An existence theorem for a singular third-order boundary value problem on \([0,+\infty)\) (Q847382)

The authors consider the singular nonlinear boundary vlue problem \[ y'''(x)= f(y(x)),\quad y(0)= 0,\quad \lim_{x\to+\infty} y(x)= 1, \] \[ \lim_{x\to+\infty} y'(x)= \lim_{x\to+\infty} y''(x)= 0. \] Here \(f(y)= (1-y)^\lambda g(y)\) with \(\lambda> 0\) and \(g\) is continuous on \((0,1]\). Under some assumptions, the existence of at least one positive increasing concave solution is shown. Kneser's theorem is employed in the proof, which considers an auxiliary second-order terminal value problem.