Positive solutions of nonlinear differential equations with prescribed decay of the first derivative (Q703418)

The author considers existence and uniqueness of bounded positive solutions of the problem \[ u''+f(t, u, u')=0, t\geq t_0\geq 0,\quad u(t)>0, t\geq t_0, \] \[ \lim_{t\to+\infty}u(t)=M>0,\quad \alpha(t)\leq u'(t)\leq \beta(t), t\geq t_0, \] where \(f\), \(\alpha\), \(\beta\) are continuous functions. The proof is based on Banach's contraction principle. This result complements some others known in the literature.