Stieltjes differential in impulse nonlinear problems

An impulse nonlinear problem admitting discontinuous solutions that are functions of bounded variation is studied. This problem models the deformation of a discontinuous string (chains of strings fastened together by springs) with elastic supports in the form of linear and nonlinear springs (for example, springs with different turns, whose deformations do not obey Hooke’s law). The model is described by a second-order differential equation with derivatives in special measures and Dirichlet boundary conditions. Existence theorems are proved, and conditions for the existence of nonnegative solutions are obtained.