Approximate solutions for a class of first order nonlinear difference equations Akbar H. Borzabadi * , Mortaza Gachpazan Department of Applied Mathematics, School of Mathematics and Computer Science, Damghan University of Basic Sciences, Damghan, Iran Abstract In this paper, a measure-theoretical approach to find the approximate solutions for a class of first order nonlinear dif- ference equations is introduced. In this method the problem is transformed to an equivalent optimization problem. Then, by considering it as a calculus of variations problem, some concepts in measure theory are used to approximate the solu- tion. The procedure of constructing approximate solution in form of an algorithm is shown. Finally a numerical example is given. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Difference equation; Calculus of variations; Measure theory; Linear programming 1. Introduction Consider the following difference equation: F ðx k ; x kþ1 Þ¼ 0; k ¼ 0; 1; 2; ... ; N  1; x 0 ¼ a;  ð1Þ where F ð; Þ is a nonlinear function and a is known. We know there are various methods for solving the above difference equation but the efficiency of these methods will be decreased by assuming the nonlinearity of F ð; Þ (see [4]). In this paper, our aim is to convert this problem to a calculus of variations problem (CVP) and to find an approximate solution for the new problem using some concepts of measure theory. In this manner, the non- linearity of the function F ð; Þ does not have any serious effect on solving process of the problem. Using the concepts in measure theory based on the idea of Young [7], has been theoretically established by Rubio in [6]. Rubio used some concepts of measure theory for solving the optimal control problem with bounded-state variables. One can see in [1–3], some extensions of Rubio’s idea to find the approximate solutions for some 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.096 * Corresponding author. E-mail addresses: borzabadi@dubs.ac.ir (A.H. Borzabadi), gachpazan@dubs.ac.ir (M. Gachpazan). Applied Mathematics and Computation 190 (2007) 1108–1115 www.elsevier.com/locate/amc