Fixed point theorems in partially ordered metric spaces and applications. (English) Zbl 1106.47047

This paper gives some coupled fixed point theorems for a monotone mapping in a metric space endowed with a partial order, using a weak contractivity type assumption. Besides including several recent developments, the theorems can be used to investigate a class of problems. As an application, the existence and uniqueness of solutions for a periodic boundary value problem are discussed.


47H10 Fixed-point theorems
34B15 Nonlinear boundary value problems for ordinary differential equations
54H25 Fixed-point and coincidence theorems (topological aspects)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
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[1] Gouze, J.-L.; Hadeler, K.P., Monotone flows and order intervals, Nonlin. world, 1, 23-34, (1994) · Zbl 0803.65076
[2] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045
[3] Heikkila, S.; Lakshmikantham, V., Monotone iterative techniques for discontinuous nonlinear differential equations, (1994), Marcel Delker New York · Zbl 0804.34001
[4] Ladde, G.S.; Lakshmikantham, V.; Vatsala, A.S., Monotone iterative techniques for nonlinear differential equations, (1985), Pitman Advanced Publishing Program · Zbl 0658.35003
[5] Lakshmikantham, V.; Gnana Bhaskar, T.; Vasundhara Devi, J., Theory of set differential equations in metric spaces, (2005), Cambridge. Sci Pub. · Zbl 1156.34003
[6] Lakshmikantham, V.; Mohapatra, R.N., Theory of fuzzy differential equations and inclusions, (2003), Taylor&Francis London · Zbl 1072.34001
[7] Lakshmikantham, V.; Koksal, S., Monotone flows and rapid convergence for nonlinear partial differential equations, (2003), Taylor& Francis · Zbl 1017.35001
[8] J.J. Nieto, R.R. Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order (in press) · Zbl 1095.47013
[9] J.J. Nieto, R.R. Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta. Math. Sinica (in press) · Zbl 1140.47045
[10] Ran, A.C.M.; Reurings, M.C.B., A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. amer. math. soc., 132, 1435-1443, (2003) · Zbl 1060.47056
[11] Schroder, J., Operator inequalities, (1980), Academic Press · Zbl 0455.65039
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