The first non-zero Neumann $p-$fractional eigenvalue

In this work we study the asymptotic behavior of the first non-zero Neumann $p-$fractional eigenvalue ${\lambda }_1(s,p)$ as $s\to 1^{-}$ and as $p\to \mathrm{\infty }\mathrm{.}$ We show that there exists a constant $\mathcal{K}$ such that $\mathcal{K}(1-s){\lambda }_1(s,p)$ goes to the first non-zero Neumann eigenvalue of the $p-$Laplacian. While in the limit case $p\to \mathrm{\infty },$ we prove that ${\lambda }_1(1,s{)}^{1\mathrm{/}p}$ goes to an eigenvalue of the H"older $\mathrm{\infty }-$Laplacian.