On positive solutions for a nonlinear boundary value problem with impulse. (English) Zbl 1164.34371

Bereketoglu, Huseyin; Huseynov, Aydin

Czech. Math. J. 56, No. 1, 247-265 (2006).

Summary: We study nonlinear second order differential equations subject to separated linear boundary conditions and to linear impulse conditions. Sign properties of an associated Green’s function are investigated and existence results for positive solutions of the nonlinear boundary value problem with impulse are established. Upper and lower bounds for positive solutions are also given.

MSC:

34B37	Boundary value problems with impulses for ordinary differential equations
34B27	Green’s functions for ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

impulse conditions; Green’s function; completely continuous operator; fixed point theorem in cones

References:

[1]	F. M. Atici and G. Sh. Guseinov: On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions. J. Comput. Appl. Math. 132 (2001), 341–356. · Zbl 0993.34022 · doi:10.1016/S0377-0427(00)00438-6
[2]	D. D. Bainov and P. S. Simeonov: Impulsive Differential Equations: Asymtotic Properties of the Solutions. World Scientific, Singapore, 1995. · Zbl 0828.34002
[3]	P. W. Eloe and J. Henderson: Positive solutions of boundary value problems for ordinary differential equations with impulse. Dynam. Contin. Discrete Impuls. Systems 4 (1998), 285–294. · Zbl 0903.34013
[4]	P. W. Eloe and M. Sokol: Positive solutions and conjugate points for a boundary value problem with impulse. Dynam. Systems Appl. 7 (1998), 441–449. · Zbl 0945.34012
[5]	L. H. Erbe, S. Hu and H. Wang: Multiple positive solutions of some boundary value problems. J. Math. Anal. Appl. 184 (1994), 640–648. · Zbl 0805.34021 · doi:10.1006/jmaa.1994.1227
[6]	L. H. Erbe and H. Wang: On the existence of positive solutions of ordinary differential equations. Proc. Amer. Math. Soc. 120 (1994), 743–748. · Zbl 0802.34018 · doi:10.1090/S0002-9939-1994-1204373-9
[7]	D. Guo and V. Lakshmikantham: Nonlinear Problems in Abstract Cones. Academic Press, San Diego, 1998. · Zbl 0661.47045
[8]	M. A. Krasnosel’skii: Positive Solutions of Operator Equations. Noordhoff, Groningen, 1964.
[9]	M. A. Neumark: Lineare Differential Operatoren. Akademie-Verlag, Berlin, 1967.
[10]	A. M. Samoilenko and N. A. Perestyuk: Impulsive Differential Equations. World Scientific, Singapore, 1995.
[11]	Š. Schwabik, M. Tvrdý and O. Vejvoda: Differential and Integral Equations: Boundary Value Problems and Adjoint. Academia and Reidel, Praha and Dordrecht, 1979.
[12]	Š. Schwabik: Generalized Ordinary Differential Equations. World Scientific, Singapore, 1992.
[13]	M. Tvrdý: Differential and integral equations in the space of regulated functions. Memoirs on Differential Equations and Mathematical Physics 25 (2002), 1–104. · Zbl 1081.34504
[14]	M. Tvrdý: Linear distributional differential equations of the second order. Math. Bohem. 119 (1994), 415–436. · Zbl 0819.34007