A FUNCTIONAL-ANALYTIC METHOD FOR THE STUDY OF DIFFERENCE EQUATIONS EUGENIA N. PETROPOULOU AND PANAYIOTIS D. SIAFARIKAS Received 29 October 2003 and in revised form 10 February 2004 We will give the generalization of a recently developed functional-analytic method for studying linear and nonlinear, ordinary and partial, difference equations in the 1 and 2 p p spaces, p ∈ N, p ≥ 1. The method will be illustrated by use of two examples concerning a nonlinear ordinary difference equation known as the Putnam equation, and a linear partial difference equation of three variables describing the discrete Newton law of cooling in three dimensions | A FUNCTIONAL-ANALYTIC METHOD FOR THE STUDY OF DIFFERENCE EQUATIONS EUGENIA N. PETROPOULOU AND PANAYIOTIS D. SIAFARIKAS Received 29 October 2003 and in revised form 10 February 2004 We will give the generalization of a recently developed functional-analytic method for studying linear and nonlinear ordinary and partial difference equations in the ep and ep spaces p e N p 1. The method will be illustrated by use of two examples concerning a nonlinear ordinary difference equation known as the Putnam equation and a linear partial difference equation of three variables describing the discrete Newton law of cooling in three dimensions. 1. Introduction The aim of this paper is to present the generalization of a functional-analytic method which was recently developed for the study of linear and nonlinear difference equations of one two three and four variables in the Hilbert space . c - ep 1 f i1 . ip Np C .ỵ I f i1 . ip 2 and the Banach space 00 00 ep 1 f i1 . ip Np C ỵ I f i1 . ip wj where Np N X X NT and p 1 2 3 4. p-times More precisely this method was introduced for the first time by Ifantis in 5 for the study of linear and nonlinear ordinary difference equations. Later this method was extended by the authors in 7 9 10 in order to study a class of nonlinear ordinary difference equations more general than the one studied in 5 . For the study of linear and Copyright 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004 3 2004 237-248 2000 Mathematics Subject Classification 39A10 39A11 URL http S1687183904310101 238 Functional-analytic method for difference equations nonlinear partial difference equations of two variables we developed a similar functional-analytic method in 11 12 which was extended in 8 in order to study partial difference equations of three and four variables. The aim of this paper is to present the generalization of this functional-analytic method for the study of linear and nonlinear partial difference .
