When do all trajectories in the Liénard plane cross the vertical isocline? (Q1902876)

Consider trajectories of the system \(\dot x= y- F(x)\), \(\dot y= -g(x)\). The system is said to have the property \(X^+\) if, for every point \((x_0, y_0)\) with \(y_0> F(x_0)\) and \(x_0\geq 0\), the positive semitrajectory passing through \((x_0, y_0)\) crosses the vertical isocline \(y= F(x)\). The authors investigate the property \(X^+\) in the critical case \(\lim_{x\to \infty} F(x)=- \infty\) and \[ \lim_{x\to \infty} {1\over F(x)} \int^a_b {g(\xi)\over F(\xi)} d\xi= {1\over 4}. \]