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

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.