We consider a certain type of second-order neutral delay differential systems and we establish two results concerning the oscillation of solutions after the system undergoes controlled abrupt perturbations (called impulses). As a matter of fact, some particular non-impulsive cases of the system are oscillatory already. Thus, we are interested in finding adequate impulse controls under which our...

This work deals with a scalar nonlinear neutral delay differential equation issued from the study of wave propagation. A critical value of the coefficients is considered, where only few results are known. The difficulty follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. An analysis based on the energy method provides new results about ...

and Applied Analysis 3 In the following, we will investigate the stability of (c, 0), (0, c ± ), and (c, c). Firstly, we consider the stability of (c, 0) when a > μ. (It has similar results with (c, 0) and (0, c).) The characteristic equation of linearization of (1) at (c, 0) is Δ (c + ,0) (λ) = (λ + μ − ae −λτ

where p≥ 2, φp(x) = |x|p–2x for x = 0 and φp(0) = 0; g :R4 →R is a continuous periodic function with g(t + T , ·, ·, ·)≡ g(t, ·, ·, ·), and g(t,a,a, 0) – e(t) ≡ 0 for all a ∈R. e :R→R is a continuous periodic function with e(t + T )≡ e(t) and ∫ T 0 e(t)dt = 0, c is a constant and |c| = 1, δ ∈ C1(R,R) and δ is a T -periodic function, T is a positive constant; n is a positive integer. By applicat...

Abstract In this paper, the successive approximations method is applied to solve multi-pantograph equations. The multi-pantograph equation is a kind of delay differential equations (DDEs). By this method, good approximate solutions can be obtained with only a few iteration. In addition, this method can be applied easily to solve neutral functional-differential equation with proportional delays....

By some analysis of the neutral operator (Ax)(t) = x(t)− ∑n i=1 cix(t− δi) and an application of the fixed-point index theorem, we obtain sufficient conditions for the existence, multiplicity and nonexistence of periodic solutions to a second-order differential equation with the prescribed neutral operator, which improve and extend some recent results of Lu-Ge, WuWang, and Zhang. An example is ...

This paper considers the asymptotic stability analysis of both exact and numerical solutions of the following neutral delay differential equation with pantograph delay. ⎧⎨ ⎩ x′(t) +Bx(t) + Cx′(qt) +Dx(qt) = 0, t > 0, x(0) = x0, where B,C,D ∈ Cd×d, q ∈ (0, 1), and B is regular. After transforming the above equation to non-automatic neutral equation with constant delay, we determine sufficient co...

Upper and lower solution method plays an important role in studying boundary value problems for nonlinear differential equations; see 1 and the references therein. Recently, many authors are devoted to extend its applications to boundary value problems of functional differential equations 2–5 . Suppose α is one upper solution or lower solution of periodic boundary value problems for second-orde...

This article concerns the behavior of the solutions to periodic linear neutral delay differential equations as well as to periodic linear neutral delay difference equations. Some new results are obtained via two appropriate distinct roots of the corresponding (so called) characteristic equation.