Estimates of solutions to a class of systems of nonlinear delay differential equations

Under study are the systems of nonlinear delay differential equations with periodic coefficients of the linear terms. Some sufficient conditions for the asymptotic stability of the zero solution are established. We obtain the estimates that characterize the decay rate of solutions at infinity and describe the attraction sets of the zero solution.     

Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations

Summary. We consider systems of delay differential equations (DDEs) of the form with the initial condition . Recently, Torelli [10] introduced a concept of stability for numerical methods applied to dissipative nonlinear systems of DDEs (in some inner product norm), namely RN-stability, which is the straighforward generalization of the wellknown concept of BN-stability of numerical methods with respect to dissipative systems of ODEs. Dissipativity means that the solutions and corresponding to different initial functions and , respectively, satisfy the inequality , and is guaranteed by suitable conditions on the Lipschitz constants of the right-hand side function . A numerical method is said to be RN-stable if it preserves this contractivity property. After showing that, under slightly more stringent hypotheses on the Lipschitz constants and on the delay function , the solutions and are such that , in this paper we prove that RN-stable continuous Runge-Kutta methods preserve also this asymptotic stability property. Received March 29, 1996 / Revised version received August 12, 1996     

Asymptotic stability of two-parameter systems of delay differential equations

We establish exact efficient asymptotic stability criteria for two-parameter systems of two autonomous delay differential equations.     

Asymptotic behavior of solutions of second-order nonlinear delay differential equations with impulses

This paper is concerned with second-order nonlinear delay differential equations with impulses of the form

     

Asymptotic behavior of solutions of forced nonlinear neutral delay differential equations with impulses

Sufficient conditions for asymptotic behavior of the solutions of nonlinear forced neutral delay differential equations with impulses are found. The results given in [2,4,6,7] are generalized and improved.     

Asymptotic solutions of nonlinear differential equations

An asymptotic expression for solutions of nonlinear differential equations is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 676–678, May, 1991.     

Asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations

This paper is concerned with the numerical solution to initial value problems of nonlinear delay differential equations of neutral type. We use A-stable linear multistep methods to compute the numerical solution. The asymptotic stability of the A-stable linear multistep methods when applied to the nonlinear delay differential equations of neutral type is investigated, and it is shown that the A-stable linear multistep methods with linear interpolation are GAS-stable. We validate our conclusions by numerical experiments.     

Asymptotic stability of solutions for some classes of impulsive differential equations with distributed delay

In this paper we show the asymptotic stability of the solutions of some differential equations with delay and subject to impulses. After proving the existence of mild solutions on the half-line, we give a Gronwall–Bellman-type theorem. These results are prodromes of the theorem on the asymptotic stability of the mild solutions to a semilinear differential equation with functional delay and impulses in Banach spaces and of its application to a parametric differential equation driving a population dynamics model.     

Asymptotic integration of a class of nonlinear differential equations

This work is concerned with the behavior of solutions of a class of second-order nonlinear differential equations locally near infinity. Using methods of the fixed point theory, the existence of solutions with different asymptotic representations at infinity is established. A novel technique unifies different approaches to asymptotic integration and addresses a new type of asymptotic behavior.     

Asymptotic behavior of solutions of nonlinear impulsive delay differential equations with positive and negative coefficients

This paper is concerned with the nonlinear impulsive delay differential equations with positive and negative coefficients

(*)

Sufficient conditions are obtained for every solution of equation (*) tending to a constant as t→∞.     

Existence and stability of oscillating solutions for a class of delay differential equations

In this paper we study a Hopf bifurcation for a general class of simple delay differential equations with a term that can describe a treatment. Conditions guaranteeing appearance of the Hopf bifurcation as well as stability of arising periodic orbits are proved. Applications of proved theorems to a general version of the logistic equation are presented.     

Convergence of bounded solutions for a class of systems of delay differential equations

In this paper, the convergence of a class of systems of delay differential equations is considered. We show that every bounded solution of such systems tends to an equilibrium under certain hypotheses. Our results extend some corresponding ones already known.     

广义中立型系统的渐近稳定性

讨论了广义中立型延迟系统理论的渐近稳定性,给出了广义中立型系统渐近稳定的一些充分条件。     

Asymptotic behavior of solutions for nonlinear delay differential equation with impulses

This paper studies the asymptotic behavior of solutions of the nonlinear delay differential equation with impulses