MR3379136
Hideaki MATSUNAGA, Satoru MURAKAMI, Yutaka NAGABUCHI and Nguyen VAN MINH
Hideaki MATSUNAGA, Satoru MURAKAMI, Yutaka NAGABUCHI and Nguyen VAN MINH
Center Manifold Theorem and Stability for Integral Equations with Infinite Delay
Funkcialaj Ekvacioj. Serio Internacia
58
2015
87--134
http://fe.math.kobe-u.ac.jp/FE/FullPapers/58-1/58_87.pdf
http://www.ams.org/mathscinet-getitem?mr=MR3379136
The present paper deals with autonomous integral equations with infinite delay via dynamical system approach. Existence, local exponential attractivity, and other properties of center manifold are established by means of the variation-of-constants formula in the phase space that is obtained in our previous paper [22] (Funkcial. Ekvac. 55 (2012), 479--520). Furthermore, we prove a stability reduction principle by which the stability of an autonomous integral equation is implied by that of an ordinary differential equation which we call the central equation.
Integral equations with infinite delay, Center manifolds, Stability properties, Phase space, A variation-of-constants formula.
Primary: 45M10; Secondary: 34K20.
58-87
2015
Center Manifold Theorem and Stability for Integral Equations with Infinite Delay
Hideaki MATSUNAGA, Satoru MURAKAMI, Yutaka NAGABUCHI and Nguyen VAN MINH
Hideaki MATSUNAGA, Satoru MURAKAMI, Yutaka NAGABUCHI and Nguyen VAN MINH
1
Aulbach, B.; Van Minh, N.
Nonlinear semigroups and the existence, stability of semilinear nonautonomous evolution equations
Abstr. Appl. Anal.
1
1996
351-380
MR1481548
2
Carr, J. L.
Applications of Centre Manifold Theory
Applied Mathematical Sciences, 35, Springer-Verlag, New York-Berlin
1981
MR0635782
3
Chen, X.-Y.; Hale, J. K.; Tan, B.
Invariant foliations for $C^1$ semigroups in Banach spaces
J. Differential Equations
139
1997
283-318
MR1472350
4
Chicone, C.; Latushkin, Y.
Center manifolds for infinite dimensional nonautonomous differential equations
J. Differential Equations
141
1997
356-399
MR1488358
5
Coddington, E. A.; Levinson, N.
Theory of Ordinary Differential Equations
Krieger, Marbar, Florida
1984
MR0069338
6
Diekmann, O.; van Gils, S. A.; Verduyn Lunel, S. M.; Walther, H.-O.
Delay Equations
Applied Mathematical Sciences, 110, Springer-Verlag, New York
1995
MR1345150
7
Diekmann, O.; Gyllenberg, M.
Equations with infinite delay: blending the abstract and the concrete
J. Differential Equations
252
2012
819-851
MR2853522
8
Faria, T.
Normal forms and Hopf bifurcations for partial differential equations with delays
Trans. Amer. Math. Soc.
352
2000
2217-2238
MR1491862
9
Faria, T.; Huang, W.; Wu, J.
Smoothness of center manifolds for maps and formal adjoints for semilinear FDEs in general Banach spaces
SIAM J. Math. Anal.
34
2002
173-203
MR1950831
10
Hale, J. K.
Ordinary Differential Equations
Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience, New York-London-Sydney
1969
MR0419901
11
Hale, J. K.; Verduyn Lunel, S. M.
Introduction to Functional Differential Equations
Applied Mathematical Sciences, 99. Springer-Verlag, New York
1993
MR1243878
12
Henry, D.
Geometric Theory of Semilinear Parabolic Equations
Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York
1981
MR0610244
13
Hino, Y.; Murakami, S.; Naito, T.
Functional Differential Equations with Infinite Delay
Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin
1991
MR1122588
14
Hino, Y.; Murakami, S.; Naito, T.; Van Minh, N.
A variation of constants formula for abstract functional differential equations in Banach spaces
J. Differential Equations
179
2002
336-355
MR1883747
15
Hino, Y.; Murakami, S.; Van Minh, N.
Decomposition of variation of constants formula for abstract functional
differential equations
Funkcial. Ekvac.
45
2002
341-372
MR1975051
1975051
16
Hirsh, M. W.; Pugh, C.; Shub, M.
Invariant Manifolds
Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York
1977
MR0501173
17
Krisztin, T.
Invariance and noninvariance of center manifolds of time-$t$ maps with respect to the semiflow
SIAM J. Math. Anal.
36
2005
717-739
MR2111913
18
Lang, S.
Fundamentals of Differential Geometry
Graduate Texts in Mathematics, 191, Springer-Verlag, New York
1999
MR1666820
19
Magal, P.; Ruan, S.
Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models
Mem. Amer. Math. Soc.
202
2009
no. 951, 71pp
MR2559965
20
Mallet-Paret, J.; Sell, G.
Inertial manifolds for reaction diffusion equations in higher space dimensions
J. Amer. Math. Soc.
1
1988
805-866
MR0943276
21
Matsunaga, H.; Murakami, S.
Some invariant manifolds for functional difference equations with infinite delay
J. Difference Equ. Appl.
10
2004
661-689
MR2064815
22
Matsunaga, H.; Murakami, S.; Van Minh, N.
Decomposition and variation-of-constants formula in the phase space for integral equations
Funkcial. Ekvac.
55
2012
479-520
MR3052749
3052749
23
Matsunaga, H.; Murakami, S.; Nagabuchi, Y.; Nakano, Y.
Formal adjoint equations and asymptotic formula for solutions of Volterra difference equations with infinite delay
J. Difference Equ. Appl.
18
2012
57-88
MR2864813
24
Matsunaga, H.; Murakami, S.; Nagabuchi, Y.
Formal adjoint operators and asymptotic formula for solutions of autonomous linear integral equations
J. Math. Anal. Appl.
410
2014
807-826
MR3111868
25
Memory, M. C.
Stable and unstable manifolds for partial functional differential equations
Nonlinear Anal.
16
1991
131-142
MR1090786
26
Van Minh, N.; Wu, J.
Invariant manifolds of partial functional differential equations
J. Differential Equations
198
2004
381-421
MR2039148
27
Murakami, S.; Nagabuchi, Y.
Invariant manifolds for abstract functional differential equations and related Volterra difference equation in a Banach space
Funkcial. Ekvac.
50
2007
133-170
MR2332082
2332082
28
Murakami, S.; Van Minh, N.
Some invariant manifolds for abstract functional differential equations and linearized stabilities
Vietnam J. Math.
30
2002
suppl., 437-458
MR1964235
29
Pliss, V. A.
A reduction principle in the theory of stability of motion
Izv. Akad. Nauk SSSR Ser. Mat.
28
1964
1297-1324
MR0190449
30
Shub, M.
Global Stability of Dynamical Systems
Springer-Verlag, New York
1987
MR0869255
31
Vanderbauwhed, A.; van Gils, S. A.
Center manifolds and contractions on a scale of Banach spaces
J. Funct. Anal.
72
1987
209-224
MR0886811
32
Wu, J.
Theory and Applications of Partial Functional Differential Equations
Applied Mathematical Sciences, 119. Springer-Verlag, New York
1996
MR1415838
33
Yoshizawa, T.
Stability Theory by Liapunov's Second Method
Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo
1966
MR0208086