On the generalized pantograph functional-differential equation. (English) Zbl 0767.34054

Summary: The generalized pantograph equation \(y'(t)=Ay(t)+By(qt)+Cy'(qt)\), \(y(0)=y_ 0\), where \(q\in (0,1)\), has numerous applications, as well as being a useful paradigm for more general functional-differential equations with monotone delay. Although many special cases have been already investigated extensively, a general theory for this equation is lacking — its development and exposition is the purpose of the present paper. After deducing conditions on \(A,B,C\in\mathbb{C}^{d\times d}\) that are equivalent to well-posedness, we investigate the expansion of \(y\) in Dirichlet series. This provides a very fruitful form for the investigation of asymptotic behaviour, and we duly derive conditions for \(\lim_{t\to\infty}y(t)=0\). The behaviour on the stability boundary possesses no comprehensive explanation, but we are able to prove that, along an important portion of that boundary, \(y\) is almost periodic and, provided that \(q\) is rational, it is almost rotationally symmetric. The paper also addresses itself to a detailed analysis of the scalar equation \(y'(t)=by(qt)\), \(y(0)=1\), to high-order pantograph equations, to a phenomenon, similar to resonance, that occurs for specific configurations of eigenvalues of \(A\), and to the equation \(Y'(t)=AY(t)+Y(qt)B\), \(Y(0)=Y_ 0\).


34K20 Stability theory of functional-differential equations
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
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