On bounded solutions of the balanced generalized pantograph equation

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dc.creatorBogachev, Leonid
dc.creatorDerfel, Gregory
dc.creatorMolchanov, Stanislav
dc.creatorOckendon, John
dc.date2007-03-29
dc.date2007-03-29
dc.date.accessioned2026-07-07T07:55:05Z
dc.date.available2026-07-07T07:55:05Z
dc.descriptionThe 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.
dc.identifierhttps://arxiv.org/abs/math/0703897
dc.identifierhttp://arxiv.org/abs/math/0703897
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/126848
dc.subjectProbability
dc.subjectClassical Analysis and ODEs
dc.subjectPrimary 34K06, 45J05, 60Jxx; secondary 34K12
dc.titleOn bounded solutions of the balanced generalized pantograph equation
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