Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Caratheodory Perturbed Term

By employing a well-known fixed point theorem, we establish the existence of multiple positive solutions for the following fourth-order singular differential equation Lu = p(t) f(t,u(t),u ''(t)) - g (t, u(t), u ''(t)), 0 <t < 1, alpha(1)u(0) - beta(1)u'(0) = 0, gamma(1)u(1) + delta(1)u'(1) = 0, alpha(2)u ''(0) - beta(2)u'''(0) = 0, gamma(2)u ''(1) + delta(2)u'''(1) = 0, with alpha(i), beta(i), gamma(i), delta(i) >= 0 and beta(i)gamma(i) + alpha(i)gamma(i) + alpha(i)delta(i) > 0, i = 1, 2, where L denotes the linear operator Lu := (ru''') - qu '', r is an element of C-1([0, 1], (0, +infinity 8)), and q is an element of C([0, 1], [0, +8 infinity)). This equation is viewed as a perturbation of the fourth-order Sturm-Liouville problem, where the perturbed term g : (0, 1) x [0, +infinity) x (-infinity, +infinity) -> (-infinity, +infinity) only satisfies the global Caratheodory conditions, which implies that the perturbed effect of g on f is quite large so that the nonlinearity can tend to negative infinity at some singular points.