A functional-analytic method for the study of difference equations
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We will give the generalization of a recently developed functional-analytic method for studying linear and nonlinear, ordinary and partial, difference equations in the 1p and 2p 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. 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
2p
p
f i1 ,...,i p : N −→ C :
=
∞
i1 =1
∞ f i1 ,...,i p 2 < +∞ ···
(1.1)
i p =1
and the Banach space
1p
=
p
f i1 ,...,i p : N −→ C :
∞
i1 =1
∞ ··· f i1 ,...,i p < +∞ ,
(1.2)
i p =1
where N p = N · · × N 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://dx.doi.org/10.1155/S1687183904310101
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Functional-analytic method for difference equations
nonlinear partial difference equations of two variables, we developed a similar functionalanalytic 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 equations of p variables in the Hilbert space 2p , defined by (1.1), and the Banach space 1p , defined by (1.2), respectively, with p ∈ N, p ≥ 1. The motivation for seeking solutions of partial difference equations in the spaces 2p and 1p arises from various problems of mathematics, physics, and biology, such as probability problems, problems concerning integral equations, generating analytic functions, Laurent or z-transforms, numerical schemes, boundary value problems of partial differential equations, problems of quantum mechanics, and problems of population dynamics and epidemiology (for more details, see [11] and the references therein). Also, by assuring the existence of a solution of a difference equation in the space 2p or 1p , we obtain information regarding the asymptotic behavior of the unknown sequence for initial conditions which are in general complex numbers. We would like, at this point, to give an outline of the functional-analytic method that we will present in details in Section 2. (For a sketch of the main ideas used in the
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