Existence theorems of solutions for nonlinear impulsive Volterra integral equations in Banach spaces (Q5915446)

The authors study the nonlinear impulsive Volterra integral equation \[ x(t) = x_0(t) + \int_{t_0}^t H(t,s,x(s)) ds + \sum_{t_0< t_k< t} a_k(t) I_k(x(t_k)), \] in a Banach space and proves the existence of a piecewise continuous solution. In addition the existence of maximal and minimal solutions is established when the ordering is defined in terms of some cone. The assumptions include conditions involving the measure of noncompactness and restrictions on how the functions \(H(t,s,x)\) and \(I_k(x)\) may grow when \(||x||\to \infty\).