Abstract
Existence and localization results are derived from Krasnoselskii’s compressionexpansion fixed point theorem in cones, for operator equations in spaces of continuous functions from a compact real interval to an abstract space. The main idea, first used in [12], is to handle two equivalent operator forms of the equation, one of fixed point type giving the operator to which Krasnoselskii’s theorem applies and an other one of coincidence type which is used to localize a positive solution in a shell. An application is presented for a boundary value problem associated to a fourth order partial differential equation on a rectangular domain.
Authors
Radu Precup
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Keywords
Positive solution; Boundary value problem; Fixed point theorem in cones.
Paper coordinates
R. Precup, Positive solutions of evolution operator equations, Austral. J. Math. Anal. Appl. 2 (2005) no. 1, 1-10.
About this paper
Journal
Australian Journal of Mathematical Anaysis and Applications
Publisher Name
paper on journal website
Print ISSN
Online ISSN
1449-5910
MR2133376, Zbl 1078.47059,
google scholar link
[1] R. AGARWAL, M. MEEHAN, D. O’REGAN and R. PRECUP, Location of nonnegative solutions for differential equation on finite and semi-infinite intervals, Dynam. Systems Appl. 12 (2003), 323-341.
[2] L. H. ERBE, S. HU and H. WANG, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl. 184 (1994), 640-648.
[3] L. H. ERBE and H. WANG, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994), 743-748.
[4] A. GRANAS and J. DUGUNDJI, Fixed Point Theory, Springer, New York, 2003.
[5] A. HORVAT–MARC and R. PRECUP, Nonnegative solutions of nonlinear integral equations in ordered Banach spaces, Fixed Point Theory 5 (2004), 65-70.
[6] M. A. KRASNOSELSKII, Positive Solutions of Operator Equations, Noordhoff, Groninge 1964.
[7] K. LAN and J. R. L. WEBB, Positive solutions of semilinear differential equations with singularities, J. Differential Equations 148 (1998), 407-421.
[8] W. LIAN, F. WONG and C. YEH, On the existence of positive solutions of nonlinear second order differential equations, Proc. Amer. Math. Soc. 124 (1996), 1117-1126.
[9] J. L. LIONS, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod & Gauthier–Villars, Paris, 1969.
[10] M. MEEHAN and D. O’REGAN, Multiple nonnegative solutions of nonlinear integral equation on compact and semi-infinite intervals, Appl. Anal. 74 (2000), 413-427.
[11] D. O’REGAN and R. PRECUP, Theorems of Leray–Schauder Type and Applications, Taylor and Francis, London, 2002.
[12] R. PRECUP, Existence and localization results for the nonlinear wave equation, Fixed Point Theory, submitted.