An Inverse Bifurcation Problem and a generelization of Abel's equation

本研究旨在解决以下问题：问题1（逆分岔问题）。求边值问题的非线性项f =u''+ [lambda-q (x)] u=f (u), 0<小于等于>x<小于等于>pi/2, '=d/u' (0) =u (pi/2) =0。在求解逆分岔问题时，出现什么样的积分方程也是本研究的目的之一。关于问题1，我们得到了两个结果：一个局部存在性结果和一个唯一性结果。在我们的研究过程中，我们已经意识到，我们的方法用于获得上述结果是有效的解决类似的反问题，确定未知的非线性项出现在边值问题中，从其谱参数的信息。其中最重要的是以下问题：问题2 （Denisov-Lorenzi问题）。给定区间[0，a]上的函数a (lambda), b (lambda)，确定一个非线性项g，使（过定）边值问题=u' =lambdag (u), 0<x<1, '=d/u (0) =1, u' (0) =a (lambda), u (1) =b （lambda）对每个lambda [0， a]都有一个解u。对这一问题进行了讨论，并对工作中的局部存在性定理作了改进。对于问题1和问题2在建立局部存在性结果时出现的积分方程，我们也给出了答案。在这里处理的一类积分方程将使我们能够介绍由非线性逆问题引起的积分方程的一个基本方面。

This research was intended to solve the following problem :Problem 1 (Inverse Bifurcation Problem). Determine a nonlinear term f of the boundary value problem=u''+ [lambda-q (x)] u=f (u), 0<less than or equal>x<less than or equal>pi/2, '=d/u' (0) =u (pi/2) =0.from its first bifurcating branch.It was also a purpose of this research to find what kind of integral equation appears when we try to solve the inverse bifurcation problem.Concerning Problem 1, two results have been established : a local existence result and a uniqueness result. In the course of our research we have realized that our method used for obtaining the above results is effective in solving similar inverse problems of determining unknown nonlinear terms appearing in boundary value problems from information on their spectral parameters. Foremost among those is the following problem :Problem 2 (Denisov-Lorenzi Problem). Given functions a (lambda), b (lambda) on the interval [0, A], determine a nonlinear term g, with which the (overdetermined) boundary value problem=u''=lambdag (u), 0<x<1, '=d/u (0) =1, u' (0) =a (lambda), u (1) =b (lambda)admits a solution u for each lambda [0, A].This problem has been discussed and an improvement of the local existence theorem in the work is given. We have also given an answer to the question : what kind of integral equations appear in establishing local existence results for Problems 1 and 2. A class of integral equations treated there would enable us to present a basic aspect of integral equations arising from nonlinear inverse problems.