  We study the existence of solutions of mixed Riemann-Hilbert or Cherepanov
boundary value problem with simply connected fibers on the unit disk
${\Delta}$. Let $L$ be a closed arc on $\partial{\Delta}$ with the end points
$\omega_{-1}, \omega_1$ and let $a$ be a smooth function on $L$ with no zeros.
Let $\gamma_{\xi}$, $\xi\in\partial{\Delta}\setminus\mathring{L}$, be a smooth
family of smooth Jordan curves in the complex plane which all contain point $0$
in their interiors and such that $\gamma_{\omega_{-1}}$, $\gamma_{\omega_{1}}$
are strongly starshaped with respect to $0$. Then under condition that for each
$w\in\gamma_{\omega_{\pm 1}}$ the angle between $w$ and the normal to
$\gamma_{\omega_{\pm 1}}$ at $w$ is less than $\frac{\pi}{10}$, there exists a
H\"{o}lder continuous function $f$ on $\overline{\Delta}$, holomorphic on
${\Delta}$, such that ${\rm Re}(\overline{a(\xi)} f(\xi)) = 0$ on $L$ and
$f(\xi)\in\gamma_{\xi}$ on $\partial{\Delta}\setminus\mathring{L}$.
