Nonlinear impulsive Hahn—Sturm—Liouville problems on the whole line

Abstract

Impulsive Hahn—Sturm—Liouville problems in singular cases are discussed. The existence of solutions of such equations on the whole axis and in the case of Weyl’s limit-circle has been investigated. First, we construct the corresponding Green’s function. This boundary-value problem is thus reduced to a fixed point problem. Later, we demonstrate the existence and uniqueness of the solutions to this problem by using the traditional Banach fixed point theorem. Finally, we derive an existence theorem without considering the solution’s uniqueness. We apply the well-known Schauder fixed point to obtain this result.