LYAPUNOV-TYPE INEQUALITIES FOR A SECOND-ORDER DIFFERENTIAL EQUATION WITH A POTENTIAL HAVING TWO SINGULARITIES

We establish Lyapunov-type inequalities for the second-order differential equation u00(x)+V(x)u(x) = 0 in (a, b), under the Dirichlet boundary condition u(a) = u(b) = 0, where the potential function V admits two singularities at x = a and x = b, and V is not an element of L1((a, b)). Notice that due to the nature of the function V, the standard Lyapunov inequality is not applicable in our case. Two kinds of Lyapunov-type inequalities are obtained. For the first one, we use the integral formulation of the problem. For the second inequality, we introduce a new approach based on the choice of an appropriate test function and integral estimates. Next, the obtained inequalities are applied to estimate the eigenvalues of a singular boundary value problem.