45 research outputs found
ANTI-INTERFERON ANTIBODIES IN ALPHA INTERFERON TREATED PATIENTS WITH CHRONIC MYELOID LEUKAEMIA
The temporomandibular joint in young adults at autopsy: a morphologic classification and evaluation
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 ā²(x) = ay(qx) + by(x). In the present paper, we answer this question for the balanced generalized pantograph equation of the form āa2y ā²ā²(x)+a1y ā²(x)+y(x) = ā« ā 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: = ā« ā 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 āmultiplication ā jumps. The paper also includes three āelementary ā proofs for the simple prototype equation y ā²(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