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The question about the existence and characterization of bounded solutions to linear functional-differential equations with both advanced and delayed arguments was posed in the 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 −a 2 y″(x + a 1 y′(x) + y(x) = ∫0∞ μ (dα), where a 1 ≥ 0, a 2 ≥ 0 a 12 + a 22 > 0, and μ is a probability measure. By setting K:=∫ 0∞ ln ± 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)=1/2y(qx)+1/2y(x/q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools.

On Bounded Solutions of the Balanced Generalized Pantograph Equation