  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.
