Asymptotics of analytic difference equations.

*(English)*Zbl 0548.39001
Lecture Notes in Mathematics. 1085. Berlin etc.: Springer-Verlag. V, 134 p. DM 21.50 (1984).

The author studies classes of difference equations of the type \((1)\quad\phi (s,y(s),y(s+1))=0,\) where s is a complex variable and \(\phi\) and y are n-dimensional vector functions. The functions \(\phi\) that are considered are characterized approximately by the following properties. (i) \(\phi\) is holomorphic in a set \(S\times U\times U\), where S is an open sector and U is a neighborhood of a point \(y_ 0\in {\mathbb{C}}^ n\). (ii) \(\phi\) is represented asymptotically by a series of the form
\[
{\hat\phi }(s,y,z)=\sum^{\infty}_{h=0}\phi_ h(y,z)s^{-h/p}\quad (p\in N)
\]
as \(s\to\infty \) in S and this asymptotic expansion is uniformly valid on all sets S’\(\times U\times U\), where S’ is a closed subsector of S. (iii) Equation (1) possesses a formal solution \(f=\sum^{\infty}_{h=0}f_ hs^{-h/p}\) such that \(f_ 0=y_ 0\). The author derives existence theorems for analytic solutions of (1) that are represented asymptotically by the given formal solution. Solutions of nonlinear equations of the type (1) are used to simplify linear systems of difference equations. One of the primary objectives of this work is to give a complete analytic theory for the homogeneous linear system (2) \(y(s+1)=A(s)y(s)\), where A is an \(n\times n\) matrix function which is meromorphic at infinity. A general result concerning this problem is stated in Chapter III. Chapter I contains a selection of known results from the formal theory of linear difference equations and from the theory of Gevrey classes of holomorphic functions and series. The last section deals with the existence of right inverses of linear difference operators on Banach space of holomorphic functions with different types of asymptotic behavior.

Reviewer: F.Gross

##### MSC:

39A10 | Additive difference equations |

47B39 | Linear difference operators |

39A70 | Difference operators |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

39A11 | Stability of difference equations (MSC2000) |

30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |