Asymptotic behaviour of a two‐dimensional differential system with non‐constant delay

The asymptotic behaviour and stability properties are studied for a real two‐dimensional system x^′(t) = A(t)x (t) + B(t)x (θ (t)) + h (t, x (t), x (θ (t))), with a nonconstant delay t ‐ θ (t) ≥ 0. It is supposed that A,B and h are matrix functions and a vector function, respectively. The method of investigation is based on the transformation of the considered real system to one equation with complex‐valued coefficients. Stability and asymptotic properties of this equation are studied by means of a suitable Lyapunov‐Krasovskii functional. The results generalize the great part of the results of J. Kalas and L. Baráková [J. Math. Anal. Appl. 269 , No. 1, 278–300 (2002)] for two‐dimensional systems with a constant delay (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and asymptotic behaviour of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms

In this paper we consider the existence of a local solution in time to a weakly damped wave equation of Kirchhoff type with the damping term and the source term:

     

Stability switches in a system of linear differential equations with diagonal delay

This paper deals with the stability problem of a delay differential system of the form x^′(t)=-ax(t-τ)-by(t), y^′(t)=-cx(t)-ay(t-τ), where a, b, and c are real numbers and τ is a positive number. We establish some necessary and sufficient conditions for the zero solution of the system to be asymptotically stable. In particular, as τ increases monotonously from 0, the zero solution of the system switches finite times from stability to instability to stability if ; and from instability to stability to instability if . As an application, we investigate the local asymptotic stability of a positive equilibrium of delayed Lotka-Volterra systems.     

Stability and boundedness of solutions of neutral functional differential equations with finite delay

Sufficient and necessary criteria are established for the uniform stability and uniformly asymptotic stability of solutions of neutral functional differential equations (NFDEs) with finite delay by using the Liapunov functional approach. We also prove that the uniformly asymptotic stability of solutions implies the existence of bounded solution.     

Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions

In this paper a nonlocal phase-field model for non-isothermal phase transitions with a non-conserved order parameter is studied. The paper extends recent investigations to the non-isothermal situation, complementing results obtained by H. Gajewski for the non-isothermal case for conserved order parameters in phase separation phenomena. The resulting field equations studied in this paper form a system of integro-partial differential equations which are highly nonlinearly coupled. For this system, results concerning global existence, uniqueness and large-time asymptotic behaviour are derived. The main results are proved using techniques that have been recently developed by P. Krej?í and the authors for phase-field systems involving hysteresis operators.     

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

     

Stability and bifurcation in a non-kolmogorov type prey-predator system with time delay

The purpose of this paper is to study a non-Kolmogrov type prey-predator system. First, we investigate the linear stability of the model by analyzing the associated characteristic equation of the linearized system. Second, we show that the system exhibits the Hopf bifurcation. The stability and direction of the Hopf bifurcation are determined by applying the norm form theory and center manifold theorem. Finally, numerical simulations are performed to illustrate the obtained results.     

Non-autonomous systems: asymptotic behaviour and weak invariance principles

Results pertaining to asymptotic behaviour of solutions of non-autonomous ordinary differential equations with locally integrably bounded right-hand sides are presented. Ramifications for weakly asymptotically autonomous systems and adaptively controlled systems are highlighted.     

Precise asymptotic behavior of solutions of the sublinear Emden-Fowler differential equation

The aim of this paper is to show that if the sublinear Emden-Fowler differential equation

(A)     

Nonoscillation and asymptotic behaviour for third order nonlinear differential equations

In this paper we consider the equation y +q(t)y + p(t)h(y)=0, where p, q are real valued continuous functions on [0, ) such that q(t) 0, p(t) 0 and h(y) is continuous in (–, ) such that h(y)y > 0 for y 0. We obtain sufficient conditions for solutions of the considered equation to be nonoscillatory. Furthermore, the asymptotic behaviour of these nonoscillatory solutions is studied.     

On the Hénon equation: asymptotic profile of ground states, I

This paper is concerned with the qualitative property of the ground state solutions for the Hénon equation. By studying a limiting equation on the upper half space , we investigate the asymptotic energy and the asymptotic profile of the ground states for the Hénon equation. The limiting problem is related to a weighted Sobolev type inequality which we establish in this paper.     

Boundedness of solutions for second order differential equations with asymmetric nonlinearity

In this paper we are concerned with the boundedness of all solutions of the second order differential equations

     

Oscillatory and asymptotic behaviour of fourth order nonlinear difference equations

This paper considers a class of fourth order nonlinear difference equations Δ²(r _n Δ²(y _n ) + Δ²(r _n ,f(n,n _n )=0,n ∈N(n ₀) wheref(n, y) may be classified as superlinear, sublinear, strongly superlinear and strongly sublinear. In superlinear and sublinear cases, necessary and sufficient conditions are obtained for the difference equation to admit the existence of nonoscillatory solutions with special asymptotic properties. In strongly superlinear and strongly sublinear cases, sufficient conditions are given for all solutions to be oscillatory. Partially Supported by the National Science Foundation of China     

On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations

In this paper second order quasilinear ordinary differential equations are considered, and a necessary and sufficient condition for the existence of a slowly growing positive solution is established. Moreover, the precise asymptotic forms as t→∞ of slowly growing positive solutions and slowly decaying positive solutions are obtained.     

Free energies and asymptotic behaviour for incompressible viscoelastic fluids

The existence, uniqueness and asymptotic stability for an incompressible, linear viscoelastic fluid is studied using a new free energy, the representation of which is based on the concept of a minimal state. A restriction imposed by thermodynamics is also used. Furthermore, an expression for the minimum free energy in the time domain is derived, which shows explicitly its dependence on the minimal state.     

Two classes of piecewise-linear difference equations with eventual periodicity six

In this paper, we will study two classes of difference equations which are piecewise-linear and of similar forms. We will show that all non-trivial solutions of both equations are eventually periodic with prime period six.     

On existence and asymptotic behaviour of solutions of a functional integral equation

The paper contains a result on the existence and asymptotic behaviour of solutions of a functional integral equation. That result is proved under rather general hypotheses. The main tools used in our considerations are the concept of a measure of noncompactness and the classical Schauder fixed point principle. The investigations of the paper are placed in the space of continuous and tempered functions on the real half-line. We prove an existence result which generalizes several ones concerning functional integral equations and obtained earlier by other authors. The applicability of our result is illustrated by some examples.     

Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation

In this paper we study the existence, the uniqueness, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation x_n+1=∑_i=0^kA_i/x_n−i^p_i, where k{1,2,…,}, A_i, i{0,1,…,k}, are positive fuzzy numbers, p_i, i{0,1,…,k}, are positive constants and x_i, i{−k,−k+1,…,0}, are positive fuzzy numbers.     

Global existence and asymptotic behaviour for systems of nonlinear hyperbolic equations

We consider the initial-boundary value problem for a class of nonlinear hyperbolic equations system in a bounded domain. Using the potential well theory, the existence of global solutions is investigated. We also established the asymptotic behaviour of global solutions as t?→?+?∞ by applying the multiplier method.