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Abstract  

We establish the existence of mild solutions and periodic mild solutions for a class of abstract first-order non-autonomous neutral functional differential equations with infinite delay in a Banach space.

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Abstract  

A mathematical model consisting of a system of two ordinary differential equations is formulated to represent the interrelationship between healthy and radiated cells at a given cite. Three different modes of radiation are considered: constant, decaying, and periodic radiation. For the constant case, precise criteria for persistence and extinction are obtained. In the decaying case, it is shown that the radiated cells always become extinct. Finally in the periodic case, criteria are obtained for a perturbed positive periodic solution.

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Abstract  

We consider the delay differential equation
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\dot x(t) = - \mu x(t) + f(x(t - \tau ))$$ \end{document}
, where µ, τ are positive parameters and f is a strictly monotone, nonlinear C 1-function satisfying f(0) = 0 and some convexity properties. It is well known that for prescribed oscillation frequencies (characterized by the values of a discrete Lyapunov functional) there exists τ* > 0 such that for every τ > τ* there is a unique periodic solution. The period function is the minimal period of the unique periodic solution as a function of τ > τ*. First we show that it is a monotone nondecreasing Lipschitz continuous function of τ with Lipschitz constant 2. As an application of our theorem we give a new proof of some recent results of Yi, Chen and Wu [14] about uniqueness and existence of periodic solutions of a system of delay differential equations.
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principles and duality for periodic solutions of Lagrange Equations with superlinear nonlinearities, J. Math. Analysis App ., 264 (2001), 168–181. Rogowski A. On the new variational

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Summary In a recent paper we derived a stability criterion for a Volterra equation which is based on the contraction mapping principle. It turns out that this criterion has significantly wider application. In particular, when we use Becker’s form of the resolvent it readily establishes critical resolvent properties which have been very illusive when investigated by other techniques. First, it enables us to show that the resolvent is L 1. Next, it allows us to show that the resolvent satisfies a uniform bound and that it tends to zero. These properties are then used to prove boundedness of solutions of a nonlinear problem, establish the existence of periodic solutions of a linear problem, and to investigate asymptotic stability properties. We also apply the results to a Liénard equation with distributed delay and possibly negative damping so that relaxation oscillations may occur.

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linéaires , Les Presses de l’Université de Montréal, 1987. 168 pp. MR 89d :58024 Nowakowski, A. and Rogowski, A. , On the new variational principles and duality for periodic solutions of

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Cai, X. C., Yu, J. S. and Guo, Z. M. , Existence of periodic solutions for fourthorder difference equations, Comput. Math. Appl. , 50 (1–2) (2005), 49–55. Z. M G. Existence of

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