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(逆分叉问题)。确定边值问题=u‘’+[lambda-q(X)]u=f(U)，0&lt；小于或等于&gt；x&lt；小于或等于&gt；pi/2，‘=d/u’(0)=u(pi/2)=0的非线性项f。在研究过程中，我们认识到我们用来获得上述结果的方法对于解决类似的由谱参数信息确定边值问题中出现的未知非线性项的反问题是有效的。其中最重要的是以下问题：问题2(杰尼索夫-洛伦齐问题)。给定区间[0，A]上的函数a(Lambda)，b(Lambda)，确定一个非线性项g，利用它，(超定)边值问题=u‘’=lambdag(U)，0&lt；x&lt；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.讨论了这个问题，并改进了工作中的局部存在定理.我们还给出了一个问题的答案：在建立问题1和问题2的局部存在性结果时，出现了什么样的积分方程组。在那里处理的一类积分方程组将使我们能够给出由非线性反问题产生的积分方程组的基本方面。

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.