Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T04:09:36.499Z Has data issue: false hasContentIssue false
  • English
  • Français

ASYMPTOTIC EQUIVALENCE OF ALMOST PERIODIC SOLUTIONS FOR A CLASS OF PERTURBED ALMOST PERIODIC SYSTEMS

Published online by Cambridge University Press:  25 August 2010

MANUEL PINTO ,
VICTOR TORRES  and
GONZALO ROBLEDO
MANUEL PINTO
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago 7800024, Chile e-mail: [email protected]
VICTOR TORRES
Affiliation:
Departamento de Ciencias Físicas y Matemáticas, Universidad Arturo Prat, Avda. Arturo Prat 2120, Iquique 1110939, Chile e-mail: [email protected]
GONZALO ROBLEDO
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago 7800024, Chile e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The solutions of a perturbed linear ordinary differential equation (ODE) system are studied. Provided that some integrability and oddness conditions are satisfied, we show that they are asymptotically equivalent at t = ±∞ to the solutions of the unperturbed one. This fact is used to determine the existence of almost periodic or pseudo-almost periodic solutions of the perturbed system.

Keywords

34E1034C2734C41
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 3 , September 2010 , pp. 583 - 592
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Akhmet, M. U., Tleubergenova, M. A. and Zafer, A., Asymptotic equivalence of differential equations and asymptotically almost periodic solutions, Nonlinear Anal. 67 (2007), 1870–1877.CrossRefGoogle Scholar
2.Alonso, A., Hong, J. and Obaya, R., Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic sequences, Appl. Math. Lett. 13 (2000), 131–137.CrossRefGoogle Scholar
3.Bodine, S. and Lutz, D. A., On asymptotic equivalence of perturbed linear systems of differential and difference equations, J. Math. Anal. Appl. 326 (2007), 1174–1189.CrossRefGoogle Scholar
4.Brauer, F. and Wong, J. S. W., On asymptotic behavior of perturbed linear systems, J. Differ. Equ. 6 (1969), 142–153.CrossRefGoogle Scholar
5.Burton, T. A., Linear differential equations with periodic coefficients, Proc. Amer. Math. Soc. 17 (1966), 327–329.Google Scholar
6.Coppel, W. A.. Dichotomies in stability theory. Lecture Notes in Mathematics Vol. 629 (Springer–Verlag, Berlin, New York, 1978).CrossRefGoogle Scholar
7.Cuevas, C. and Pinto, M., Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non dense domain, Nonlinear Anal. 45 (2001), 73–83.Google Scholar
8.Delsarte, J., Essai sur l'application de la théorie des fonctions presque périodiques à l'arithmétique, Ann. Sci. l'École Normale Supérieure Ser. 3, 62 (1945), 185–204.CrossRefGoogle Scholar
9.Epstein, I. J., Periodic solutions of systems of differential equations, Proc. Amer. Math. Soc. 13 (1962), 690–694.Google Scholar
10.Fink, A. M., Almost periodic differential equations. Lecture Notes in Mathematics (Springer-Verlag, Berlin, New York, 1974).Google Scholar
11.Gramain, F., Fonctions presque périodiques (au sens de Bohr) et fonction zéta de Riemann. Séminaire Delange-Pisot-Poitou. Théorie des nombres, 14 no. 2 (1972–1973), Exposé G17. 6p.Google Scholar
12.Hu, Z. S. and Mingarelli, A. B., On a question in the theory of almost periodic differential equations, Proc. Amer. Math. Soc. 127 (1999), 2665–2670.Google Scholar
13.Jiang, H., Zhang, L. and Teng, Z., Existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time varying delays, IEEE Trans. Neural Netw. 16 (2005), 1340–1351.Google Scholar
14.Lenas, P. and Pavlou, S., Periodic, quasi–periodic and chaotic coexistence of two competing microbial populations in a periodically operated chemostat, Math. Biosci. 121 (1994), 61–110.CrossRefGoogle Scholar
15.Liu, B. and Huang, L., Existence and exponential stability of almost periodic solutions for cellular neural networks with continuously distributed delays, J. Korean Math. Soc. 43 (2006), 445–459.Google Scholar
16.Pankov, A. A., Bounded and almost periodic solutions of nonlinear operator differential Equations, Mathematics and its applications 55 (Kluwer Academic Publishers, Dordrecht, 1990).CrossRefGoogle Scholar
17.Piao, D., Pseudo almost periodic solutions for differential equations involving reflection of the argument, J. Korean Math. Soc. 41 (2004), 747–754.CrossRefGoogle Scholar
18.Pinto, M., Pseudo almost periodic solutions of neutral integral and differential equations with applications, Nonlinear Anal. Theory Methods Appl. 72 (2010), 4377–4383.Google Scholar
19.Van Vleck, F. S., A note on the relation between periodic and orthogonal fundamental solutions of linear systems II, Amer. Math. Mon. 71 (1964), 774–776.CrossRefGoogle Scholar
20.Wolkowicz, G. S. K. and Zhao, X. Q., N–specie competition in a periodic chemostat, Differ. Integ. Equ. 11 (1998), 465–491.Google Scholar
21.Zhang, C., Pseudo almost periodic solutions of some differential equations II, J. Math. Anal. Appl. 192 (1995), 543–561.CrossRefGoogle Scholar
22.Zhang, C., Almost periodic type functions and ergodicity (Science Press, Beijing; Kluwer Academic Publishers, Dordrecht, 2003).Google Scholar