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 $p\to\infty.$ 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 $p\to \infty,$ we prove that $\lambda_1(1,s)^{1/p}$ goes to an eigenvalue of the H\"older $\infty-$Laplacian.