The generalized pantograph equation y′(t) = Ay(t) + By(qt) + Cy′(qt), y(0) = y0, 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 ∈ ℂd×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 limt⋅→∞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) = Y0.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0956792500000966

On the generalized pantograph functional-differential equation

/pdf/the-functional-differential-equation-y-left-x-right-ay-left-39nz20p14f.pdf

The functional-differential equation $y'\left( x \right) = ay\left( {\lambda x} \right) + by\left( x \right)$

On Neutral Functional)Differential Equations with Proportional Delays

https://iopscience.iop.org/article/10.1070/RM1977v032n02ABEH001623/pdf

On certain problems in the theory of differential equations with deviating argument

Publisher Summary This chapter focuses on the asymptotic behavior of solutions of the functional differential equation y′(x) = ay (λx) + by (x); 0 ≤ x ≤ x ∞. In this equation, λ, a, b are constants, with 0 1 each solution y(x) is defined on an interval [;0, R), R ≤ ∞. However, the asymptotic behavior of solutions is defined on x → ∞. In this case y(x) is C∞. Conversely, any C∞ solution on [;0, R) can be continued to [;0,∞). The theorems generalize almost all results of the equation to the general case in which b is complex. The only major exception is the case λ 1, Re b = 0 is incomplete compared with other cases.

Paul O. Frederickson

Papers

Dirichlet series solutions for certain functional differential equations

Global solutions to certain nonlinear functional differential equations