## 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 $$Y'(t)=AY(t)+Y(qt)B$$, $$Y(0)=Y_ 0$$.

### MSC:

 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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### References:

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