Positive solutions for a nonlocal boundary-value problem with vector-valued response (Q1420676)

By using variational methods, sufficient conditions are given for the solvability of the nonlocal nonlinear boundary value problem \[ \frac{d}{dt}\bigl(k(t)x'(t)\bigr)+V_x\bigl(t,x(t)\bigr)=0,\;t\in[0,T],\quad x(0)=0,\;x'(T)=\int^T_\eta x'(s) \,dg(s), \] where the integral is understood in the sense of Riemann-Stieltjes with \(0<\eta<T\). The solution \(x:[0,T]\to \mathbb{R}^n\) should satisfy the equation almost everywhere. The function \(k\) is absolutely continuous and positive.