The existence of solutions to a class of two point quasilinear boundary value problems (Q1206669)

The authors show that the boundary value problem \[ x'' + x^\alpha + 1 = 0, \quad x(0) = x'(1) =0, \] possesses for any \(\alpha > 0\) a solution \(x\in C^2[0,1]\) such that \(x(t)\geq 0\) for \(0\leq t<1\) and \(x(t)>0\) for \(0< t<1\).