On the first curve of Fu\v{c}ik Spectrum Of $p$-fractional Laplacian Operator with nonlocal normal boundary conditions

In this article, we study the Fu\v{c}ik spectrum of the $p$-fractional Laplace operator with nonlocal normal derivative conditions which is defined as the set of all $(a,b)\in \text{mb}{R}^2$ such that $$\text{mc}(F_p)\{\begin{array}{l}{\mathrm{\Lambda }}_{n,p}(1-\text{al})(-\mathrm{\Delta }{)}_p^{\text{al}}u+\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=\frac{{\chi }_{{\mathrm{\Omega }}_{\text{e}}}}{\text{e}}(a(u^{+}{)}^{p-1}-b(u^{-}{)}^{p-1})\text{in}\mathrm{\Omega },\\ \text{mc}{N}_{\text{al},p}u=0in\text{mb}{R}^n\setminus \bar{\mathrm{\Omega }},\end{array}$$ has a non-trivial solution $u$, where $\mathrm{\Omega }$ is a bounded domain in $\text{mb}{R}^n$ with Lipschitz boundary, $p\ge 2$, $n>p\text{al}$, $\text{e},\text{al}\in (0,1)$ and $\Omega{_\e}:={x \in \Omega: d(x,\pa \Omega)\leq \e }$\mathrm{.}Weshowedexistenceofthefirstnon-trivialcurve$\mc C$ of this spectrum which is used to obtain the variational characterization of a second eigenvalue of the problem $\text{mc}(F_p)$. We also discuss some properties of this curve $\text{mc}C$, e.g. Lipschitz continuous, strictly decreasing and asymptotic behaviour and nonresonance with respect to the Fu\v{c}ik spectrum.