Existence theorems for a nonlinear second-order distributional differential equation

Abstract

In this work, we are concerned with existence of solutions for a nonlinear second-order distributional differential equation, which contains measure differential equations and stochastic differential equations as special cases. The proof is based on the Leray–Schauder nonlinear alternative and Kurzweil–Henstock–Stieltjes integrals. Meanwhile, examples are worked out to demonstrate that the main results are sharp. Keywords: Distributional differential equation, Measure differential equation, Stochastic differential equation, Regulated function, Kurzweil–Henstock–Stieltjes integral, Leray–Schauder nonlinear alternative