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Stability of a critical nonlinear neutral delay differential equation. (English) Zbl 1315.34083

This work studies existence, uniqueness and stability of periodic solutions of the neutral delay-differential equation \[ y'(t) + c y'(t-1) + f(y(t)) + g(y(t-1)) = s(t). \] This problem is classically addressed in the literature for the linear version of this equation, under the assumption that \(|c| < 1\), as the exponential stability may then be derived from the analysis of the characteristic function, see e.g. [J. K. Hale and S. M. Verduyn Lunel, Introduction to functional differential equations. New York, NY: Springer-Verlag (1993; Zbl 0787.34002)]. In this article, the authors consider instead the more difficult case \(|c|=1\). Energy estimates are derived and used as the main tool to study linear and nonlinear versions of this equation.
The analysis of the stability of the solutions \(y(t)\) is first performed when the equation has a constant right-hand side, under several sets of assumptions (\(s=0\), \(g=0\), etc.). When the right-hand side \(s(t)\) is periodic and a periodic solution \(r(t)\) does exist, the authors derive a result of convergence of the solution \(y(t)\) to \(r(t)\), under a restrictive condition that depends in a complex way on the ranges of \(y\) and \(r\) and a measure of the nonlinearity of \(f\).
In the linear homogeneous case – \(f(y)=ay\), \(g(y) = by\) and \(s=0\) – a detailed stability diagram of the system in the \((a,b)\)-plane is established; the occurrence of small divisors is characterized geometrically in the non-homogeneous case. For the nonlinear equation, a variant of this geometric condition is used to derive the existence and uniqueness of periodic solutions for small initial conditions. Periodic functions \(s(t)\) with rational periods are a special case where this condition holds; the more general case is related to a Diophantine condition on the period.

MSC:

34K40 Neutral functional-differential equations
34K20 Stability theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations

Citations:

Zbl 0787.34002
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References:

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