Assume that $M^n(n\geq3)$ is a complete hypersurface in $\mathbb{R}^{n+1}$ with zero scalar curvature. Assume that $B, H, g$ is the second fundamental form, the mean curvature and the induced metric of $M$, respectively. We prove that $M$ is a hyperplane if $$-P_1(\nabla H,\nabla|H|)\leq-\delta|H||\nabla H|^2$$ for some positive constant $\delta$, where $P_1=nHg-B$ which denotes the first order Newton transformation, and $$\left(\int_M|H|^ndv\right)^\frac{1}{n}<\alpha$$ for some small enough positive constant $\alpha$ which depends only on $n$ and $\delta$. We also derive similar result for complete hypersurfaces in $\mathbb{S}^{n+1}$ with constant scalar curvature $R=n(n-1)$.

In the author's previous paper, we considered the equivalent conditions with $W^{-m, p(\cdot )} $-version ($m\ge 0$ integer) of the J. L. Lions Lemma,  where $p(\cdot)$ is a variable exponent.  In this paper, we directly derive $W^{-m,p(\cdot)}$-version of the J. L. Lions Lemma. Therefore, we can use all of the equivalent conditions. As an application, we derive the generalized  Korn inequality. Furthermore, we consider the relation to other fundamental results.

In this paper, we consider a system of set-valued nonlinear generalized variational inclusion problems (SNGVIP) in $q$-uniformly smooth Banach spaces. We define a class of $H(·,·)$-mixed mappings and its associated class of resolvent operators. Further, using proximal-point mapping method, we suggest an iterative algorithm for finding the solution of the system of set-valued nonlinear generalized variational inclusion problems. Furthermore, we discuss the convergence criteria of the sequences generated by the iterative algorithm.