Stability of solutions to differential equations with several variable delays. II

We consider a class of scalar linear differential equations with several variable delays and constant coefficients. We treat coefficients and maximum admissible values of delays as parameters that define a family of equations from the class under consideration. We study domains in the parameter space, where fundamental solutions of all equations of the family are uniformly or exponentially stable and have a fixed sign. We establish explicit necessary and sufficient conditions for the stability and sign-definiteness of the equations family.     

Asymptotics of solutions of difference equations with delays

We consider a linear scalar difference equation with several variable delays and constant coefficients. The coefficients and maximum admissible values of delays are supposed to be the set of parameters that define a family of equations of the investigated class. We obtain effective necessary and sufficient conditions of the uniform and exponential stability of solutions to all equations of the family, as well as the conditions of the sign-definiteness and monotonicity of stable solutions.     

Stability of solutions to differential equations with several variable delays. III

We consider a class of scalar linear differential equations with several variable delays and constant coefficients. We treat the coefficients and maximum admissible values of delays as parameters that define a family of equations of the considered class. Using the necessary and sufficient stability conditions established in preceding papers, we obtain an analytic form and a geometric interpretation of boundaries of stability domains for families of equations with a small number of independent parameters.     

On exponential stability of second order delay differential equations

We propose a new method for studying stability of second order delay differential equations. Results we obtained are of the form: the exponential stability of ordinary differential equation implies the exponential stability of the corresponding delay differential equation if the delays are small enough. We estimate this smallness through the coefficients of this delay equation. Examples demonstrate that our tests of the exponential stability are essentially better than the known ones. This method works not only for autonomous equations but also for equations with variable coefficients and delays.     

Global exponential convergence for a class of neutral functional differential equations with proportional delays

This paper is concerned with a class of non‐autonomous neutral functional differential equations with multi‐proportional delays. It is shown that all solutions of the addressed system are globally exponentially convergent by employing the differential inequality technique and a novel argument. The obtained results improve and supplement existing ones. We also use numerical simulations to demonstrate our theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Oscillations and limit cycles in Lotka-Volterra systems with delays

In this paper, competitive Lotka-Volterra systems are studied that have distributed delays and constant coefficients on interaction terms and have time dependent growth rate vectors with an asymptotically constant average. Algebraic conditions are found to rule out non-vanishing oscillations for such systems and heteroclinic limit cycles for autonomous systems. As a supplement to these results, simple sufficient conditions are provided for certain components of all solutions to vanish and a criterion is given for partial permanence. An outstanding feature of all these results is that the conditions are irrelevant of the size and distribution of the delays.     

Asymptotic stability of exact and discrete solutions for neutral multidelay-integro-differential equations

This paper concerns the long-time behavior of the exact and discrete solutions to a class of nonlinear neutral integro-differential equations with multiple delays. Using a generalized Halanay inequality, we give two sufficient conditions for the asymptotic stability of the exact solution to this class of equations. Runge–Kutta methods with compound quadrature rule are considered to discretize this class of equations with commensurate delays. Nonlinear stability conditions for the presented methods are derived. It is found that, under suitable conditions, this class of numerical methods retain the asymptotic stability of the underlying system. Some numerical examples that illustrate the theoretical results are given.     

Estimates of solutions to the linear differential equations of neutral type with several delays of the argument

Under study is a systemof linear differential equations of neutral type with several delays of the argument. We obtain the conditions on the matrix coefficients of the system under which all solutions decrease with an exponential rate at infinity. Using some functionals of Lyapunov-Krasovskii type, the uniform estimates of solutions are established.     

Sharp conditions for global stability of Lotka-Volterra systems with distributed delays

We give a criterion for the global attractivity of a positive equilibrium of n-dimensional non-autonomous Lotka-Volterra systems with distributed delays. For a class of autonomous Lotka-Volterra systems, we show that such a criterion is sharp, in the sense that it provides necessary and sufficient conditions for the global asymptotic stability independently of the choice of the delay functions. The global attractivity of positive equilibria is established by imposing a diagonal dominance of the instantaneous negative feedback terms, and relies on auxiliary results showing the boundedness of all positive solutions. The paper improves and generalizes known results in the literature, namely by considering systems with distributed delays rather than discrete delays.     

Exponential stability of energy solutions to stochastic partial differential equations with variable delays and jumps

In this work, we investigate stochastic partial differential equations with variable delays and jumps. We derive by estimating the coefficients functions in the stochastic energy equality some sufficient conditions for exponential stability and almost sure exponential stability of energy solutions, and generalize the results obtained by Taniguchi [T. Taniguchi, The exponential stability for stochastic delay partial differential equations, J. Math. Anal. Appl. 331 (2007) 191-205] and Wan and Duan [L. Wan, J. Duan, Exponential stability of non-autonomous stochastic partial differential equations with finite memory, Statist. Probab. Lett. 78 (5) (2008) 490-498] to cover a class of more general stochastic partial differential equations with jumps. Finally, an illustrative example is established to demonstrate our established theory.     

Solutions of singular elliptic equations via the oblique derivative problem

We study a class of second elliptic equations whose highest order coefficients vanish everywhere on the boundary. Under suitable conditions on the lower order coefficients, Langlais proved in 1985 that such equations have unique smooth solutions up to the boundary provided the data are smooth enough. Our goal here is to prove some Schauder estimates for these equations and to obtain results even in Lipschitz domains. In addition, we show that bounded solutions of such problems are as smooth as the data allow. A key step is to observe that smooth solutions must satisfy an oblique derivative boundary condition.     

Existence of bounded and unbounded non-oscillatory solutions for partial difference equations

In this paper, the existence of bounded and unbounded non-oscillatory solutions for high-order neutral partial difference equations with variable coefficients and delays are studied. Some new sufficient conditions for bounded and unbounded non-oscillatory solutions are given.     

Globally exponential stability and periodic solutions of CNNS with variable coefficients and variable delays

In this Letter, by using the inequality method and the Lyapunov functional method, we analyze the globally exponential stability and the existence of periodic solutions of a class of cellular neutral networks with delays and variable coefficients. Some simple and new sufficient conditions ensuring the existence and uniqueness of globally exponential stability of periodic solutions for cellular neutral networks with variable coefficients and delays are obtained. In addition, one example is also worked out to illustrate our theory.     

Exponential stability and estimates for monotone and differential-difference systems

We present necessary and sufficient conditions for the exponential stability in the nonnegative cone and refine exponential estimates for solutions of systems of autonomous difference equations with monotone nondecreasing right-hand sides, including discontinuous ones, as well as for solutions of some class of systems of differential-difference equations with monotonicity. Unlike well-known criteria, the new ones are free of some additional assumptions on the right-hand sides of the considered models other than the original monotonicity conditions. We show that, in the nonsmooth and discontinuous cases, the traditional exponential stability conditions based on ??linearization?? can lead to negative or very coarse results.     

Stochastic Retarded Evolution Equations: Green Operators,Convolutions, and Solutions

Abstract

In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established.     

On the system governed by parabolic partial delay-differential equations with first boundary conditions

Summary In this paper, we consider a system governed by second order parabolic partial delay-differential equations with first boundary conditions. All the coefficients of the system are assumed bounded measurable and contain delays in their arguments. The second order parabolic partial delay-differential equation is in ? divergence form ?. In Theorem4.1, we present results on the existence and uniqueness of weak solution in the sense of Aronson for this class of systems. In Theorem4.2, we prove that whenever the coefficients and forcing terms converge in the almost everywhere topology the corresponding solutions converge weakly in an appropriate Sobolev space. Entrata in Redazione il 9 novembre 1977.     

Uniformly attracting solutions of nonautonomous differential equations

Understanding the structure of attractors is fundamental in nonautonomous stability and bifurcation theory. By means of clarifying theorems and carefully designed examples we highlight the potential complexity of attractors for nonautonomous differential equations that are as close to autonomous equations as possible. We introduce and study bounded uniform attractors and repellors for nonautonomous scalar differential equations, in particular for asymptotically autonomous, polynomial, and periodic equations. Our results suggest that uniformly attracting or repelling solutions are the true analogues of attracting or repelling fixed points of autonomous systems. We provide sharp conditions for the autonomous structure to break up and give way to a bewildering diversity of nonautonomous bifurcations.     

Positivity and stabilisation for nonlinear stochastic delay differential equations

We use state dependent Gaussian perturbations to stabilise the solutions of differential equations with coefficients that take, as arguments, averaged sets of information from the history of the solution, as well as isolated past and present states. The properties that guarantee stability also guarantee positivity of solutions as long as the initial value is nonzero.

We do not require that any component of the coefficients of the equations satisfy Lipschitz conditions. Instead, we require that the functional part of each coefficient which feeds back the present state of the process admit to bounds imposed by a member of a particular class of concave functions. Lipschitz conditions are included as a special case of these bounds.

We generalise these results to the finite dimensional case, also constructing perturbations that can destabilise the otherwise stable solutions of a deterministic system of equations.     

On Global Existence of Solutions of Initial Boundary Value Problem for a System of Semilinear Parabolic Equations with Nonlinear Nonlocal Neumann Boundary Conditions

We establish conditions for the existence and nonexistence of global solutions of initial boundary value problem for a system of semilinear parabolic equations with nonlinear nonlocal Neumann boundary conditions. We show that these conditions are determined by the behavior of the problem coefficients as t→∞.     

Asymptotic behavior of a linear difference equation with continuous time

We consider a class of linear homogeneous difference equations with constant coefficients and commensurate delays. We prove an asymptotic formula for the solutions as t → ∞ under the assumption of the existence of a simple “dominant” real characteristic value. Research supported by FONDECYT Grant, Chile, No. 1.070.980.