On the generalized pantograph functional-differential equation

The generalized pantograph equation y ′( t ) = Ay ( t ) + By ( qt ) + Cy ′( qt ), y (0) = y 0 , where q ∈ (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 ∈ ℂ d × 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 ⋅→∞ 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 .