On the existence of a smooth solution of a multidimensional boundary value problem (Q2459773)

Consider the boundary value problem \[ \ddot z_k+\sum^m_{i=1} \sum^m_{j=1}a^k_{ij}(z)\dot z_i\dot z_j=0,\quad z_k(0)=0, \quad z_k(1)=1, \quad k=1, \dots,m,\tag{*} \] with the additional condition \[ 0\leq z_k(s)\leq 1\quad \text{for}\quad 0\leq s\leq 1,\;k=1,\dots,m,\tag{**} \] where \(a^k_{ij}\) are smooth scalar functions. The author derives sufficient conditions for the functions \(a^k_{ij}\) such that (*) has a \(C^2\)-solution satisfying (**).