System of impulsive differential equations with a product of two nonlinear functions and nonlinear boundary conditions

A nonlocal two-point boundary value problem for pulse systems of ordinary differential equations of the first order with nonlinear conditions, including derivatives of an unknown vector function, is investigated. The system of differential equations contains the product of two nonlinear vector functions, for each of which the Lipschitz condition is satisfied. The existence, uniqueness and continuous dependence of the solution on the given functions are proved. The problem is reduced to a system of nonlinear functional integral equations in a Banach space. The method of successive approximations in combination with the method of contraction mappings is applied in proving the existence and uniqueness of a solution of nonlinear systems of functional integral equations.