Analysis of the archetypal functional equation in the non-critical case

We study the archetypal functional equation of the form $y(x)=\iint_{R^2} y(a(x-b))\,\mu(da,db)$ ($x\in R$), where $\mu$ is a probability measure on $R^2$; equivalently, $y(x)=E\{y(\alpha (x-\beta))\}$, where $E$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of non-trivial (i.e. non-constant) bounded continuous solutions is governed by the value $K:=\iint_{R^2}\ln |a| \mu(da,db) =E \{\ln |\alpha|\}$; namely, under mild technical conditions no such solutions exist whenever $K0$ (and $\alpha>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(\alpha,\beta)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $P(\alpha0$ is drastically different from that with $\alpha>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x\rightsquigarrow\alpha (x-\beta)$

On Bounded Solutions of the Balanced Generalized Pantograph Equation

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