćirić Lj.B., Italian J. Pure Applied Math. 23 pp 37– (2008)
[10] Guo D., Nonlinear Problems in Abstract Cones (1988) · Zbl 0661.47045
[11] Hadzić O., Mat. Vesnik 3 (16) 31 pp 397– (1979)
[12] Han[sbreve] O., Czech. Math. J. 7 pp 154– (1957)
[13] Han[sbreve] , O. 1961 . Random operator equations . Proc. 4th Berkeley Symp. Mathematical Statistics and Probability . Vol. II, Part I , University of California Press , Berkeley , pp. 185 – 202 .
[14] Heikkila S., Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994)
[15] Himmelberg C.J., Fund. Math. 87 pp 53– (1975)
[16] Huang N.J., Applied Math. Lett. 12 pp 107– (1999) · Zbl 0945.60057
[17] Hussain N., J. Math. Anal. Appl. 338 pp 1351– (2008) · Zbl 1134.47039
[18] Itoh S., Pacific J. Math. 68 pp 85– (1977)
[19] Ladde G.S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985) · Zbl 0658.35003
[20] Lakshmikantham V., Nonlinear Anal. Theor. 70 pp 4341– (2009) · Zbl 1176.54032
[21] Lakshmikantham V., Theory of Set Differential Equations in Metric Spaces (2005)
[22] DOI: 10.1201/9780203011386
[23] Lakshmikantham V., Monotone Flows and Rapid Convergence for Nonlinear Partial Differential Equations (2003) · Zbl 1017.35001
[24] Lakshmikantham V., Appl. Math. Lett. 21 pp 828– (2008) · Zbl 1161.34031
[25] Lin T.C., Proc. Am. Math. Soc. 103 pp 1129– (1988)
[26] McShane E.J., Proc. Am. Math. Soc. 18 pp 41– (1967)
[27] DOI: 10.1007/s11083-005-9018-5 · Zbl 1095.47013
[28] DOI: 10.1090/S0002-9939-07-08729-1 · Zbl 1126.47045
[29] Papageorgiou N.S., Math. Japonica 29 pp 93– (1984)
[30] DOI: 10.1090/S0002-9939-1986-0840638-3
[31] Ran A.C.M., Proc. Amer. Math. Soc. 132 pp 1435– (2004) · Zbl 1060.47056
[32] DOI: 10.1016/0022-247X(69)90104-8 · Zbl 0202.33804
[33] DOI: 10.1090/S0002-9939-1985-0796453-1
[34] Shahzad N., J. Math. Anal. Appl. 323 pp 1038– (2006) · Zbl 1107.47042
[35] DOI: 10.1006/jmaa.2000.6772 · Zbl 0970.60074
[36] [Sbreve]paček A., Czech. Math. J. 5 pp 462– (1955)
[37] DOI: 10.1006/jmaa.1994.1256 · Zbl 0856.47036
[38] DOI: 10.1137/0315056 · Zbl 0407.28006
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