Search

We establish Pontryagin principles for a Mayer's optimal control problem governed by a functional differential equation. The control functions are piecewise continuous and the state functions are piecewise continuously differentiable. To do that, we follow the method created by Philippe Michel for systems governed by ordinary differential equations, and we use properties of the resolvent of a linear functional differential equation. [ABSTRACT FROM AUTHOR]

In this paper, the stability of some classes of linear functional differential equations was discussed by direct method, iteration method, fixed point method and open mapping theorem. It is shown that the Hyers–Ulam stability holds true for y ( n ) = g ( t ) y ( t − τ ) + h ( t ) . The stability of functional differential equations with multiple delays of first order and general delay differential equations also have been discussed. [ABSTRACT FROM AUTHOR]

Abstract: We prove existence and uniqueness of solutions for a class of infinitely delayed stochastic evolution equations with multiplicative noise term where A is the generator of an analytic semigroup on a UMD Banach space E and F and G are functions from the history of the system satisfying Lipschitz conditions. This paper is based on recent work of van Neerven et al., developing the theory of abstract stochastic evolution equations in UMD Banach spaces. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using Lyapunov method. Several stability theorems of impulsive stochastic functional differential equations with finite delay are derived. These new results are employed to impulsive stochastic equations with bounded time-varying delays and stochastically perturbed equations. Meanwhile, an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay. [Copyright &y& Elsevier]

Abstract: We consider retarded functional differential equations in the setting of Kurzweil–Henstock integrable functions and we state an averaging result for these equations. Our result generalizes previous ones. [Copyright &y& Elsevier]

Abstract: The stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors, but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay. The main aim of this paper is to close this gap. We establish criteria of pth moment -bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay, and we also illustrate the result with an example. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a class of smooth essentially strongly order-preserving semiflows and improve the limit set dichotomy for essentially strongly order-preserving semiflows. Generic convergence and stability properties of this class of smooth essentially strongly order-preserving semiflows are then developed. We also establish the generalized Krein–Rutman Theorem for a compact and eventually essentially strongly positive linear operator. By applying the main results of this paper to essentially cooperative and irreducible systems of delay differential equations, we obtain some results on generic convergence and stability, the linearized stability of an equilibrium and the existence of the most unstable manifold in these systems. The obtained results improve some corresponding ones already known. [Copyright &y& Elsevier]

Abstract: In this paper we study the asymptotic stability of the zero solution of even order linear delay differential equations of the form where and are certain constants and . Here is a constant delay. In proving our results we make use of Pontryagin''s theory for quasi-polynomials. [Copyright &y& Elsevier]

Abstract: In this paper, we study the behavior of solutions of second order delay differential equation where , , , are real numbers, τ is positive real number. A basic theorem on the behavior of solutions is established. As a consequence of this theorem, a stability criterion is obtained. [Copyright &y& Elsevier]

Abstract: A mathematical model for HIV/AIDS with explicit incubation period is presented as a system of discrete time delay differential equations and its important mathematical features are analysed. The disease-free and endemic equilibria are found and their local stability investigated. We use the Lyapunov functional approach to show the global stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The HIV/AIDS model is numerically analysed to asses the effects of incubation period on the dynamics of HIV/AIDS and the demographic impact of the epidemic using the demographic and epidemiological parameters for Zimbabwe. [Copyright &y& Elsevier]

Abstract: In this paper, we study almost periodic logistic delay differential equations. The existence and module of almost periodic solutions are investigated. In particular, we extend some results of Seifert in [G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Differential Equations 164 (2000) 451–458]. [Copyright &y& Elsevier]

Abstract: In a previous paper we gave sufficient conditions for a system of delay differential equations to present Bautin-type bifurcation. In the present work we present an example of delay equation that satisfies these conditions. [Copyright &y& Elsevier]

Abstract: This paper concerns the regularity of a functional differential equation in the form: , , where A is the generator of an analytic semigroup on a Banach space X, and are -bounded linear operator for . By spectral analysis, it is shown that the associated solution semigroup of this equation is eventually differentiable. [Copyright &y& Elsevier]

Abstract: In this paper, we employ a well-known fixed-point index theorem to study the existence and non-existence of positive periodic solutions for the periodic impulsive functional differential equations with two parameters. Several existence and non-existence results are established. [Copyright &y& Elsevier]

Abstract: We consider the boundary value problems: , , , , subject to some boundary conditions. By using a generalization of the Leggett–Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions to the above problems. [Copyright &y& Elsevier]

Abstract: The paper discusses the pth moment exponential stability for a general class of neutral stochastic functional differential equations of the Ito type. This investigation can be very complicated, even in many special cases, by using usual methods based on Lyapunov functionals. In this paper we present criteria which are relatively easy to verify the pth moment exponential stability of the solutions of such equations. [Copyright &y& Elsevier]

Abstract: This paper considers the stability problem for a class of impulsive systems of functional differential equations. By using Lyapunov functions and the Razumikhin technique, several stability criteria in terms of two measures are established. Some examples are worked out to illustrate the theorems. [Copyright &y& Elsevier]

Abstract: We consider general nonlinear dynamical systems in a Banach space with dependence on parameters in a second Banach space. An abstract theoretical framework for sensitivity equations is developed. An application to measure dependent delay differential systems arising in a class of HIV models is presented. [Copyright &y& Elsevier]

Abstract: A chemostat model with general nonmonotone response functions is considered. The nutrient conversion process involves time delay. We show that under certain conditions, when n species compete in the chemostat for a single resource that is allowed to be inhibitory at high concentrations, the competitive exclusion principle holds. In the case of insignificant death rates, the result concerning the attractivity of the single species survival equilibrium already appears in the literature several times (see [H.M. El-Owaidy, M. Ismail, Asymptotic behavior of the chemostat model with delayed response in growth, Chaos Solitons Fractals 13 (2002) 787–795; H.M. El-Owaidy, A.A. Moniem, Asymptotic behavior of a chemostat model with delayed response growth, Appl. Math. Comput. 147 (2004) 147–161; S. Yuan, M. Han, Z. Ma, Competition in the chemostat: convergence of a model with delayed response in growth, Chaos Solitons Fractals 17 (2003) 659–667]). However, the proofs are all incorrect. In this paper, we provide a correct proof that also applies in the case of differential death rates. In addition, we provide a local stability analysis that includes sufficient conditions for the bistability of the single species survival equilibrium and the washout equilibrium, thus showing the outcome can be initial condition dependent. Moreover, we show that when the species specific death rates are included, damped oscillations may occur even when there is no delay. Thus, the species specific death rates might also account for the damped oscillations in transient behavior observed in experiments. [Copyright &y& Elsevier]

Abstract: The purpose of this contribution is to give sufficient conditions for the existence of global solutions or left semi-global solutions for some classes of delayed functional differential equations. The principle of monotone sequences is used as a main tool of investigation. Inequalities for coordinates of global solutions are derived as a consequence of used method. Examples illustrate the results. [Copyright &y& Elsevier]

Abstract: First, we systematize earlier results on the global stability of the model of population growth. Second, we investigate the effect of delay on the asymptotic behavior when the nonlinearity f is a unimodal function. Our results can be applied to several population models [Elements of Mathematical Ecology, 2001 ; Appl. Anal. 43 (1992) 109–124; Math. Comput. Modelling, in press; Funkt. Biol. Med. 256 (1982) 156–164; Math. Comput. Modelling 35 (2002) 719–731; Mat. Stos. 6 (1976) 25–40] because the function f does not need to be monotone or differentiable. Specifically, our results generalize earlier result of [Delay Differential Equations with Applications in Population Dynamics, 1993], since our function f may not be differentiable. [Copyright &y& Elsevier]

Abstract: In this paper we stochastically perturb the delay Lotka–Volterra model into the stochastic delay differential equation (SDDE) The main aim is to reveal the effects of environmental noise on the delay Lotka–Volterra model. Our results can essentially be divided into two categories: [(i)] If the delay Lotka–Volterra model already has some nice properties, e.g., nonexplosion, persistence, and asymptotic stability, then the SDDE will preserve these nice properties provided the noise is sufficiently small. [(ii)] When the delay Lotka–Volterra model does not have some desired properties, e.g., nonexplosion and boundedness, the noise might make the SDDE achieve these desired properties. [Copyright &y& Elsevier]

In this paper, we study a class of partial neutral functional differential equations with infinite delay. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille–Yosida theorem. We give some sufficient conditions ensuring the existence, uniqueness and regularity of solutions. A principle of linearized stability is also established in the autonomous case. To illustrate our abstract results, we conclude this work by an example. [Copyright &y& Elsevier]

Some new results are given concerning the behavior of the oscillatory solutions of first or second order delay differential equations. These results establish that all oscillatory solutions x of a first or second order delay differential equation satisfy x(t)=O(v(t)) as t→∞, where v is a nonoscillatory solution of a corresponding first or second order linear delay differential equation. Some applications of the results obtained are also presented. [Copyright &y& Elsevier]

One-dimensional perturbed neutral delay differential equations of the form (x(t)−P(t,x(t−τ)))′=f(t,xt)+g(t,xt) are considered assuming that f satisfies −v(t)M(φ)⩽f(t,φ)⩽v(t)M(−φ), where M(φ)=max{0,maxs∈[−r,0]φ(s)}. A typical result is the following: if &z.dfnc;g(t,φ)&z.dfnc;⩽w(t)&z.dfnc;φ&z.dfnc; and ∫tt+1w(s) ds→0, then the zero solution is uniformly asymptotically stable providing that the zero solution of the corresponding equation without perturbation (x(t)−P(t,x(t−τ)))′=f(t,xt) is uniformly asymptotically stable. Some known results associated with this equation are extended and improved. [Copyright &y& Elsevier]

In this paper, by means of the method of upper and lower solutions and monotone iterative technique, the existence of maximal and minimal solutions of the boundary value problems for first order impulsive delay differential equations is established. [Copyright &y& Elsevier]

In this paper, we obtain some new oscillation criteria for the difference equation with several delays xn+1−xn+∑lower limit i=1, upper limit m pi(n)xn−ki=0, n=0,1,2,…, which improve the existing results in the literature. The mistakes in Comput. Math. Appl. 44 (2002) 817–821 are pointed out. [Copyright &y& Elsevier]

In this paper, we discuss the quasi-periodic logistic delay differential equations. As a corollary, we give a more sharp result than that in [G. Seifert, J. Differential Equations 164 (2000) 451–458] for the periodic logistic delay differential equations. [Copyright &y& Elsevier]