关键词: 平均曲率算子, 离散混合边值问题, 非平凡凸解, 锥, 不动点定理

Abstract:

In this paper, by using the fixed point theorem in cones, we discuss the existence of $ \Delta[\phi(\Delta v(t-1))]=f(t, -v(t)), {\quad} t\in[2, T-1]_{{\Bbb Z}}, $ $ \Delta v(1)=0, {\quad} v(T)=0 $ nontrivial convex solutions for a discrete mixed boundary value problem of mean curvature operator in Minkowski space, where $ \phi(s)=\frac{s}{\sqrt{1-s^{2}}}, s\in(-1, 1), $ $ [2, T-1]_{{\Bbb Z}}:=\{2, 3, \cdots, T-2, $ $ T-1\}, $ $ T\geqslant4 $ and $ T\in{\Bbb N}^{\ast} $, the nonlinear term $ f(t, u) $ is nonnegative and continuous, and singularity is allowed at $ u=1 $.

Key words: Mean curvature operator, Discrete mixed boundary value problem, Nontrivial convex solutions, Cone, Fixed point theorem