Global attractivity in a perturbed linear delay differential equation

In the main result of this paper, some sharp conditions are obtained for global attractivity in a scalar perturbed linear delay differential equation. The proof of the main theorem is based on a new estimate for the infinite integral of the absolute value of the fundamental solution of a linear delay differential equation. We also derive sufficient conditions for asymptotic stability of a system of linear delay differential equations.     

An existence theorem for a type of functional differential equation with infinite delay

Summary We prove an existence theorem for a functional differential equation with infinite delay using the Schauder fixpoint theorem. We extend a result in [19] applying the fixed point procedure in an appropriate function space.     

GLOBAL EXISTENCE FOR A SEMILINEAR DIFFERENTIAL SYSTEM

In this paper, a semilinear elliptic-parabolic PDE system which arises in a two dimensional groundwater flow problem is studied. Existence and uniqueness results are established via the L^p - L^q a priori estimates and the inverse function theorem.     

Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay

In this paper, we consider initial-boundary value problem of viscoelastic wave equation with a delay term in the interior feedback. Namely, we study the following equation $$u_{tt}(x, t) - \Delta {u}(x, t) + \int_{0}^{t} g(t - s)\,\Delta {u}(x, s){\rm d}s + \mu_{1} u_{t}(x, t) + \mu_{2} u_{t}(x, t -\tau) = 0$$ together with initial-boundary conditions of Dirichlet type in Ω × (0, + ∞) and prove that for arbitrary real numbers  μ ₁ and μ ₂, the above-mentioned problem has a unique global solution under suitable assumptions on the kernel g. This improve the results of the previous literature such as Nicaise and Pignotti (SIAM J. Control Optim 45:1561–1585, 2006) and Kirane and Said-Houari (Z. Angew. Math. Phys. 62:1065–1082, 2011) by removing the restriction imposed on μ ₁ and μ ₂. Furthermore, we also get an exponential decay results for the energy of the concerned problem in the case μ ₁ = 0 which solves an open problem proposed by Kirane and Said-Houari (Z. Angew. Math. Phys. 62:1065–1082, 2011).     

Ulam stability for a delay differential equation

We study the Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability for a delay differential equation. Some examples are given.     

Global existence for a new periodic integrable equation

We establish the local well-posedness for a new periodic integrable equation. We show that the equation has classical solutions that blowup in finite time as well as classical solutions which exist globally in time.     

Global attractivity and positive almost periodic solution for delay logistic differential equation

In this paper, we establish some sufficient conditions for the global attractivity of positive solutions and the unique existence of positive almost periodic solutions for delay logistic differential equations of the form

x^′(t)=r(t)x(t)(a(t)−b(t)x(t)−L(_xt)).$x^{\prime }\left(t\right)=r\left(t\right)x\left(t\right)(a\left(t\right)-b\left(t\right)x\left(t\right)-L\left(x_t\right))\text{.}$

非线性时滞差分方程的全局吸引性

得到了具有多重时滞非线性差分方程x(n 1)-x(n) ∑i=1^k pi(n)fi(x(gi(n)))=0,n=0,1,2,…的每个解趋于零的充分条件。     

Global stability and the Hopf bifurcation for some class of delay differential equation

In this paper, we present an analysis for the class of delay differential equations with one discrete delay and the right‐hand side depending only on the past. We extend the results from paper by U. Fory? (Appl. Math. Lett. 2004; 17 (5):581–584), where the right‐hand side is a unimodal function. In the performed analysis, we state more general conditions for global stability of the positive steady state and propose some conditions for the stable Hopf bifurcation occurring when this steady state looses stability. We illustrate the analysis by biological examples coming from the population dynamics. Copyright © 2007 John Wiley & Sons, Ltd.     

非线性时滞差分议程的全局渐近稳定性

In this paper,a sufficient condition for the global asymptotic stability of the solutions of the following nonlinear delay difference equation is obtained,xn 1=xn xn-1xn-2 a/xmxm-1 xn-2 a,n=0,1…,where a∈(0,∞) and the initial values x-2,x-1,x0∈(0,∞).As a special case,a conjecture by Ladas is confirmed.     

Oscillation theorems for a second order delay differential equation

A generalized version is proved of the following inequality, arising in a study of invertible measure preserving transformations: $\text{(∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}{}^{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{(∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}{}^{\mathrm{m}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>m</mtext>}}\text{? (∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}^{\mathrm{mn}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>mn</mtext>}}\text{(∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}\text{),}\text{where}\text{x}{}_{\mathrm{i}}\text{? 0, i = 1, 2,…, N,}\text{and}\text{(m ? 1)(n ? 1) > 0}$.     

Global existence results for impulsive differential equations

This paper studies the global existence of solutions of the impulsive differential equation

     

Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain

In this paper, we study the global dynamics of a class of differential equations with temporal delay and spatial non-locality in an unbounded domain. Adopting the compact open topology, we describe the delicate asymptotic properties of the nonlocal delayed effect and establish some a priori estimate for nontrivial solutions which enables us to show the permanence of the equation. Combining these results with a dynamical systems approach, we determine the global dynamics of the equation under appropriate conditions. Applying the main results to the model with Ricker?s birth function and Mackey-Glass?s hematopoiesis function, we obtain threshold results for the global dynamics of these two models. We explain why our results on the global attractivity of the positive equilibrium in C₊?{0} under the compact open topology becomes invalid in C₊?{0} with respect to the usual supremum norm, and we identify a subset of C₊?{0} in which the positive equilibrium remains attractive with respect to the supremum norm.     

Global stability of a partial differential equation with distributed delay due to cellular replication

In this paper, we investigate a nonlinear partial differential equation, arising from a model of cellular proliferation. This model describes the production of blood cells in the bone marrow. It is represented by a partial differential equation with a retardation of the maturation variable and a distributed temporal delay. Our aim is to prove that the behaviour of primitive cells influences the global behaviour of the population.     

Convergence in a neutral delay differential equation

By using some facts from limiting equations theory we prove that the solution x(.;?), with continuous initial condition ?, of the neutral functional differential equation [x(t)-cx(t-r)]' =-F(x(t))+F(x(t-r)), t>0, where c ε [0,1), r≧0 and F is (not necessarily strictly) increasing. satisfies lim x(t;?) = &;, where &; is the unique root of the algebraic equation [math001]