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    On Bounded Solutions of the Balanced Generalized Pantograph Equation

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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 ā€²(x) = ay(qx) + by(x). In the present paper, we answer this question for the balanced generalized pantograph equation of the form āˆ’a2y ā€²ā€²(x)+a1y ā€²(x)+y(x) = āˆ« āˆž 0 y(Ī±x)Āµ(dĪ±), where a1 ā‰„ 0, a2 ā‰„ 0, a2 1 +a22> 0, and Āµ is a probability measure. Namely, setting K: = āˆ« āˆž 0 ln Ī± Āµ(dĪ±), we prove that if K ā‰¤ 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 ā€œmultiplication ā€ jumps. The paper also includes three ā€œelementary ā€ proofs for the simple prototype equation y ā€²(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
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