On an inverse problem for the equation $u''=f(u)$ with the unknown f

Abstract

We consider the equation $u''=f(u)$ under the conditions $u(0)=p$, $u'(0)=0$, $u(\overline{x}(p))=A$, $u(x)\in (A,B)$ for any $x\in (0,\overline{x}(p))$, with the unknown function f, where p is a parameter that runs over the interval [A, B), $A<B$, and $\overline{x}=\overline{x}(p)$ is a given function strictly positive in (A, B). For any continuously differentiable function f and p fixed the differential equation can have at most one solution u(x) that obeys all these conditions. We prove that if $f_1$ and $f_2$ are two continuously differentiable functions each of which satisfies all the conditions above, then $f_1\equiv f_2$ in [A, B).