Cesàro summability in a linear autonomous difference equation

Author:
Mihály Pituk

Journal:
Proc. Amer. Math. Soc. **133** (2005), 3333-3339

MSC (2000):
Primary 39A11; Secondary 34K40

DOI:
https://doi.org/10.1090/S0002-9939-05-08154-2

Published electronically:
May 4, 2005

MathSciNet review:
2161157

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a linear autonomous difference equation with a unique real eigenvalue , it is shown that for every solution the ratio of and the eigensolution corresponding to is Cesàro summable to a limit which can be expressed in terms of the initial data. As a consequence, for most solutions the Lyapunov characteristic exponent is equal to . The proof is based on a Tauberian theorem for the Laplace transform.

**[1]**G. Doetsch,*Handbuch der Laplace-Transformation I*, Verlag Birkhäuser, Basel, 1950 (in German). MR**0043253 (13:230f)****[2]**R. D. Driver,*Some harmless delays*, Delay and Functional Differential Equations and Their Applications (Proc. Conf., Park City, Utah, 1972), Academic Press, New York, 1972, pp. 103-119. MR**0385277 (52:6141)****[3]**R. D. Driver, D. W. Sasser, and M. L. Slater,*The equation**with ``small" delay*, Amer. Math. Monthly**80**(1973), 990-995. MR**0326104 (48:4449)****[4]**R. D. Driver, G. Ladas, and P. N. Vlahos,*Asymptotic behavior of a linear delay difference equation*, Proc. Amer. Math. Soc.**115**(1992), 105-112. MR**1111217 (92i:39004)****[5]**M. V. S. Frasson and S. M. Verduyn Lunel,*Large time behaviour of linear functional differential equations*, Integral Equations Operator Theory**47**(2003), 91-121. MR**2015849 (2004j:34141)****[6]**J. Hale,*Theory of Functional Differential Equations*, Springer-Verlag, New York, 1977. MR**0508721 (58:22904)****[7]**I.-G. E. Kordonis and Ch. G. Philos,*On the behavior of the solutions for linear autonomous neutral delay difference equations*, J. Differ. Equations Appl.**5**(1999), 219-233. MR**1697057 (2000b:39001)****[8]**Ch. G. Philos and I. K. Purnaras,*An asymptotic result for some delay difference equations with continuous variable*, Advances in Difference Equations**2004:1**(2004), 1-10. MR**2059199****[9]**D. V. Widder,*Introduction to Transform Theory*, Academic Press, New York, 1971.

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Additional Information

**Mihály Pituk**

Affiliation:
Department of Mathematics and Computing, University of Veszprém, P. O. Box 158, 8201 Veszprém, Hungary

Email:
pitukm@almos.vein.hu

DOI:
https://doi.org/10.1090/S0002-9939-05-08154-2

Keywords:
Difference equation,
Ces\`{a}ro summability,
Lyapunov exponent,
Laplace transform,
Tauberian theorems

Received by editor(s):
June 21, 2004

Published electronically:
May 4, 2005

Additional Notes:
This research was supported in part by the Hungarian National Foundation for Scientific Research (OTKA) Grant No. T 046929

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.