  In this article, we study the Fu\v{c}ik spectrum of fractional Laplace
operator which is defined as the set of all $(\al,\ba)\in \mb
  R^2$ such that
  \begin{equation*}
  \quad \left. \begin{array}{lr}
  \quad (-\De)^s u = \al u^{+} - \ba u^{-} \; \text{in}\; \Om
  \quad \quad \quad \quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om.\\
\end{array} \quad \right\} \end{equation*} has a non-trivial solution $u$,
where $\Om$ is a bounded domain in $\mb R^n$ with Lipschitz boundary, $n>2s$,
$s\in(0,1)$. The existence of a first nontrivial curve $\mc C$ of this
spectrum, some properties of this curve $\mc C$, e.g. Lipschitz continuous,
strictly decreasing and asymptotic behavior are studied in this article. A
variational characterization of second eigenvalue of the fractional eigenvalue
problem is also obtained. At the end, we study a nonresonance problem with
respect to Fu\v{c}ik spectrum.
