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 -a_2 y''(x)+a_1 y'(x)+y(x)=int_0^infty y(qx) m(dq), where a_1 > or = 0, a_2 > or = 0, a_1^2+a_2^2>0, and m is a probability measure. Namely, setting K:=int_0^infty ln(q) m(dq), we prove that if K < or = 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)=(1/2)y(qx)+(1/2)y(x/q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools.