Exponential dichotomy and invariant manifolds of semi-linear differential equations on the line

Trinh Viet Duoc; Nguyen Ngoc Huy

doi:10.24193/subbmath.2024.1.09

Authors

Trinh Viet Duoc University of Science, Vietnam National University, Hanoi
Nguyen Ngoc Huy Thuyloi University

DOI:

https://doi.org/10.24193/subbmath.2024.1.09

Keywords:

Exponential dichotomy, invariant manifolds, semi-linear differential equations

Abstract

In this paper we investigate the homogeneous linear differential equation \(v'(t)=A(t)v(t)\) and the semi-linear differential equation \(v'(t)=A(t)v(t)+g(t,v(t))\) in Banach space \(X\), in which \(A:\mathbb{R} \to \mathcal{L}(X)\) is a strongly continuous function, \(g:\mathbb{R} \times X\to X\) is continuous and satisfies \(\varphi\)-Lipschitz condition. The first we characterize the exponential dichotomy of the associated evolution family with the homogeneous linear differential equation by space pair \((\mathcal{E},\mathcal{E}_{\infty})\), this is a Perron type result. Applying the achieved results, we establish the robustness of exponential dichotomy. The next we show the existence of stable and unstable manifolds for the semi-linear differential equation and prove that each a fiber of these manifolds is differentiable submanifold of class \(C^1\).

Author Biographies

Trinh Viet Duoc, University of Science, Vietnam National University, Hanoi

Faculty of Mathematics, Mechanics, and Informatics
Nguyen Ngoc Huy, Thuyloi University

Department of Mathematics

References

Barreira, L., Valls, C., Strong and weak admissibility of L1 spaces, Electron. J. Qual. Theory Di er. Equ., 78(2017), 1-22.

Chicone, C., Ordinary Differential Equations with Applications, Springer, New York, 2006.

Chicone, C., Latushkin, Y., Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys and Monographs vol. 70, Providence RI, Amer. Math. Soc., 1999.

Coppel, W.A., Dichotomies in Stability Theory, Lecture Notes in Math., 629, Springer, Berlin, 1978.

Daleckii, Ju.L., Krein, M.G., Stability of Solutions of Differential Equations in Banach Space, Transl. Math. Monogr., 43, Amer. Math. Soc., Provindence, RI, 1974.

Duoc, T.V., Bounded and periodic solutions of inhomogeneous linear evolution equations, Commun. Korean Math. Soc., 36(2021), no. 4, 759-770.

Elaydi, S., H ajek, O., Exponential dichotomy and trichotomy of nonlinear differential equations, Differential Integral Equations, 3(1990), 1201-1224.

Hirsch, N., Pugh, C., Shub, M., Invariant Manifolds, Lecture Notes in Math., vol. 183, Springer, New York, 1977.

Huy, N.T., Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235(2006), 330-354.

Huy, N.T., Invariant manifolds of admissible classes for semi-linear evolution equations, J. Differential Equations, 246(2009), 1820-1844.

Huy, N.T., Duoc, T.V., Integral manifolds and their attraction property for evolution equations in admissible function spaces, Taiwanese J. Math., 16(2012), 963-985.

Latushkin, Y., Randolph, T., Schnaubelt, R., Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces, J. Dynam. Differential Equations, 10(1998), 489-510.

Massera, J.L., Scha er, J.J., Linear Differential Equations and Function Spaces, Academic Press, New York, 1966.

Perko, L., Differential Equations and Dynamical Systems, Springer, New York, 2001.

Perron, O., Die stabilitatsfrage bei differentialgeighungen, Math. Z., 32(1930), 703-728.

Rabiger, F., Schnaubelt, R., The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions, Semigroup Forum, 52(1996), 225-239.

Sasu, A.L., Integral equations on function spaces and dichotomy on the real line, Integral Equations Operator Theory, 58(2007), 133-152.

Sasu, A.L., Pairs of function spaces and exponential dichotomy on the real line, Adv. Di er. Equ., (2010), Article ID 347670, 1-15.

Sasu, A.L., Sasu, B., Exponential dichotomy on the real line and admissibility of function spaces, Integral Equations Operator Theory, 54(2006), 113-130.