On the existence of solutions for singular boundary value problem of third-order differential equations

The singular boundary value problems of third-order differential equations -u '''(t) = h(t)f(t, u(t)), t is an element of (0, 1), u(0) = u '(0) = 0, u '(1) = alpha u '(eta) are considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators, where h(t) is allowed to be singular at both t = 0 and t = 1, and f is not necessary to be nonnegative. The existence results of nontrivial solutions and positive solutions are given by means of the topological degree theory.