The Stieltjes string model with external load

The following model p(t)u′(t)−p(0)φ1(u)−∫0t(gu)(s)dQ(s)=F(t)of the Stieltjes string on segment [0, l] with a nonlocal boundary value condition u(0)=φ0(u) is considered. Optimal conditions on a unique solvability of the mentioned problem are established.