Monotone-iterative method for a mixed nonlinear boundary value problem for differential equations with "maxima"

The main object of investigation of the paper is so called differential equation with "maxima". These equations, for example, find a wide application in the theory of automatic regulation. The main characteristic of this type of equations is the presence of the maximum of the unknown function over a past time interval. A nonlinear boundary value problem mixed with an initial condition for a nonlinear equation is studied. An algorithm for constructing two sequences of successive approximations of the solution is given. This algorithm is based on the monotone iterative technique combined by the method of lower/upper solutions. Each term of the constructed sequences is a solution of an initial value problem for linear differential equations with "maxima". Also the terms of the sequences are lower/upper solutions of the given problem. It is proved both sequences are monotonically convergent. The suggested procedure is illustrated on some examples.