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

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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.