Cesàro summability in a linear autonomous difference equation
Author:
Mihály Pituk
Journal:
Proc. Amer. Math. Soc. 133 (2005), 33333339
MSC (2000):
Primary 39A11; Secondary 34K40
Published electronically:
May 4, 2005
MathSciNet review:
2161157
Fulltext PDF Free Access
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]
Gustav
Doetsch, Handbuch der LaplaceTransformation. Band I. Theorie der
LaplaceTransformation, Verlag Birkhäuser, Basel, 1950 (German).
MR
0043253 (13,230f)
 [2]
Rodney
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
𝑥′(𝑡)=𝑎𝑥(𝑡)+𝑏𝑥(𝑡𝜏)
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), no. 1,
105–112. MR 1111217
(92i:39004), http://dx.doi.org/10.1090/S00029939199211112170
 [5]
Miguel
V. S. Frasson and Sjoerd
M. Verduyn Lunel, Large time behaviour of linear functional
differential equations, Integral Equations Operator Theory
47 (2003), no. 1, 91–121. MR 2015849
(2004j:34141), http://dx.doi.org/10.1007/s000200031155x
 [6]
Jack
Hale, Theory of functional differential equations, 2nd ed.,
SpringerVerlag, New YorkHeidelberg, 1977. Applied Mathematical Sciences,
Vol. 3. 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), no. 3, 219–233. MR 1697057
(2000b:39001), http://dx.doi.org/10.1080/102361908808184
 [8]
Ch.
G. Philos and I.
K. Purnaras, An asymptotic result for some delay difference
equations with continuous variable, Adv. Difference Equ.
1 (2004), 1–10. MR 2059199
(2005d:39071), http://dx.doi.org/10.1155/S1687183904310058
 [9]
D. V. Widder, Introduction to Transform Theory, Academic Press, New York, 1971.
 [1]
 G. Doetsch, Handbuch der LaplaceTransformation 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. 103119. 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), 990995. 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), 105112. 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), 91121. MR 2015849 (2004j:34141)
 [6]
 J. Hale, Theory of Functional Differential Equations, SpringerVerlag, 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), 219233. 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), 110. MR 2059199
 [9]
 D. V. Widder, Introduction to Transform Theory, Academic Press, New York, 1971.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
39A11,
34K40
Retrieve articles in all journals
with MSC (2000):
39A11,
34K40
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:
http://dx.doi.org/10.1090/S0002993905081542
PII:
S 00029939(05)081542
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.
