An algorithm for approximate solving of differential equations with “maxima”

An algorithm for constructing two sequences of successive approximations of the solution of the initial value problem for nonlinear differential equations with “maxima” is given. This algorithm is based on the monotone iterative technique. It is proved that both sequences are monotonically convergent. Each term of the constructed sequences is a solution of an initial value problem for linear differential equations with “maxima” and it is a lower/upper solution of the given problem. Both the scalar case and the multidimensional case are studied. An example, solved by computer realization of the suggested algorithm, illustrates the practical application of the method.