On the existence of periodic solutions for a kind of high-order neutral functional differential equation

By using the coincidence degree theory of Mawhin, we study a kind of high-order neutral functional differential equation with distributed delay as follows:

     

具偏差变元的高阶中立型泛函微分方程的周期解存在性问题

利用 Fourier 级数理论,伯努利数理论和重合度理论研究了一类具偏差变元的高阶中立型泛函微分方程的周期解问题,得到了周期解存在的充分条件.     

Convergence in a neutral delay differential equation

By using some facts from limiting equations theory we prove that the solution x(.;?), with continuous initial condition ?, of the neutral functional differential equation [x(t)-cx(t-r)]' =-F(x(t))+F(x(t-r)), t>0, where c ε [0,1), r≧0 and F is (not necessarily strictly) increasing. satisfies lim x(t;?) = &;, where &; is the unique root of the algebraic equation [math001]     

On solutions of a neutral differential equation with deviating argument

We prove a theorem on the existence and asymptotic behaviour of solutions of a differential equation with a deviating argument of neutral type. The considered equation contains both delayed and advanced arguments. The method used in the proof of our main result depends on conjunction of the classical Schauder fixed point theorem with the technique of measures of noncompactness.     

Bifurcation analysis in a neutral differential equation

The dynamics of a neural network model in neutral form is investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. and a Bendixson's criterion for higher dimensional ordinary differential equations due to Li and Muldowney.     

On a class of abstract neutral functional differential equations

By using the theory of semigroups of growth α, we discuss the existence of mild solutions for a class of abstract neutral functional differential equations. A concrete application to partial neutral functional differential equations is considered.     

Existence and asymptotic behavior of a functional differential equation in Banach space

Existence and asymptotic behavior of solutions are given for the equation u′(t) = ?A(t)u(t) + F(t,u_t) (t ? 0) and u₀ = ? ? C([?r,0]; X)  C. The space X is a Banach space; the family $\text{{A(t) ¦ 0 ? t ? T}}$ of unbounded linear operators defined on D(A) ? X → X generates a linear evolution system and F: C → X is continuous with respect to a fractional power of A(t₀) for some t₀ ? [0, T].     

Norm continuity and stability for a functional differential equation in Hilbert space

A functional differential equation in Hilbert space with initial data on [−h,0] is considered. An unbounded operator A and a square integrable weight function are acting in the distributed delay term. For a not necessarily continuous weight function the norm continuity of the associated solution semigroup is established at every t>h. In the case that the spectrum of A is real and negative, the asymptotic stability of the solution is obtained.     

Periodic solutions to a kind of neutral functional differential equation in the critical case

In this paper, we consider a kind of neutral functional differential equation as follows:

     

On stability of the second order neutral differential equation

There exist a well-developed stability theory for neutral differential equations of the first order and only a few results on functional differential equations of the second order. One of the aims of this paper is to fill this gap. Explicit tests for stability of linear neutral delay differential equations of the second order are obtained.     

Periodic solutions to a second order nonlinear neutral functional differential equation in the critical case

In this paper, the authors study the existence of periodic solutions for a second order neutral functional differential equation

(x(t)-cx(t-τ))^″=f(x(t))x^′(t)+g(t,x(t-μ(t)))+e(t)$(x(t)-\mathit{cx}(t-\tau ){)}^{″}=f(x(t))x^{\prime }(t)+g(t,x(t-\mu (t)))+e(t)$

On convergence of the solutions of a functional differential equation

A sufficient condition is given for the solutions of a functionally perturbed linear system of ordinary differential equations to have limits at ± ∞.     

On the existence of positive periodic solutions for neutral functional differential equation with multiple deviating arguments

By means of the abstract continuation theory for k-contractions, some criteria are established for the existence and nonexistence of positive periodic solutions of the following neutral functional differential equation:

     

On the solvability of a complete second-order differential equation in Banach space

The paper deals with the Cauchy problem for a complete second-order differential equation with unbounded operator coefficientsu+A(t)u+B(t)u=f, u(0)=u0, u(0)=u ₁ . By using the commutant method, we construct a coercive solution of this problem in Holder space in the case where the operatorB is as strong as the operator A².Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1449–1454, October, 1993.     

On the null-controllability in function space of nonlinear systems of neutral functional differential equations with limited controls

Consider the following functional equations of neutral type: $$\begin{gathered} (i) (d/dt)D(t,x_t ) = L(t,x_t ), \hfill \\ (ii) (d/dt)D(t,x_t ) = L(t,x_t ) + B(t)u(t), \hfill \\ (iii) (d/dt)D(t,x_t ) = L(t,x_t ) + B(t)u(t) + f(t,x(t),u(t)), \hfill \\ \end{gathered} $$ whereD, L are bounded linear operators fromC([?h, 0],E ⁿ) intoE ⁿ for eacht?(σ, ∞) =J, B is ann ×m continuous matrix function,u:J →C ^m is square integrable with values in the unitm-dimensional cubeC ^m, andf(t, 0, 0)=0. We prove that, if the system (i) is uniformly asymptotically stable and if the controlled system (ii) is controllable, then the system (iii) is null-controllable with constraints, provided that $$f = f_1 + f_2 $$ , where $$\begin{gathered} |f_1 (t,\phi ,0)| \leqslant \varepsilon \parallel \phi \parallel , |f_2 (t,\phi ,0)| \leqslant \pi (t)\parallel \phi \parallel , t \geqslant \sigma , \hfill \\ \Pi = \int_0^\infty {\pi (t)dt< \infty .} \hfill \\ \end{gathered} $$      

Linear autonomous neutral differential equations in a Banach space

The existence of solutions for a class of linear functional differential equations defined on a general Banach space is established; the solutions are shown to generate a semigroup of class C₀; a representation of the solutions in terms of a particular family of linear transformations is developed.