Existence and uniqueness of solutions to first-order systems of nonlinear impulsive boundary-value problems with sub-, super-linear or linear growth

Abstract

In this work we present some new results concerning the existence and uniqueness of solutions to an impulsive first-order, nonlinear ordinary differential equation with "non-periodic" boundary conditions. These boundary conditions include, as a special case, so-called "anti-periodic" boundary conditions. Our methods to prove the existence and uniqueness of solutions involve new differential inequalities, the classical fixed-point theorem of Schaefer, and the Nonlinear Alternative. Our new results apply to systems of impulsive differential equations where the right-hand side of the equation may grow linearly, or sub- or super-linearly in its second argument.