Mixed Riemann-Hilbert boundary value problem with simply connected fibers

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 $\partial{\Delta}\setminus\mathring{L}$.