  Let $M$ be an $m (\ge2)$-dimensional closed orientable submanifold in an
$n$-dimensional complete simply-connected Riemannian manifold $N$, where the
sectional curvature of $N$ is bounded above by $\delta$. When $\delta<0$,
inspired by Niu-Xu (arXiv:2106.01912), we give new upper bounds for the first
nonzero eigenvalues of the $p$-Laplacian and the $L_T$ operator, respectively.
These generalize Niu-Xu's work for the Laplacian (arXiv:2106.01912) and improve
the estimates due to Chen (Nonlinear Anal.,196, 111833, 2020) for the
$p$-Laplacian and Grosjean (Hokkaido Math. J., 33(2) , 319-339, 2004) for the
$L_T$ operator, respectively. We also obtain several Reilly-type inequalities
for the weighted manifolds and some boundary value problems.

Differential Geometry