×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.