Exponential dichotomy and invariant manifolds of semi-linear differential equations on the line
DOI:
https://doi.org/10.24193/subbmath.2024.1.09Keywords:
Exponential dichotomy, invariant manifolds, semi-linear differential equationsAbstract
In this paper we investigate the homogeneous linear differential equation vi(t) = A(t)v(t) and the semi-linear differential equation vi(t) = A(t)v(t) + g(t, v(t)) in Banach space X, in which A : R → L(X) is a strongly continuous function, g : R × X → X is continuous and satisfies ϕ-Lipschitz condition. The first we characterize the exponential dichotomy of the associated evolution family with the homogeneous linear differential equation by space pair (E, E∞), 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 C1.
Mathematics Subject Classification (2010): 34C45, 34D09, 34D10.
Received 14 June 2021; Accepted 09 September 2022
References
Barreira, L., Valls, C., Strong and weak admissibility of L∞ spaces, Electron. J. Qual. Theory Differ. 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., Providence, 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., Hájek, 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., Schäffer, J.J., Linear Differential Equations and Function Spaces, Aca- demic Press, New York, 1966.
Perko, L., Differential Equations and Dynamical Systems, Springer, New York, 2001.
Perron, O., Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32(1930), 703-728.
Räbiger, 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. Differ. 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.
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