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