Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations
- 764 Downloads
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, 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.
Keywordsfixed point partially ordered set first–order differential equation lower and upper solutions
MR (2000) Subject Classification47H10 34B15
Unable to display preview. Download preview PDF.
- 5.Zeidler, E.: Nonlinear functional analysis and its applications, Vol I: Fixed-Point Theorems, Springer-Verlag, New York, 1986Google Scholar
- 6.Amann, H.: Order structures and fixed points, Bochum: Mimeographed lecture notes, Ruhr–Universität, 1977Google Scholar
- 7.Heikkilä, S., Lakshmikantham, V.: Monotone iterative techniques for discontinuous nonlinear differential equations, Marcel Dekker, Inc., New York, 1994Google Scholar
- 8.Ladde, G. S., Lakshmikantham, V., Vatsala, A. S.: Monotone iterative techniques for nonlinear differential equations, Pitman, Boston, 1985Google Scholar