tailieunhanh - Positive periodic solutions to impulsive delay differential equations

In this paper we discuss the existence of positive periodic solutions for nonautonomous second order delay differential equations with singular nonlinearities in the presence of impulsive effects. Simple sufficient conditions are provided that enable us to obtain positive periodic solutions. Our approach is based on a variational method. | Turk J Math (2017) 41: 969 – 982 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Positive periodic solutions to impulsive delay differential equations Naima DAOUDI-MERZAGUI, Fatima DIB∗ Department of Mathematics, University of Tlemcen, Tlemcen, Algeria Received: • Accepted/Published Online: • Final Version: Abstract: In this paper we discuss the existence of positive periodic solutions for nonautonomous second order delay differential equations with singular nonlinearities in the presence of impulsive effects. Simple sufficient conditions are provided that enable us to obtain positive periodic solutions. Our approach is based on a variational method. Key words: Delay differential equation, periodic solution, singular nonlinearities, impulses, mountain-pass theorem 1. Introduction The impulsive differential equations characterize various processes of the real world, described by models that are subject to sudden changes in their states. Essentially, impulsive differential equations correspond to a smooth evolution that may change instantaneously or even abruptly; this type of equation allows the study of models in physics, population dynamics, ecology, industrial robotics, economics, biotechnology, optimal control, and chaos theory. Due to its significance, a great deal of work has been done in the theory of impulsive differential equations; see for example [3, 13, 28], and for an introduction of the basic theory of impulsive differential equations in Rn we refer to [5, 12, 17]. Recently, variational methods and critical point theory have been successfully employed to investigate impulsive differential equations when the nonlinearity is regular; the existence and multiplicity of solutions for impulsive boundary value problems have been considered in [9, 16, 19–28]. However, few papers have investigated the case of impulsive boundary value problems with .