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impulse conditions; Green’s function; completely continuous operator; fixed point theorem in cones
In this paper 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.
[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. DOI 10.1016/S0377-0427(00)00438-6 | MR 1840633
[2] D. D. Bainov and P. S. Simeonov: Impulsive Differential Equations: Asymtotic Properties of the Solutions. World Scientific, Singapore, 1995. MR 1331144
[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. MR 1621834
[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. MR 1664566
[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. DOI 10.1006/jmaa.1994.1227 | MR 1281534
[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. DOI 10.1090/S0002-9939-1994-1204373-9 | MR 1204373
[7] D. Guo and V. Lakshmikantham: Nonlinear Problems in Abstract Cones. Academic Press, San Diego, 1998. MR 0959889
[8] M. A. Krasnosel’skii: Positive Solutions of Operator Equations. Noordhoff, Groningen, 1964. MR 0181881
[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. MR 1355787
[11] Š. Schwabik, M. Tvrdý and O. Vejvoda: Differential and Integral Equations: Boundary Value Problems and Adjoint. Academia and Reidel, Praha and Dordrecht, 1979. MR 0542283
[12] Š. Schwabik: Generalized Ordinary Differential Equations. World Scientific, Singapore, 1992. MR 1200241 | Zbl 0781.34003
[13] M. Tvrdý: Differential and integral equations in the space of regulated functions. Memoirs on Differential Equations and Mathematical Physics 25 (2002), 1–104.
[14] M. Tvrdý: Linear distributional differential equations of the second order. Math. Bohem. 119 (1994), 415–436. MR 1316594
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