Abstract:

The existence of positive solution was studied for the
nonlinear Sturm-Liouville boundary value problem, where the
nonlinear term $f(t,u)$ may be singular at $t = 0,\,t = 1$. By
introducing the integrations of height functions of nonlinear term
on bounded set the growths of nonlinear term were described. By
applying the Krasnoselskii fixed point theorem in degree theory and
the dominated convergence theorems in real variable, an existence
theorem of positive solution was proved when there are limit
functions $\mathop {\lim }\limits_{u \to + 0} f(t,u) / u$ and
$\mathop {\lim }\limits_{u \to + \infty } f(t,u) / u$.

Key words: boundary value problem, positive solution, existence, nonlinear ordinary differential equation, boundary value problem, positive solution, existence