Exponential dichotomies and homoclinic orbits in functional differential equations

Suppose an autonomous functional differential equation has an orbit Γ which is homoclinic to a hyperbolic equilibrium point. The purpose of this paper is to give a procedure for determining the behavior of the solutions near Γ of a functional differential equation which is a nonautonomous periodic perturbation of the original one. The procedure uses exponential dichotomies and the Fredholm alternative. It is also shown that any smooth function p(t) defined on the reals which approaches zero monotonically as t → ± ∞ is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero. Examples illustrating the results are also given.     

Singular linear systems of differential equations with delays

The differential equation Ax + Bx = Cx(t-l) + f is studied where A, B, C are square matrices. All matrices are allowed to be singular. Examples are given to show that not all initial functions are consistent and that for consistent initial conditions, solutions may be continuous for only a finite time period. Solutions are given explicitly by a recursion formula and consistent initial conditions are explicitly characterized     

Exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps

In this paper, we consider a class of neutral stochastic partial differential equations with delays and Poisson jumps. Sufficient conditions for the existence and exponential stability in mean square as well as almost surely exponential stability of mild solutions are derived by means of the Banach fixed point principle. An example is provided to illustrate the effectiveness of the proposed result.     

Exponential stability of impulsive nonlinear stochastic differential equations with mixed delays

This paper is concerned with the problem of exponential stability for a class of impulsive nonlinear stochastic differential equations with mixed time delays. By applying the Lyapunov–Krasovskii functional, Dynkin formula and Razumikhin technique with a stochastic version as well as the linear matrix inequalities (LMIs) technique, some novel sufficient conditions are derived to ensure the exponential stability of the trivial solution in the mean square. The obtained results generalize and improve some recent results. In particular, our results are expressed in terms of LMIs, and thus they are more easily verified and applied in practice. Finally, a numerical example and its simulation are given to illustrate the theoretical results.     

Periodic solutions for nonlinear nth order differential equations with delays

By applying the continuation theorem of coincidence degree theory, we establish the existence of 2π-periodic solutions for a class of nonlinear nth order differential equations with delays.     

Exponential estimates for singularly perturbed linear fuctional differential equations

Exponential estimates on the fundamental matrix, uniform on the perturbation parameter, are obtained for singularly perturbed systems of linear retarded functional differential equations, under the assumption that the eigenvalues of a certain coefficient matrix in the system have negative real parts. The exponential rates in the estimates are computable from upper bounds on the real parts of the characteristic values of the system or of associated simpler equations. Differences between differential-difference equations and equations with distributed delays are emphasized.     

Explicit exponential stability conditions for linear differential equations with several delays

New explicit conditions of exponential stability are obtained for the nonautonomous linear equation

     

Periodic solutions of some nonlinear autonomous functional differential equations

Summary We develop here some new fixed point theorems and apply them to the question of existence of nontrivial periodic solutions of nonlinear, autonomous functional differential equations. We prove that the standard results of G. S. Jones and R. B. Grafton can be obtained by our methods, and we prove periodicity results for some equations, for instance a neutral functional differential equation, which appear inaccessible by previous techniques. Partially supported by NSFGP 20228 and a Rutgers Research Council Faculty Fellowship. Entrata in Redazione il 10 gennaio 1973.     

Riccati differential inequalities and dichotomies for linear systems

A general notion of dichotomy for linear differential systems is investigated. It is well known that a system x′=A(t)x in which the matrix A(t) is bounded and diagonally dominant by rows or columns has an invariant splitting of its solution space into two subspaces each uniformly asymptotically stable, one for increasing time and the other for decreasing time. Similar results are obtained here where the concept of diagonal dominance is weakened using Riccati inequalities.     

Periodic boundary-value problem for third-order linear functional differential equations

For a linear functional differential equation of the third order {fx481-01}, we establish the theorems on existence and uniqueness of a solution satisfying the conditions {fx481-02}. Here, ℓ is a linear continuous operator transforming the space C([0, ω]; R) into the space L([0, ω]; R) and q ∈ L([0, ω]; R). The problem of nonnegativity of the solution of the analyzed boundary-value problem is also studied. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 413–425, March, 2008.     

Exponential stability of mild solutions of stochastic partial differential equations with delays

A semiiinear stochastic partial differential equation with variable delays is considered. Sufficient conditions for the exponential stability in the p-th mean of mild solutions are obtained. Also, pathwise exponential stability is proved. Since the technique ofLyapunov functions is not suitable for delayed equations, the results have been proved by using the properties of the stochastic convolution. As the sufficient conditions obtained are also valid for the case without delays, one can ensure exponential stability of mild solution in some cases where the sufficient conditions in Ichikawa [11] do not give any answer. The results are illustrated with some examples     

On exponential stability of linear differential equations with several delays

New explicit conditions of exponential stability are obtained for the nonautonomous linear equation