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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 for nonlinear operators on topological linear spaces 34B15 Nonlinear boundary value problems for ODE 54H25 Fixed-point and coincidence theorems in topological spaces 54F05 Linearly, generalized, and partial ordered topological spaces
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##### 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) · Zbl 0661.47045 [3] Heikkila, S.; Lakshmikantham, V.: Monotone iterative techniques for discontinuous nonlinear differential equations. (1994) [4] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S.: Monotone iterative techniques for nonlinear differential equations. (1985) · Zbl 0658.35003 [5] Lakshmikantham, V.; Bhaskar, T. Gnana; Devi, J. Vasundhara: Theory of set differential equations in metric spaces. (2005) · Zbl 1156.34003 [6] Lakshmikantham, V.; Mohapatra, R. N.: Theory of fuzzy differential equations and inclusions. (2003) · Zbl 1072.34001 [7] Lakshmikantham, V.; Koksal, S.: Monotone flows and rapid convergence for nonlinear partial differential equations. (2003) [8] J.J. Nieto, R.R. Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order (in press) [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) [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)