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On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems
Dichotomy and periodic solutions to partial functional differential equations
1. | School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Vien Toan ung dung va Tin hoc, Dai hoc Bach khoa Hanoi, 1 Dai Co Viet, Hanoi, Viet Nam |
2. | Thai Binh College of Education and Training, Cao dang Su pham Thai Binh, Chu Van An, Quang Trung, Thai Binh, Viet Nam |
We establish a sufficient condition for existence and uniqueness of periodic solutions to partial functional differential equations of the form $\dot{u}=A(t)u+F(t)(u_t)+g(t,u_t)$ on a Banach space $X$ where the operator-valued functions $t\mapsto A(t)$ and $t\mapsto F(t)$ are $1$-periodic, the nonlinear operator $g(t,φ)$ is $1$-periodic with respect to $t$ for each fixed $φ∈ {\mathcal{C}}:=C([-r,0],X)$, and satisfying $\|g(t,φ_1)-g(t,φ_2)\|≤\varphi(t)\|φ_1-φ_2\|_C$ for $φ_1, φ_2∈ {\mathcal{C}}$ with $\varphi$ being a positive function such that $\sup_{t≥0}∈t_{t}^{t+1}\varphi(τ)dτ < ∞$. We then apply the results to study the existence, uniqueness, and conditional stability of periodic solutions to the above equation in the case that the family $(A(t))_{t≥ 0}$ generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.
References:
[1] |
T. Burton,
Stability and Periodic Solutions of Ordinary and Functional Differential Equations Academic Press, Orlando, Florida. 1985. |
[2] |
Ju. L. Daleckii and M. G. Krein,
Stability of Solutions of Differential Equations in Banach Spaces Transl. Amer. Math. Soc. Provindence RI, 1974. |
[3] |
K. J. Engel and R. Nagel,
One-parameter Semigroups for Linear Evolution Equations, Graduate Text Math. , 194 Springer-Verlag, Berlin-Heidelberg, 2000. |
[4] |
M. Geissert, M. Hieber and N.T. Huy,
A general approach to time periodic incompressible viscous fluid flow problems, Arch. Ration. Mech. Anal., 220 (2016), 1095-1118.
doi: 10.1007/s00205-015-0949-8. |
[5] |
N. T. Huy,
Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354.
doi: 10.1016/j.jfa.2005.11.002. |
[6] |
N. T. Huy,
Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line, J. Math. Anal. Appl., 354 (2009), 372-386.
doi: 10.1016/j.jmaa.2008.12.062. |
[7] |
N. T. Huy,
Periodic motions of Stokes and Navier-Stokes flows around a rotating obstacle, Arch. Ration. Mech. Anal., 213 (2014), 689-703.
doi: 10.1007/s00205-014-0744-y. |
[8] |
N. T. Huy and T. V. Duoc,
Integral manifolds for partial functional differential equations in admissibility spaces on a half-line, J. Math. Anal. Appl., 411 (2014), 816-828.
doi: 10.1016/j.jmaa.2013.10.027. |
[9] |
N. T. Huy and N. Q. Dang,
Existence, uniqueness and conditional stability of periodic solutions to evolution equations, J. Math. Anal. Appl., 433 (2016), 1190-1203.
doi: 10.1016/j.jmaa.2015.07.059. |
[10] |
N. T. Huy and N. Q. Dang,
Periodic solutions to evolution equations: Existence, conditional stability and admissibility of function spaces, Ann. Polon. Math., 116 (2016), 173-195.
|
[11] |
J. H. Liu, G. M. N'Guerekata and N. V. Minh,
Topics on Stability and Periodicity in Abstract Differential Equations Series on Concrete and Applicable Mathematics -Vol. 6, World Scientific Publishing, Singapore, 2008.
doi: 10.1142/9789812818249. |
[12] |
J. L. Massera,
The existence of periodic solutions of systems of differential equations, Duke Math. J., 17 (1950), 457-475.
|
[13] |
J. L. Massera and J. J. Schäffer,
Linear Differential Equations and Function Spaces Academic Press, New York, 1966. |
[14] |
J. L. Massera and J. J. Schäffer,
Linear differential equations and functional analysis, Ⅰ, Ann. of Math., 67 (1958), 517-573.
doi: 10.2307/1969871. |
[15] |
J. L. Massera and J. J. Schäffer,
Linear differential equations and functional analysis, Ⅱ. Equations with periodic coefficients, Ann. of Math., 69 (1959), 88-104.
doi: 10.2307/1970095. |
[16] |
N. V. Minh, F. Räbiger and R. Schnaubelt,
Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integr. Eq. and Oper. Theory, 32 (1998), 332-353.
doi: 10.1007/BF01203774. |
[17] |
R. Miyazaki, D. Kim, T. Naito and J. S. Shin,
Fredholm operators, evolution semigroups, and periodic solutions of nonlinear periodic systems, J. Differential Equations, 257 (2014), 4214-4247.
doi: 10.1016/j.jde.2014.08.007. |
[18] |
R. Nagel and G. Nickel,
Well-posedness for non-autonomous abstract Cauchy problems, Prog. Nonl. Diff. Eq. Appl., 50 (2002), 279-293.
|
[19] |
A. Pazy,
Semigroup of Linear Operators and Application to Partial Differential Equations Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[20] |
J. Prüss,
Periodic solutions of semilinear evolution equations, Nonlinear Anal., 3 (1979), 601-612.
doi: 10.1016/0362-546X(79)90089-0. |
[21] |
J. Prüss, Periodic solutions of the thermostat problem, Differential equations in Banach Spaces (Book's Chapter), 216-226, Lecture Notes in Math. , 1223, Springer, Berlin, 1986.
doi: 10.1007/BFb0099195. |
[22] |
J. Serrin,
A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 3 (1959), 120-122.
doi: 10.1007/BF00284169. |
[23] |
J. S. Shin and T. Naito,
Representations of solutions, translation formulae and asymptotic behavior in discrete linear systems and periodic continuous linear systems, Hiroshima Math. J., 44 (2014), 75-126.
|
[24] |
T. Yoshizawa,
Stability Theory and the Existence Of Periodic Solutions and Almost Periodic Solutions Applied Mathematical Sciences, 14. Springer-Verlag, New York-Heidelberg, 1975. |
[25] |
O. Zubelevich,
A note on theorem of Massera, Regul. Chao. Dyn., 11 (2006), 475-481.
doi: 10.1070/RD2006v011n04ABEH000365. |
show all references
References:
[1] |
T. Burton,
Stability and Periodic Solutions of Ordinary and Functional Differential Equations Academic Press, Orlando, Florida. 1985. |
[2] |
Ju. L. Daleckii and M. G. Krein,
Stability of Solutions of Differential Equations in Banach Spaces Transl. Amer. Math. Soc. Provindence RI, 1974. |
[3] |
K. J. Engel and R. Nagel,
One-parameter Semigroups for Linear Evolution Equations, Graduate Text Math. , 194 Springer-Verlag, Berlin-Heidelberg, 2000. |
[4] |
M. Geissert, M. Hieber and N.T. Huy,
A general approach to time periodic incompressible viscous fluid flow problems, Arch. Ration. Mech. Anal., 220 (2016), 1095-1118.
doi: 10.1007/s00205-015-0949-8. |
[5] |
N. T. Huy,
Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354.
doi: 10.1016/j.jfa.2005.11.002. |
[6] |
N. T. Huy,
Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line, J. Math. Anal. Appl., 354 (2009), 372-386.
doi: 10.1016/j.jmaa.2008.12.062. |
[7] |
N. T. Huy,
Periodic motions of Stokes and Navier-Stokes flows around a rotating obstacle, Arch. Ration. Mech. Anal., 213 (2014), 689-703.
doi: 10.1007/s00205-014-0744-y. |
[8] |
N. T. Huy and T. V. Duoc,
Integral manifolds for partial functional differential equations in admissibility spaces on a half-line, J. Math. Anal. Appl., 411 (2014), 816-828.
doi: 10.1016/j.jmaa.2013.10.027. |
[9] |
N. T. Huy and N. Q. Dang,
Existence, uniqueness and conditional stability of periodic solutions to evolution equations, J. Math. Anal. Appl., 433 (2016), 1190-1203.
doi: 10.1016/j.jmaa.2015.07.059. |
[10] |
N. T. Huy and N. Q. Dang,
Periodic solutions to evolution equations: Existence, conditional stability and admissibility of function spaces, Ann. Polon. Math., 116 (2016), 173-195.
|
[11] |
J. H. Liu, G. M. N'Guerekata and N. V. Minh,
Topics on Stability and Periodicity in Abstract Differential Equations Series on Concrete and Applicable Mathematics -Vol. 6, World Scientific Publishing, Singapore, 2008.
doi: 10.1142/9789812818249. |
[12] |
J. L. Massera,
The existence of periodic solutions of systems of differential equations, Duke Math. J., 17 (1950), 457-475.
|
[13] |
J. L. Massera and J. J. Schäffer,
Linear Differential Equations and Function Spaces Academic Press, New York, 1966. |
[14] |
J. L. Massera and J. J. Schäffer,
Linear differential equations and functional analysis, Ⅰ, Ann. of Math., 67 (1958), 517-573.
doi: 10.2307/1969871. |
[15] |
J. L. Massera and J. J. Schäffer,
Linear differential equations and functional analysis, Ⅱ. Equations with periodic coefficients, Ann. of Math., 69 (1959), 88-104.
doi: 10.2307/1970095. |
[16] |
N. V. Minh, F. Räbiger and R. Schnaubelt,
Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integr. Eq. and Oper. Theory, 32 (1998), 332-353.
doi: 10.1007/BF01203774. |
[17] |
R. Miyazaki, D. Kim, T. Naito and J. S. Shin,
Fredholm operators, evolution semigroups, and periodic solutions of nonlinear periodic systems, J. Differential Equations, 257 (2014), 4214-4247.
doi: 10.1016/j.jde.2014.08.007. |
[18] |
R. Nagel and G. Nickel,
Well-posedness for non-autonomous abstract Cauchy problems, Prog. Nonl. Diff. Eq. Appl., 50 (2002), 279-293.
|
[19] |
A. Pazy,
Semigroup of Linear Operators and Application to Partial Differential Equations Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[20] |
J. Prüss,
Periodic solutions of semilinear evolution equations, Nonlinear Anal., 3 (1979), 601-612.
doi: 10.1016/0362-546X(79)90089-0. |
[21] |
J. Prüss, Periodic solutions of the thermostat problem, Differential equations in Banach Spaces (Book's Chapter), 216-226, Lecture Notes in Math. , 1223, Springer, Berlin, 1986.
doi: 10.1007/BFb0099195. |
[22] |
J. Serrin,
A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 3 (1959), 120-122.
doi: 10.1007/BF00284169. |
[23] |
J. S. Shin and T. Naito,
Representations of solutions, translation formulae and asymptotic behavior in discrete linear systems and periodic continuous linear systems, Hiroshima Math. J., 44 (2014), 75-126.
|
[24] |
T. Yoshizawa,
Stability Theory and the Existence Of Periodic Solutions and Almost Periodic Solutions Applied Mathematical Sciences, 14. Springer-Verlag, New York-Heidelberg, 1975. |
[25] |
O. Zubelevich,
A note on theorem of Massera, Regul. Chao. Dyn., 11 (2006), 475-481.
doi: 10.1070/RD2006v011n04ABEH000365. |
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