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

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovasz local lemma in probabilistic combinatorics. We show that the conclusion of the Lovasz local lemma holds for dependency graph G and probabilities {p
 x
 } if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p
 x
 }. Furthermore, we show that the usual proof of the Lovasz local lemma – which provides a sufficient condition for this to occur – corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer(98) and explicitly by Dobrushin.(37,38) We also present some refinements and extensions of both arguments, including a generalization of the Lovasz local lemma that allows for ‘‘soft’’ dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

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The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lovasz Local Lemma

In this paper we study discretizations of the general pantograph equation y'(t) = ay(t) + by(6(t)) + cy'(O(t)), t > 0, y(O) = Yo, where a, b, c, and yo are complex numbers and where 0 and 0 are strictly increasing functions on the nonnegative reals with 6(0) = q(O) = 0 and 0(t) < t, 0(t) < t for positive t. Our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations, and we will identify conditions on a, b, c and the stepsize which imply that the solution sequence {yn }I??= is bounded or that it tends to zero algebraically, as a negative power of n.

/pdf/stability-of-the-discretized-pantograph-differential-5dfbtzl84h.pdf

Stability of the discretized pantograph differential equation

Approximate solution of multi-pantograph equation with variable coefficients

On Neutral Functional)Differential Equations with Proportional Delays

Publisher Summary This chapter discusses the Phragmen-Lindelof principle and a class of functional differential equations. The experimental mathematics and the computerhave played a role in motivating the work on this principle. One added fruitful result of using the computer is that many graphs were produced.Since graphs of solutions to retarded ordinary differential equations are rather rare, one include some of the more interesting oscillatory ones. The work was motivated by the results of systematic computer studies made of particular cases. These experiments suggested the likelihood of unbounded oscillatory solutions to equation.

Alan Feldstein

Papers

The phragmén-lindelöf principle and a class of functional differential equations