# zbMATH — the first resource for mathematics

Summability of formal solutions of a class of nonlinear difference equations. (English) Zbl 0973.39003
The author shows that the solutions of the equation $\Phi(z,y(z),y(z+1))=0$ can be characterized by their asymptotic behaviour in slightly large domains. Here $$\Phi$$ is an $$n$$-dimensional vector-function, analytical in a domain of the form $$D\times U\times U$$, where $$D$$ is some unbounded domain of $$C$$ and $$U$$ is a neighbourhood of a given point in $$C^n$$. The notion of weak Borel-sum is used. It is proven that, under additional hypotheses on $$\Phi$$, the above equation has a unique formal solution $\sum_{n=0}^{\infty}a_n z^{-n}.$

##### MSC:
 39A10 Additive difference equations 39A11 Stability of difference equations (MSC2000)
Full Text:
##### References:
 [1] Braaksma B.L.J., In Ordinary and Partial Differential Equations, Proceedings Dundee 1978, Lecture 827 pp 25– (1980) [2] Braaksma B.L.J., In Equations différentielles dans pp 1– (1988) [3] Duval A., Funkcialaj Ekvacioj 26 pp 349– (1983) [4] Ecalle J., Les Fonctions Resurgentes, tome III. Publ (1985) · Zbl 0602.30029 [5] Harris Jr. W.A., Arch. Rat. Mech. An 15 pp 377– (1964) [6] DOI: 10.1515/crll.1966.222.120 · Zbl 0142.05703 [7] DOI: 10.1515/crll.1916.146.95 [8] Immink G.K., A particular type of summability of divergent power series, with an application to linear difference equations · Zbl 0981.30002 [9] Immink G.K., Asymptotics of analytic difference equations. 1085 (1984) · Zbl 0548.39001 [10] Immink G.K, In Equations differentielles dans le champ complexe Equations 1 pp 35– (1988) [11] Immink G.K., SI AM J. Math. Anal 22 pp 239– (1991) [12] Immink G.K., Funk. Ekv. 39 pp 469– (1996) [13] Malgrange B., I’Enseignement Mathématique 20 pp 149– (1974) [14] Praagman C., In Proceedings Kon. Nederl Ac. van Wetensck, Ser. A 86 pp 249– (1983) [15] van der Put M., Galois theory of difference equations. In Lecture Notes in Mathematics 1666 (1997) · Zbl 0930.12006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.