Bounded Oscillation Theorem for Unstable-type Neutral Impulsive Differential Equations of the Second Order
U. A. Abasiekwere *
Department of Mathematics and Statistics, University of Uyo, P.M.B. 1017, Uyo, Akwa Ibom State, Nigeria.
E. Eteng
Department of Mathematics, University of Calabar, P.M.B. 1115, Calabar, Cross River State, Nigeria.
I. O. Isaac
Department of Mathematics and Statistics, Akwa Ibom State University, P.M.B. 1167, Ikot Akpaden, Akwa Ibom State, Nigeria.
Z. Lipcsey
Department of Mathematics, University of Calabar, P.M.B. 1115, Calabar, Cross River State, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
The oscillations theory of neutral impulsive differential equations is gradually occupying a central place among the theories of oscillations of impulsive differential equations. This could be due to the fact that neutral impulsive differential equations plays fundamental and significant roles in the present drive to further develop information technology. Indeed, neutral differential equations appear in networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits). In this paper, we study the behaviour of solutions of a certain class of second-order linear neutral differential equations with impulsive constant jumps. This type of equation in practice is always known to have an unbounded non-oscillatory solution. We, therefore, seek sufficient conditions for which all bounded solutions are oscillatory and provide an example to demonstrate the applicability of the abstract result.
Keywords: Second-order, impulsive, neutral delay differential equation, oscillation