On Bounded Solutions of the Balanced Generalized Pantograph Equation

Abstract

The question about the existence and characterization of bounded solutions to linear functional-differential equations with both advanced and delayed arguments was posed in early 1970s by T. Kato in connection with the analysis of the pantograph equation, y ′(x) = ay(qx) + by(x). In the present paper, we answer this question for the balanced generalized pantograph equation of the form −a2y ′′(x)+a1y ′(x)+y(x) = ∫ ∞ 0 y(αx)µ(dα), where a1 ≥ 0, a2 ≥ 0, a2 1 +a22> 0, and µ is a probability measure. Namely, setting K: = ∫ ∞ 0 ln α µ(dα), we prove that if K ≤ 0 then the equation does not have nontrivial (i.e., nonconstant) bounded solutions, while if K> 0 then such a solution exists. The result in the critical case, K = 0, settles a long-standing problem. The proof exploits the link with the theory of Markov processes, in that any solution of the balanced pantograph equation is an L-harmonic function relative to the generator L of a certain diffusion process with “multiplication ” jumps. The paper also includes three “elementary ” proofs for the simple prototype equation y ′(x) + y(x) = 1 1 2 y(qx) + 2 y(x/q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools. Key words: Pantograph equation, functional-differential equations, integrodifferential equations, balance condition, bounded solutions, WKB expansion