Asymptotics of Analytic Difference Equations
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1085
Geertrui K.lmmink
Asymptotics of Analytic Difference Equations
Spri nger-Verlag Berlin Heidelberg New York Tokyo 1984
Author Geertrui K. Immink Mathematisch Instituut, Rijksuniversiteit Utrecht 3508 TA Utrecht, The Netherlands
AMS Subject Classification (1980): 39A ISBN 3-540-13867-6 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13867-6 Springer-Verlag New York Heidelberg Berlin Tokyo
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PREFACE
The present monograph is concerned with classes of difference equations of the type ,y(s + 1» = 0,
(0.1)
where s is a complex variable and
and yare n-dimensional vector
functions. The functions
that are considered are characterized
approximately
by the following properties. is holomorphic in a set S x U x U; where S is an open sector and U
(i)
is a neighbourhood of a point y E ¢n. o
is represented asymptotically by a series of the form
(it) A
as s
-e-e-co
_
0>
- L 4l (y , z) s h=o h
-hlp
,
(p E :N)
in S, and this asymptotic expansion is uniformly valid on all
sets S' x Il x U, where S' is a c Los.ed subsector of S. (ii i ) The equation (0.1) possesse:s a formal solution f
such that fo=yo'
f s h=o h
L
-hlp
We have derived existence theorems for analytic solutions of (0.1) that are represented asymptotically by the given formal solution. The method we have used is closely related to that employed for example by Wasow in his book on differential equations «(40]) and, in a much improved version, by Malgrange in (21]. Recent results of Ramis on differential equations «29], [30]) have led us to examine the existence of solutions belonging to certain Gevrey classes of holomorphic functions. Solutions of certain nonlinear equations of the type (0.1) may be used to siflplify linear systems of difference equations. (A similar technique is
in the theory of differential equations. See for example
[34]). One of the main purposes of this study was to give a complete analyt.ic theory for the homogeneous linear system y(s + 1) = A(s)y(s), where A is an n x n matrix function which is meromorphic at infinity. This involves establishing the existence of a number of sectors and a corresponding number of fundamental matrices of (0.2) such that
(0.2)
IV
(i)
the sectors cover a full neighbourhood of infinity,
(ii) each fundamental matrix is holomorphic in the corresponding sector and is represented asymptotically by a given (and fixed) formal fundamental matrix as s tends to infinity in this sector. A general result concern
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