Positive solutions of the three-point boundary value problem for second order differential equations with an advanced argument (Q855480)

Consider the boundary value problem \[ u''(t)+a(t)f(u(h(t)))=0,\; t\in (0,1), \quad u(0)=0,\quad \;\alpha u(\eta)=u(1), \] where \(\eta \in (0,1),\eta>0,\) and \(1-\alpha\eta>0;f:[0,\infty)\rightarrow (0,\infty)\) is continuous; \(a\in C([0,1],[0,\infty))\) and a does not vanish identically on any subinterval; the advanced argument \(h\in C((0,1),(0,1])\) and satisfies \(t\leq h(t)\leq 1\) for \(t\in (0,1).\) Sufficient conditions for the existence of positive solutions are established.