On local uniqueness of the solution of boundary-value problems

In this paper we present conditions under which differentiability of the mappings $F:AC^n(I)\to L^n(I)$ and $\Phi:AC^n(I)\to R^n$ at $x_0\in AC^n(I)$ and the uniqueness of the solution of the boundaryvalue problem $u'=F'(x_0)(u)$, $\Phi'(x_0)(u)=0$ imply local uniqueness of the solution $x_0$ of the boundary-value problem $x'=F(x)$, $\Phi(x)=0$.