Abstract
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.
Highlights
In the last few decades, great attention has been paid to automatic control systems and their applications to computational mathematics and modeling
In this paper the authors consider a class of nonlinear difference equations with both delays and maximum of the unknown function over a discrete past time interval
Any successive approximation is a solution of an initial value problem for a linear difference equation, and a practical algorithm for its obtaining is given and computer realized
Summary
In the last few decades, great attention has been paid to automatic control systems and their applications to computational mathematics and modeling. The properties of solutions of some special types of difference equations with maxima have been studied in [7,8,9] Such nonlinear difference equations can hardly be solved in explicit forms, and we need to develop some approximate methods for their solutions. Note in several papers various types of boundary value problems for difference equations have been studied, and monotone iterative method has been applied. In [21], monotone iterative technique is applied to a periodic boundary value problem for difference equations with maxima but the successive approximation are solutions of periodic boundary value problems, which is practically difficult to be obtained. In connection with the applications of difference equations as models of real world problems it is necessary to study difference equations with both delays and maximum over a discrete past time interval. Any successive approximation is a solution of an initial value problem for a linear difference equation, and a practical algorithm for its obtaining is given and computer realized
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
