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A nonlinear generalized difference equation with both delays and the maximum value of the unknown function over a discrete past time interval are studied. A nonlinear boundary value problem of antiperiodic type for the given difference equation is set up. One of the main characteristics of the considered difference equation is the presence of the unknown function in both sides of the equation. It leads to impossibility for using the step method for explicit solving of the nonlinear difference equation. In this paper, an approximate method, namely, the monotone iterative technique, is applied to solve the problem. An important feature of the given algorithm is that each successive approximation of the unknown solution is equal to the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima,” and an algorithm for its explicit solving is given. Also, each approximation is a lower/upper solution of the given nonlinear boundary value problem. The suggested scheme for approximate solving is computer realized, and it is applied to a particular example, which is a generalization of a model in population dynamics.

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The object of investigation of the paper is a special type of functional differential equations containing the maximum value of the unknown function over a past time interval. An improved algorithm of the monotone-iterative technique is suggested to nonlinear differential equations with “maxima.” The case when upper and lower solutions of the given problem are known at different initial time is studied. Additionally, all initial value problems for successive approximations have both initial time and initial functions different. It allows us to construct sequences of successive approximations as well as sequences of initial functions, which are convergent to the solution and to the initial function of the given initial value problem, respectively. The suggested algorithm is realized as a computer program, and it is applied to several examples, illustrating the advantages of the suggested scheme.