平衡广义比例方程的有界解
On bounded solutions of the balanced generalized pantograph equation
摘要：
关于具有超前变元和滞后变元的线性泛函微分方程有界解的存在性和特征问题是T. Kato结合缩放仪方程的分析，y '（x）=ay（qx）+by（x）。本文对平衡的广义比例方程-a_2y '（x）+a_1y'（x）+y（x）=int_0^infty y（qx）m（dq）回答了这个问题，其中a_1 > or = 0，a_2 > or = 0，a_1^2+a_2^2>0，m是一个概率测度.也就是说，设K：= int_0 ^infty ln（q）m（dq），我们证明了如果K <或= 0，则方程不具有非平凡的（即，非常数）有界解，而如果K>0，则存在这样的解。临界情况下的结果K=0解决了一个长期存在的问题。证明利用了与马尔可夫过程理论的联系，因为平衡缩放方程的任何解都是相对于具有“乘法”跳跃的某个扩散过程的生成元L的L-调和函数。本文还包括简单原型方程y '（x）+y（x）=（1/2）y（qx）+（1/2）y（x/q）的三个“初等”证明，分别基于扰动、分析和概率技术，这在其他情况下可能作为有效的探索工具出现。
英文摘要：
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 -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.
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