MR2108678
Takeshi NISHIKAWA, NGUYEN Van Minh and Toshiki NAITO
Takeshi NISHIKAWA, NGUYEN Van Minh and Toshiki NAITO
On the Asymptotic Periodic Solutions of Abstract Functional Differential Equations
Funkcialaj Ekvacioj. Serio Internacia
47
2004
307--327
http://fe.math.kobe-u.ac.jp/FE/FullPapers/47-2/47_307.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2108678
The paper is concerned with conditions for all mild solutions of abstract functional differential equations with finite delay in a Banach space to be periodic and asymptotic periodic, where forcing term is a continuous 1-periodic function. The obtained results extend various recent ones on the subject.
Abstract functional differential equation, Asymptotic periodic solutions.
Primary 34K14, 34K30; Secondary 34G10, 34C27.
47-307
2004
On the Asymptotic Periodic Solutions of Abstract Functional Differential Equations
Takeshi NISHIKAWA, NGUYEN Van Minh and Toshiki NAITO
Takeshi NISHIKAWA, NGUYEN Van Minh and Toshiki NAITO
1
Arendt, W.; Batty, C. J. K.
Asymptotically almost periodic solutions of inhomogeneous cauchy problem on the half-line
Bull. London Math. Soc.
31
1999
291-304
MR1673408
2
Arendt, W.; Räbiger, F.; Sourour, A.
Spectral properties of the operators equations $AX+XB=Y$
Quart. J. Math. Oxford (2)
45
1994
133-149
MR1280689
3
Arendt, W.; Batty, C. J. K.; Hieber, M.; Neubrander, F.
Vector-Valued Laplace Transforms and Cauchy Problems
Birkhäser, Basel-Boston-Berlin
2001
MR1886588
4
Bart, H.; Goldberg, S.
Characterizations of almost periodic strongly continuous groups and semigroups
Math. Ann.
236
1978
105-116
MR0493506
5
Cassell, J. S.; Zhan Yuan Hou
Asymptotically diagonal linear differential equations with retardation
J. London Math. Soc. (2)
47
1993
473-483
MR1214909
6
Chow, S. N.; Hale, J. K.
Strongly limit-compact maps
Funkc. Ekvac.
17
1974
31-38
MR0350529
0350529
7
Daners, D.; Medina, P. K.
Abstract Evolution Equations, Periodic Problems and Applications
Pitman Research Notes in Math. Ser. volume 279, Longman. New York
1992
MR1204883
8
Engel, K. J.; Nagel, R.
One-parameter Semigroups for linear Evolution Equations
Springer, Berlin
1999
MR1721989
9
Fink, A. M.
Almost Periodic Differential Equations
Lecture Notes in Math., 377, Springer, Berlin-New York
1974
MR0460799
10
Furumochi, T.; Naito, T.; Nguyen Van Minh
Boundedness and almost periodicity of solutions of partial functional differential equations
J. Differential Equations
180
2002
125-152
MR1890601
11
Gonzalez, C.; Jimenez-Melado, A.
Asymptotic behavior of solutions of difference equations in Banach spaces
Proc. Amer. Math. Soc.
128
2000
1743-1749
MR1695135
12
Gyori, I.; Krisztin, T.
Oscillation results for linear autonomous partial delay differential equations
J. Math. Anal. Appl.
174
1993
204-217
MR1212927
13
Haddock, J. R.; Nkashama, M. N.; Wu, J.
Asymptotic constancy for pseudomonotone dynamical systems on function spaces
J. Differential Equations
100
1992
292-311
MR1194812
14
Hale, J. K.; Verduyyn Lunel, S. M.
Introdcution to Functional Differential Equations
Springer-Verlag
1993
MR1243878
15
Hara, T.; Miyazaki, R.; Morii, T.
Asymptotic behavior solutions for differential-difference equations with N delays and Hopf bifurcation
Funkcialaj Ekvacioj
43
2000
193-212
MR1795970
1795970
16
Hatvani, L.
Asymptotic constancy of the solutions of a nonlinear delay equation
Sivasundaram, S. (ed.) et al., Advances in nonlinear dynamics. Langhorne, PA: Gordon and Breach. Stab. Control Theory Methods Appl.
5
1997
63-70
MR1479419
17
Hatvani, L.; Krisztin, T.
On the existence of periodic solutions for linear inhomogeneous and quasilinear functional differential equations
J. Differential Equations
97
1992
1-15
MR1161308
18
Hino, Y.; Murakami, S.
Periodic solutions of a linear volterra system
Differential equations (Xanthi, 1987), Lecture Notes in Pure and Appl. Math., 118, Dekker, New York
1987
319-326
MR1021730
19
Hino, Y.; Murakami, S.; Naito, T.
Functional Differential Equations with Infinite Delay
Lecture Notes in Math., 1473, Springer-Verlag, Berlin-New York
1991
MR1122588
20
Hino, Y.; Naito, T.; Nguyen Van Minh; Shin, J. S.
Almost Periodic Solutions of Differential Equations in Banach Spaces
Gordon and Breach, Amsterdam-London
2002
MR1933682
21
Hino, Y.; Murakami, S.; Naito, T.; Nguyen Van Minh
A variation of constants formula for functional differential equations in the phase space
J. Differential Equations
179
2002
336-355
MR1883747
22
Langenhop, C.
Periodic and almost periodic solutions of volterra integral differential equations with infinite memory
J. Differential Equations
58
1985
391-403
MR0797318
23
Levitan, B. M.; Zhikov, V. V.
Almost Periodic Functions and Differential Equations
Cambridge University Press
1982
MR0690064
24
Li, Y.; Lin, Z.; Li, Z.
A Massera type criterion for linear functional differential equations with advanced and delay
J. Math. Anal. Appl.
200
1996
715-725
MR1393112
25
Li, Y.; Cong, F.; Lin, Z.; Liu, W.
Periodic solutions for evolution equations
Nonlinear Anal.
36
1999
275-293
MR1688231
16
Massera, J. L.
The existence of periodic solutions of systems of differential equations
Duke Math. J.
17
1952
457-475
MR0040512
27
Murakami, S.; Naito, T.; Nguyen Van Minh
Evolution semigroups and sums of commuting operators: a new approach to the admissibility theory of function spaces
J. Differential Equations
164
2000
240-285
MR1765556
28
Murakami, S.; Naito, T.; Nguyen Van Minh, Massera's theorem for almost periodic solutions of functional differential equations, To appear in J. Math. Soc. Japan
MR2027625
29
Murakami, K.
Asymptotic constancy and periodic solutions for linear autonomous delay differential equations
Funkcial. Ekvac.
39
1996
519-540
MR1433915
1433915
30
Murakami, K.
Asymptotic constancy for systems of delay differential equations
Proceedings of the Second World Congress of Nonlinear Analysts, Part 7 (Athens, 1996). Nonlinear Anal.
30
1997
4595-4606
MR1603444
31
Naito, T.; Nguyen Van Minh
Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations
J. Differential Equations
152
1999
358-376
MR1674561
32
Naito, T.; Nguyen Van Minh; Miyazaki, R.; Shin, J. S.
A decomposition theorem for bounded solutions and the existence of periodic solutions to periodic differential equations
J. Differential Equations
160
2000
263-282
MR1734534
33
Naito, T.; Nguyen Van Minh; Shin, J. S.
New spectral criteria for almost periodic solutions of evolution equations
Studia Mathematica
145
2001
97-111
MR1827999
34
Naito, T.; Nguyen Van Minh; Liu, J., On the bounded solutions of Volterra equations, Applicable Analysis. To appear
MR2054638
35
van Neerven, J.
The asymptotic behaviour of semigroups of linear operators
Operator theory: Advances and Applications, 88. Birkhäuser Verlag, Basel
1996
MR1409370
36
Pazy, A.
Semigroups of Linear Operators and Applications to Partial Differential Equations
Applied Math. Sci., 44, Spriger-Verlag, Berlin-New York
1983
MR0710486
37
Prüss, J.
Bounded solutions of Volterra equations
SIAM Math. Anal.
19
1987
133-149
MR0924550
38
Prüss, J.
Evolutionary Integral Equations and Applications
Birkhäuser, Basel
1993
MR1238939
39
Vu, Q. P.
The operator equation $AX-XB=C$ with unbounded operators $A$ and $B$ and related abstract Cauchy problems
Math. Z.
208
1991
567-588
MR1136476
40
Shin, J. S.; Naito, T.
Semi-fredholm operators and periodic solutions for linear functional differential equations
J. Differential Equations
153
1999
407-441
MR1683628
41
Shin, J. S.; Naito, T.; Nguyen Van Minh
Existence and uniqueness of periodic solutions to periodic linear functional differential equations with finite delay
Funkcial. Ekvac.
44
2001
53-71
MR1847836
1847836
42
Travis, C. C.; Webb, G. F.
Existence and stability for partial functional differential equations
Trans. Amer. Math. Soc.
200
1974
394-418
MR0382808
43
Wu, J.
Theory and Applications of Partial Functional Differential Equations
Applied Math. Sci., 119, Springer, Berlin-New York
1996
MR1415838
44
Yoshizawa, T.
Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions
Applied Math., Sciences 14, Springer-Verlag, New York
1975
MR0466797