Dichotomy and periodic solutions to partial functional differential equations

Previous Article
On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems
DCDS-B Home
This Issue
Next Article
Stable foliations near a traveling front for reaction diffusion systems

October 2017, 22(8): 3127-3144. doi: 10.3934/dcdsb.2017167

Dichotomy and periodic solutions to partial functional differential equations

Nguyen Thieu Huy ^1, and Ngo Quy Dang ^2,

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

Received March 2016 Revised December 2016 Published June 2017

Fund Project: The authors thank the referee of this paper for his/her comments, suggestions and corrections which help to improve the paper. The work of the first author is partly supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM). This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) by Grant Number 101.02-2014.02

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.

Keywords: Partial functional differential equations, periodic solutions, exponential dichotomy, conditional stability, local stable manifolds.

Mathematics Subject Classification: 34K19, 35B10.

Citation: Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

References:

[1]	T. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations Academic Press, Orlando, Florida. 1985. Google Scholar
[2]	Ju. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces Transl. Amer. Math. Soc. Provindence RI, 1974. Google Scholar
[3]	K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Text Math. , 194 Springer-Verlag, Berlin-Heidelberg, 2000. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	J. L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J., 17 (1950), 457-475. Google Scholar
[13]	J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces Academic Press, New York, 1966. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	R. Nagel and G. Nickel, Well-posedness for non-autonomous abstract Cauchy problems, Prog. Nonl. Diff. Eq. Appl., 50 (2002), 279-293. Google Scholar
[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. Google Scholar
[20]	J. Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal., 3 (1979), 601-612. doi: 10.1016/0362-546X(79)90089-0. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	O. Zubelevich, A note on theorem of Massera, Regul. Chao. Dyn., 11 (2006), 475-481. doi: 10.1070/RD2006v011n04ABEH000365. Google Scholar

show all references

References:

[1]	T. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations Academic Press, Orlando, Florida. 1985. Google Scholar
[2]	Ju. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces Transl. Amer. Math. Soc. Provindence RI, 1974. Google Scholar
[3]	K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Text Math. , 194 Springer-Verlag, Berlin-Heidelberg, 2000. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	J. L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J., 17 (1950), 457-475. Google Scholar
[13]	J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces Academic Press, New York, 1966. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	R. Nagel and G. Nickel, Well-posedness for non-autonomous abstract Cauchy problems, Prog. Nonl. Diff. Eq. Appl., 50 (2002), 279-293. Google Scholar
[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. Google Scholar
[20]	J. Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal., 3 (1979), 601-612. doi: 10.1016/0362-546X(79)90089-0. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	O. Zubelevich, A note on theorem of Massera, Regul. Chao. Dyn., 11 (2006), 475-481. doi: 10.1070/RD2006v011n04ABEH000365. Google Scholar

[1]	Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993
[2]	Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
[3]	Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[4]	Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure & Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291
[5]	Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113
[6]	Vera Ignatenko. Homoclinic and stable periodic solutions for differential delay equations from physiology. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3637-3661. doi: 10.3934/dcds.2018157
[7]	Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27
[8]	Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031
[9]	Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
[10]	Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157
[11]	Alin Pogan, Kevin Zumbrun. Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks. Kinetic & Related Models, 2019, 12 (1) : 1-36. doi: 10.3934/krm.2019001
[12]	Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure & Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016
[13]	Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control & Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697
[14]	Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 209-226. doi: 10.3934/dcdsb.2017011
[15]	Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
[16]	Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009
[17]	Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227
[18]	Lok Ming Lui, Yalin Wang, Tony F. Chan, Paul M. Thompson. Brain anatomical feature detection by solving partial differential equations on general manifolds. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 605-618. doi: 10.3934/dcdsb.2007.7.605
[19]	Luis Barreira, Claudia Valls. Characterization of stable manifolds for nonuniform exponential dichotomies. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1025-1046. doi: 10.3934/dcds.2008.21.1025
[20]	Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048

American Institute of Mathematical Sciences