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Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. (English) Zbl 1140.47045
Summary: We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case [Order 22, No. 3, 223–239 (2005; Zbl 1095.47013)], we consider in this paper nonincreasing mappings as well as non-monotone mappings. We also present some applications to first-order ordinary differential equations with periodic boundary conditions, proving the existence of a unique solution admitting the existence of a lower solution.

MSC:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H07Monotone and positive operators on ordered topological linear spaces
47N20Applications of operator theory to differential and integral equations
34B15Nonlinear boundary value problems for ODE
References:
[1]Ran, A. C. M., Reurings, M. C. B.: A .xed point theorem in partially ordered sets and some applications to matrix equations. Proc. Amer. Math. Soc., 132, 1435–1443 (2004) · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4
[2]Nieto, J. J., Rodríguez-López, R.: Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations. Order, 22, 223–239 (2005) · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5
[3]Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math., 5, 285–309 (1955)
[4]Cousot, P., Cousot, R.: Constructive versions of Tarski’s fixed point theorems. Pacific J. Math., 82, 43–57 (1979)
[5]Zeidler, E.: Nonlinear functional analysis and its applications, Vol I: Fixed-Point Theorems, Springer-Verlag, New York, 1986
[6]Amann, H.: Order structures and fixed points, Bochum: Mimeographed lecture notes, Ruhr–Universität, 1977
[7]Heikkilä, S., Lakshmikantham, V.: Monotone iterative techniques for discontinuous nonlinear differential equations, Marcel Dekker, Inc., New York, 1994
[8]Ladde, G. S., Lakshmikantham, V., Vatsala, A. S.: Monotone iterative techniques for nonlinear differential equations, Pitman, Boston, 1985