Taiwanese Journal of Mathematics

Admissibly Stable Manifolds for a Class of Partial Neutral Functional Differential Equations on a Half-line

Thieu Huy Nguyen and Xuan Yen Trinh

Full-text: Open access

Abstract

For the following class of partial neutral functional differential equations \[ \begin{cases} \frac{\partial}{\partial t} Fu_t = B(t) u(t) + \Phi(t,u_t) &t \in (0,\infty), \\ u_0 = \phi \in \mathcal{C} := C([-r,0],X) \end{cases} \] we prove the existence of a new type of invariant stable and center-stable manifolds, called admissibly invariant manifolds of $\mathcal{E}$-class for the solutions. The existence of such manifolds is obtained under the conditions that the family of linear partial differential operators $(B(t))_{t \geq 0}$ generates the evolution family $\{U(t,s)\}_{t \geq s \geq 0}$ (on Banach space $X$) having an exponential dichotomy or trichotomy on the half-line and the nonlinear delay operator $\Phi$ satisfies the $\varphi$-Lipschitz condition, i.e., $\|\Phi(t,\phi)-\Phi(t,\psi)\| \leq \varphi(t) \|\phi-\psi\|_{\mathcal{C}}$ for $\phi,\psi \in \mathcal{C}$, where $\varphi(t)$ belongs to some admissible function space on the half-line. Our main method is based on Lyapunov-Perrons equations combined with the admissibility of function spaces and fixed point arguments.

Article information

Source
Taiwanese J. Math., Volume 23, Number 4 (2019), 897-923.

Dates
Received: 27 July 2018
Revised: 9 December 2018
Accepted: 17 December 2018
First available in Project Euclid: 18 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1563436874

Digital Object Identifier
doi:10.11650/tjm/181209

Mathematical Reviews number (MathSciNet)
MR3982067

Zentralblatt MATH identifier
07088953

Subjects
Primary: 34C45: Invariant manifolds 34G20: Nonlinear equations [See also 47Hxx, 47Jxx] 35B40: Asymptotic behavior of solutions 37D10: Invariant manifold theory

Keywords
exponential dichotomy partial neutral functional differential equations stable and center-stable manifolds admissibility of function spaces

Citation

Nguyen, Thieu Huy; Trinh, Xuan Yen. Admissibly Stable Manifolds for a Class of Partial Neutral Functional Differential Equations on a Half-line. Taiwanese J. Math. 23 (2019), no. 4, 897--923. doi:10.11650/tjm/181209. https://projecteuclid.org/euclid.twjm/1563436874


Export citation

References

  • R. Benkhalti, K. Ezzinbi and S. Fatajou, Stable and unstable manifolds for nonlinear partial neutral functional differential equations, Differential Integral Equations 23 (2010), no. 7-8, 773–794.
  • N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-linear Oscillations, Translated from the second revised Russian edition, International Monographs on Advanced Mathematics and Physics Hindustan, Gordon and Breach Science Publishers, New York, 1961.
  • ––––, The method of integral manifolds in nonlinear mechanics, Contributions to Differential Equations 2 (1963), 123–196.
  • J. L. Dalec'kiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Spaces, Translations of Mathematical Monographs 43, American Mathematical Society, Provindence, R.I., 1974.
  • J. Hadamard, Sur l'itération et les solutions asymptotiques des équations différentielles, Bull. Soc. Math. France 29 (1901), 224–228.
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, 1981.
  • N. T. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal. 235 (2006), no. 1, 330–354.
  • ––––, Invariant manifolds of admissible classes for semi-linear evolution equations, J. Differential Equations 246 (2009), no. 5, 1820–1844.
  • N. T. Huy and P. V. Bang, Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on A half-line, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 9, 2993–3011.
  • N. T. Huy and V. T. N. Ha, Admissibly integral manifolds for semilinear evolution equations, Ann. Polon. Math. 112 (2014), no. 2, 127–163.
  • J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics 21, Academic Press, New York, 1966.
  • R. Nagel and G. Nickel, Well-posedness for nonautonomous abstract Cauchy problems, in: Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), 279–293, Progr. Nonlinear Differential Equations Appl. 50, Birkhäuser, Basel, 2002.
  • T. H. Nguyen, Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line, J. Math. Anal. Appl. 354 (2009), no. 1, 372–386.
  • T. H. Nguyen and V. D. Trinh, Integral manifolds for partial functional differential equations in admissible spaces on a half-line, J. Math. Anal. Appl. 411 (2014), no. 2, 816–828.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, Berlin, 1983.
  • O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, (German) Math. Z. 29 (1929), no. 1, 129–160.
  • ––––, Die Stabilitätsfrage bei Differentialgleichungen, (German) Math. Z. 32 (1930), no. 1, 703–728.
  • H. Petzeltová and O. J. Staffans, Spectral decomposition and invariant manifolds for some functional partial differential equations, J. Differential Equations 138 (1997), no. 2, 301–327
  • F. Räbiger and R. Schnaubelt, The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions, Semigroup Forum 52 (1996), no. 2, 225–239.
  • N. Van Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory 32 (1998), no. 3, 332–353.
  • N. Van Minh and J. Wu, Invariant manifolds of partial functional differential equations, J. Differential Equations 198 (2004), no. 2, 381–421.
  • J. Wu, Theory and Applications of Partial Functional-differential Equations, Applied Mathematical Sciences 119, Springer-Verlag, New York, 1996.