带有Stieltjes积分边界条件的Minkowski平均曲率问题正解的存在性

Existence of Positive Solutions for the Minkowski Mean Curvature Problem with Stieltjes Integral Boundary Conditions

摘要: 本文运用锥上的不动点指数理论研究了在含有Stieltjes积分边值条件下Minkowski平均曲率问题$\{\begin{array}{l}-{\left[\phi \left(u^{\prime }\left(t\right)\right)\right]}^{\prime }=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right)，t\in \left[0,1\right],\hfill \\ u^{\prime }\left(0\right)=0，u\left(1\right)=\alpha \left[u\right]\hfill \end{array}$ 正解的存在性。其中$f:\left[0,1\right]\times \left[0,+\infty \right)\times \left(-1,0\right]\to \left[0,+\infty \right)$ 连续，$\alpha \left[u\right]={\displaystyle {\int }_0^{1}u\left(t\right)\text{d}A\left(t\right)}$ ，$A$ 是有界变差函数，$\phi \left(s\right)=\frac{s}{\sqrt{1-s^2}}$ ，$s\in \left(-1,1\right)$ 。

Abstract: In this paper, by using the theory of fixed point index on cones, we discuss the existence of positive solutions for the following second-order differential equations with mean curvature operator in Minkowski space under Stieltjes boundary condition $\{\begin{array}{l}-{\left[\phi \left(u^{\prime }\left(t\right)\right)\right]}^{\prime }=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right)，t\in \left[0,1\right],\hfill \\ u^{\prime }\left(0\right)=0，u\left(1\right)=\alpha \left[u\right]\hfill \end{array}$ where $f:\left[0,1\right]\times \left[0,+\infty \right)\times \left(-1,0\right]\to \left[0,+\infty \right)$ is continuous and $\alpha \left[u\right]={\displaystyle {\int }_0^{1}u\left(t\right)\text{d}A\left(t\right)}$ , $A$ is a function of bounded variation $\phi \left(s\right)=\frac{s}{\sqrt{1-s^2}}$ , $s\in \left(-1,1\right)$ .