Fixed-time and state-dependent time discontinuities in the theory of Stieltjes differential equations

Abstract

In the present paper, we are concerned with a very general problem, namely the Stieltjes differential Cauchy problem involving state-dependent discontinuities. Given that the theory of Stieltjes differential equations covers the framework of impulsive problems with fixed-time impulses, in the present work we generalize this setting by allowing the occurrence of fixed-time impulses, as well as the occurrence of state-dependent impulses. Along with an existence result obtained under an overarching set of assumptions involving Stieltjes integrals, it is showed that a least and a greatest solution can be found.

Keywords

• stieltjes differential equation
• state-dependent impulsive equation
• stieltjes integral
• extremal solution