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Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. (English) Zbl 1095.47013
The main results of this interesting paper are existence and uniqueness theorems for a periodic boundary value problem. The approach is based on some fixed point theorems on a partially ordered set. The fixed point results are in close connections with some results given by A. C. M. Ran and M. C. B. Reurings [Proc. Am. Math. Soc. 132, No. 5, 1435–1443 (2004; Zbl 1060.47056)] and by A. Petruşel and I. A. Rus [ibid. 134, No. 2, 411–418 (2006; Zbl 1086.47026)].

MSC:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
34B15Nonlinear boundary value problems for ODE
References:
[1]Amann, H.: Order Structures and Fixed Points. Mimeographed Lecture Notes, Ruhr-Universität, Bochum, 1977.
[2]Cousot, P. and Cousot, R.: Constructive versions of Tarski’s fixed point theorems, Pacific J. Math. 82 (1979), 43–57.
[3]Heikkilä, S. and Lakshmikantham, V.: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, Inc., New York, 1994.
[4]Ladde, G.S., Lakshmikantham, V. and Vatsala, A.S.: Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985.
[5]Ran, A.C.M. and Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Am. Math. Soc. 132 (2004), 1435–1443. · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4
[6]Tarski, A.: A lattice-theoretical fixpoint theorem and its applications, Pacific J. Math. 5 (1955), 285–309.
[7]Zeidler, E.: Nonlinear Functional Analysis and Its Applications, Vol. I: Fixed-Point Theorems, Springer, New York, 1986.