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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 early 1970s by T.~Kato
in connection with the analysis of the pantograph equation,
$y'(x)=ay(qx)+by(x)$. In our talk, we answer
this question for the \emph{balanced} generalized pantograph
equation of the form
$y'(x)+y(x)=\sum_{i} p_{i}\, y(\alpha_{i}x)$,
under the balance condition
$\sum_{i} p_{i}=1$ (${p_{i}\ge0}$).
Namely, setting
$K:=\sum_1^l p_{i}\ln \alpha_{i}$, we prove that if
$K\le 0$ 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 proofs are based on the link with the theory
of Markov processes and employ martingale technique.
Same approach may be applied also for other types of equations with rescaling
( i.e. functional, integral and integro-differential ones).
The talk is based on joint work with Leonid Bogachev (Leeds, UK),
Stanislav Molchanov (North Carolina at Charlotte, USA) and John Ockendon (Oxford, UK). |
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