Bounded Oscillation Theorem for Unstable-type Neutral Impulsive Differential Equations of the Second Order

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.

Existence Theorem for Impulsive Differential Equations with Measurable Right Side for Handling Delay Problems

“…Existence, uniqueness, and continuation of solutions are fundamental issues for differential equations of all kinds. ese issues, therefore, have been studied by many authors [1,2,[12][13][14][15][16][17][18][19] just to mention a few. ese articles consider initial value problems and boundary value problems for impulsive differential equations with an at least continuous right side f in Ω or in addition to the continuity f fulfills Lipschitz condition in its spatial variables; therefore, the analysis is based on Cauchy-Peano's or Piccard-Lindelöf's existence theorems [9].…”

“…e domains of the Radon-Nikodym derivatives are countable unions of pairwise disjoint atoms of the σ-algebras B ] λ,[a,b ] ],c and B ] τ,[a,b ] ],c , respectively, as defined in equations: D ] λ in (11); D ] τ in (37). e lists of these atoms are defined: D ] λ in (12) and D ] τ in (38). is means that the Radon-Nikodym derivatives are fully determined on D ] λ if they are determined on the intervals in D ] λ , and they are fully determined on D ] τ if they are determined on each interval in D ] τ .…”

Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay

“…In this study, a system of first-order impulsive differential equations with continuous time-dependent delays and fixed moments of impulse is formulated. Some recent results in impulsive delay differential equations with constant impulsive jumps can be seen in [7][8][9][10][11][12][13][14].…”