On second order differential equations with state-dependent delay

We study existence and uniqueness of solutions for a general class of second order abstract differential equations with state-dependent delay. Some examples related to partial differential equations with state dependent delay are presented.     

A Taylor collocation method for solving delay integral equations

A numerical method based on the use of Taylor polynomials is proposed to construct a collocation solution $u\in S_{m-1}^{(-1)}(\Pi _{N})$ for approximating the solution of delay integral equations. It is shown that this method is convergent. Some numerical examples are given to show the validity of the presented method.     

A condition on delay for differential equations with discrete state-dependent delay

Parabolic differential equations with discrete state-dependent delay are studied. The approach, based on an additional condition on the delay function introduced in [A.V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal. 70 (11) (2009) 3978–3986] is developed. We propose and study an analogue of the condition which is sufficient for the well-posedness of the corresponding initial value problem on the whole space of continuous functions C. The dynamical system is constructed in C and the existence of a compact global attractor is proved.     

Stability criteria for certain high even order delay differential equations

In this paper we study the asymptotic stability of the zero solution of even order linear delay differential equations of the form

     

On time transformations for differential equations with state-dependent delay

Systems of differential equations with state-dependent delay are considered. The delay dynamically depends on the state, i.e. is governed by an additional differential equation. By applying the time transformations we arrive to constant delay systems and compare the asymptotic properties of the original and transformed systems.     

Periodic solutions for general nonlinear state-dependent delay logistic equations

In this paper, by using the continuation theorem of coincidence degree theory, we investigate the existence of periodic solutions for more general state-dependent delay logistic equations. Several sufficient conditions are given, and the obtained conditions possess important significance in both theories and applications.     

Global Hopf bifurcation for differential equations with state-dependent delay

We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of S¹-equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay.     

An analysis of a method for solving singular integral equations

We establish convergence rates for a method of approximate solution of certain singular integral equations. The method considered involves an expansion of the kernel of the equation in terms of Chebyshev polynomials.     

Non-local PDEs with a state-dependent delay term presented by Stieltjes integral

Parabolic partial differential equations with state-dependent delays (SDDs) are investigated. The delay term presented by Stieltjes integral simultaneously includes discrete and distributed SDDs. The singular Lebesgue–Stieltjes measure is also admissible. The conditions for the corresponding initial value problem to be well-posed are presented. The existence of a compact global attractor is proved.     

On the problem of linearization for state-dependent delay differential equations

The local stability of the equilibrium for a general class of state-dependent delay equations of the form

has been studied under natural and minimal hypotheses. In particular, it has been shown that generically the behavior of the state-dependent delay (except the value of near an equilibrium has no effect on the stability, and that the local linearization method can be applied by treating the delay as a constant value at the equilibrium.

     

A HIGH ORDER METHOD FOR NON-SMOOTH FREDHOLM EQUATIONS

1.IntroductionConsidertheequationwherek(s,t)=k(f)tandf(s)aregiven,uistheunknownsolution.SinceitisrelatedcloselytoWiener-Hopfequationsandisveryimportantinpractice,therearemanynumericalresultsaboutit(e.g.[1--11]).Itiswellknownthattheaccuracyoftheapproximati…     

Solving neutral delay differential equations with state-dependent delays

In this paper we consider a class of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to subsequent points, called “breaking points”, where the solution derivative is still discontinuous. As a consequence, in a right neighbourhood of each such point we have to face a Cauchy problem where the equation has a discontinuous right-hand side. In this case the existence and the uniqueness of the solution is no longer guaranteed to the right of such points and hence the solution of the neutral equation may either cease to exist or bifurcate. After illustrating why uniqueness and existence of the solution is no longer guaranteed for general state-dependent problems and showing a possible way to detect these occurrences automatically, we explain how to generalize/regularize the problem in order to suitably extend the solution beyond the breaking point. This is important, for example, when exploring numerically the presence of possible periodic orbits.     

Exponential stability for a class of state-dependent delay equations via the Crandall-Liggett approach

In this paper we are concerned with the exponential asymptotic stability of the solution of a class of differential equations with state dependent delays. Our approach is based on the Crandall-Liggett approximation and the properties of semigroups.     

On state-dependent delay partial neutral functional integro-differential equations

In this paper the existence of mild solutions for a class of abstract neutral integro-differential equations with state-dependent delay is studied.     

The Poincaré–Bendixson theorem for a class of delay equations with state-dependent delay and monotonic feedback

We consider the real-valued differential equation

$x^{\prime }(t)=f(x(t),x(t?d(x_t)))$

with state-dependent delay, where f is strictly monotonic in its second argument. We describe a class of such equations for which a version of the Poincaré–Bendixson theorem holds.