On certain modified boundary value problems for ordinary differential equations with irregular singularity

We consider a linear second-order differential equation with irregularly singular point at the beginning of the interval. For the corresponding homogeneous differential equation, we obtain the asymptotics of the solutions and their derivatives near the singular point. Using some modified Green functions and taking into account the asymptotics, we consider three boundary value problems with various boundary conditions (including a weighted one) at the singular point, proving theorems on the existence and uniqueness of the solutions and giving their structure. Lithuanian Mathematical Journal, Vol. 49, No. 1, 2009, pp. 109–121