# American Institute of Mathematical Sciences

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

## 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

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.

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. [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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##### 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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