Singular Dirichlet boundary value problem for second order ODE

Abstract

This paper investigates the singular Dirichlet problem -u'' = f(t,u,u')\,,\quad u(0)=0\,,\quad u(T)=0\,, where f satisfies the Carath\'eodory conditions on the set (0,T) \times \mathbb{R}_0^2 and \mathbb{R}_0 = \mathbb{R} \setminus \{ 0\}. The function f(t,x,y) can have time singularities at t=0 and t=T and space singularities at x=0 and y=0. The existence principle for the above problem is given and its application is presented here. The paper provides conditions which guarantee the existence of a solution which is positive on (0,T) and which has the absolutely continuous first derivative on [0,T].