On the almost automorphy of bounded solutions of differential equations with piecewise constant argument

Journal of Mathematical Analysis and Applications

Volumn 326, Issue 1, 2007, Pages 165-178

  Minh, Nguyen Van ^a   Dat, Tran Tat ^b  

^a UNIVERSITY OF WEST GEORGIA   (United States)

^b VIETNAM NATIONAL UNIVERSITY   (Viet Nam)

Author keywords

Almost automorphic solutions; Differential equation piecewise constant argument

Indexed keywords

EID: 33750610961     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2006.02.079     Document Type: Article

References (47)

1. Alonso A.I., Hong J., and Rojo J. A class of ergodic solutions of differential equation with piecewise continuous delays. Dynam. Systems Appl. 7 (1998) 561-574
2. Arendt W., and Batty C.J.K. Almost periodic solutions of first and second oder Cauchy problems. J. Differential Equations 137 2 (1997) 363-383
3. Arendt W., Batty C.J.K., Hieber M., and Neubrander F. Vector-Valued Laplace Transforms and Cauchy Problems. Monogr. Math. vol. 96 (2001), Birkhäuser, Basel
4. Basit B. Generalization of two theorems of M.I. Kadets concerning the indefinite integral of abstract almost periodic functions. Mat. Zametki 9 (1971) 311-321 (in Russian)
5. Busenberg S., and Cooke K.L. Models of vertically transmitted diseases with sequential-continuous dynamics. In: Lakshmikantham V. (Ed). Nonlinear Phenomena in Mathematical Sciences (1982), Academic Press, New York 179-187
6. Cabada A., Ferreiro J.B., and Nieto J.J. Green's function and comparison principles for first order periodic differential equations with piecewise constant arguments. J. Math. Anal. Appl. 291 (2004) 690-697
7. Cooke K.L., and Wiener J. Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl. 99 (1984) 265-297
8. Cooke K.L., and Wiener J. Neutral differential equations with piecewise constant argument. Boll. Unione Mat. Ital. 7 (1987) 321-349
9. Cooke K.L., and Wiener J. A survey of differential equation with piecewise continuous delays. Lecture Notes in Math. vol. 1475 (1991), Springer-Verlag, Berlin 1-15
10. Dat T.T. On the existence of almost periodic, periodic and quasi periodic solutions of neutral differential equations with piecewise constant argument. Int. J. Evol. Equ. 1 (2005) 29-44
11. Diagana T., N'Guérékata G.M., and Minh N.V. Almost automorphic solutions of evolution equations. Proc. Amer. Math. Soc. 132 (2004) 3289-3298
12. Eduardo M.H. A Massera type criterion for a partial neutral functional differential equation. Electron. J. Differential Equations 2002 40 (2002) 1-17
13. Furumochi T., Naito T., and Minh N.V. Boundedness and almost periodicity of solutions of partial functional differential equations. J. Differential Equations 180 (2002) 125-152
14. Hino Y., and Murakami S. Almost automorphic solutions for abstract functional differential equations. J. Math. Anal. Appl. 286 (2003) 741-752
15. Hino Y., Naito T., Minh N.V., and Shin J.S. Almost Periodic Solutions of Differential Equations in Banach Spaces (2002), Taylor and Francis, London
16. Küpper T., and Yuan R. On quasi-periodic solutions of differential equations with piecewise constant argument. J. Math. Anal. Appl. 267 (2002) 173-193
17. Levitan B.M., and Zhikov V.V. Almost Periodic Functions and Differential Equations. Monogr. Math. vol. 96 (1978), Moscow Univ. Publ. House English translation by Cambridge University Press, 1982
18. Liu J., N'Guérékata G.M., and Minh N.V. A Massera type theorem for almost automorphic solutions of differential equations. J. Math. Anal. Appl. 299 2 (2004) 587-599
19. Géza Makay, On some possible extension of Massera's theorem, EJQTDE, Proc. 6th Coll., QTDE, 16, 2000
20. Massera J.L. The existence of periodic solutions of systems of differential equations. Duke Math. J. 17 (1950) 457-475
21. Minh Man N., and Minh N.V. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Commun. Pure Appl. Anal. 3 (2004) 291-300
22. Nguyen Van Minh, Toshiki Naito, G.M. N'Guérékata, A spectral countability condition for almost automorphy of solutions of abstract differential equations, Proc. Amer. Math. Soc., in press
23. Murakami S., Naito T., and Minh N.V. Evolution semigroups and sums of commuting operators: A new approach to the admissibility theory of function spaces. J. Differential Equations 164 (2000) 240-285
24. Murakami S., Naito T., and Minh N.V. Massera Theorem for almost periodic solutions of functional differential equations. J. Math. Soc. Japan 56 1 (2004) 247-268
25. Naito T., Minh N.V., and Shin J.S. New spectral criteria for almost periodic solutions of evolution equations. Studia Math. 145 (2001) 97-111
26. Naito T., Minh N.V., Miyazaki R., and Hamaya Y. Boundedness and almost periodicity in dynamical systems. J. Difference Equ. Appl. 7 (2001) 507-527
27. Naito T., Miyazaki R., Minh N.V., and Shin J.S. A decomposition theorem for bounded solutions and the existence of periodic solutions to periodic equations. J. Differential Equations 160 1 (2000) 263-282
28. N'Guérékata G.M. Almost automorphic functions and applications to abstract evolution equations. Contemp. Math. vol. 252 (1999) 71-76
29. N'Guérékata G.M. Almost Automorphic and Almost Periodic Functions in Abstract Spaces (2001), Kluwer, Amsterdam
30. N'Guérékata G.M. Topics in Almost Automorphy (2005), Springer, New York
31. Papaschinopoulos G. Some results concerning a class of differential equations with piecewise constant argument. Math. Nachr. 166 (1994) 193-206
32. Papaschinopoulos G. Exponential dichotomy, topological equivalence and structural stability for differential equations with piecewise constant argument. Analysis 14 (1994) 239-247
33. Papaschinopoulos G. On the integral manifold for a system of differential equations with piecewise constant argument. J. Math. Anal. Appl. 201 (1996) 75-90
34. Papaschinopoulos G. Linearization near the integral manifold for a system of differential equations with piecewise constant argument. J. Math. Anal. Appl. 215 (1997) 317-333
35. Prüss J. Evolutionary Integral Equations and Applications (1993), Birkhäuser, Basel
36. Seifert G. Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence. J. Differential Equation 164 (2000) 451-458
37. Seifert G. Second order neutral delay differential equations with piecewise constant time dependence. J. Math. Anal. Appl. 281 (2003) 1-9
38. Shah S.M., and Wiener J. Advanced differential equations with piecewise constant argument deviations. Int. J. Math. Math. Sci. 6 (1983) 671-703
39. Thanh N.T. Massera criterion for periodic solutions of differential equations with piecewise constant argument. J. Math. Anal. Appl. 302 (2005) 256-268
40. Wiener J. Generalized Solutions of Functional Differential Equations (1993), World Scientific, Singapore
41. Wiener J., and Cooke K.L. Oscillations in systems of differential equations with piecewise constant argument. J. Math. Anal. Appl. 137 (1989) 221-239
42. Wiener J., and Heller W. Oscillatory and periodic solutions to a diffusion equation of neutral type. Int. J. Math. Math. Sci. 22 2 (1999) 313-348
43. Yuan R., and Hong J. Almost periodic solutions of differential equations with piecewise constant argument. Analysis 16 (1996) 171-180
44. Yuan R., and Hong J. The existence of almost periodic solution for a class of differential equations with piecewise constant argument. Nonlinear Anal. 28 (1997) 1439-1450
45. Yuan R. Almost periodic solutions of a class of singularly perturbed differential equations with piecewise constant argument. Nonlinear Anal. 37 (1999) 641-659
46. Yuan R. On the logistic delay differential equations with piecewise constant argument and quasi periodic time dependence. J. Math. Anal. Appl. 274 (2002) 124-133
47. Zhang F., Zhao A., and Yan J. Monotone iterative method for differential equations with piecewise constant arguments. Port. Math. 57 Fasc. 3 (2000)