Principal and nonprincipal solutions of impulsive differential equations with applications A. Özbekler a , A. Zafer b, * a Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey b Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey article info Keywords: Principal Nonprincipal Polya factorization Trench factorization Impulse Second order abstract We introduce the concept of principal and nonprincipal solutions for second order differ- ential equations having fixed moments of impulse actions is obtained. The arguments are based on Polya and Trench factorizations as in non-impulsive differential equations, so we first establish these factorizations. Making use of the existence of nonprincipal solutions we also establish new oscillation criteria for nonhomogeneous impulsive differential equa- tions. Examples are provided with numerical simulations to illustrate the relevance of the results. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction The concept of the principal solution was introduced in 1936 by Leighton and Morse [1] in studying positiveness of cer- tain quadratic functionals associated with ðrðtÞz 0 Þ 0 þ qðtÞz ¼ 0; rðtÞ > 0: ð1Þ Since then the principal and nonprincipal solutions have been used successfully in connection with oscillation and asymp- totic theory of related equations, see for instance [1–8] and the references cited therein. For some extensions to Hamiltonian systems and half-linear differential equations, we refer in particular to [4,9]. A nontrivial solution u of (1) is said to be principal if for every solution v such that u – kv ; k 2 R, lim t!1 uðtÞ v ðtÞ ¼ 0: It is well known that a principal solution u of (1) exists uniquely up to a multiplication by a nonzero constant if and only if (1) is nonoscillatory. A solution v linearly independent of u is called a nonprincipal solution. Roughly speaking, the terms ‘‘prin- cipal” and ‘‘nonprincipal” may be replaced by ‘‘small” and ‘‘large”. For other characterizations of principal solution and non- principal solutions of (1), see [5, Theorem 6.4]. Impulsive differential equations are of particular interest in many areas such as biology, physics, chemistry, control the- ory, medicine, etc. as they model the real processes better than differential equations. Because of the lack of smoothness property of the solutions the theory of impulsive differential equations is much richer than that of differential equations without impulse. However, due to difficulties caused by impulsive perturbations, the theory is not well-developed in com- parison with that of non-impulsive differential equations. For basic theory of impulsive differential equations, we refer in particular to [10,11]. 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.02.008 * Corresponding author. E-mail addresses: aozbekler@gmail.com (A. Özbekler), zafer@metu.edu.tr (A. Zafer). Applied Mathematics and Computation 216 (2010) 1158–1168 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc