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The paper deals with an approximate method for solving a mixed boundary value problem for nonlinear difference equations containing a maximum of the unknown function over a past time interval. Every successive approximation to the unknown solution is the unique solution of an appropriately constructed initial value problem for a linear difference equation with maxima, and an algorithm for its explicit obtaining is suggested. Also, each approximation is a lower/upper solution of the given mixed problem. The rapid convergence of the successive approximations is proved. The suggested algorithm is realized as a computer program and it is applied to an example.

Quasilinearization for nonlinear boundary value problems for delay-type difference equations with maxima