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Dichotomies for systems with discontinuous coefficients

Published online by Cambridge University Press:  17 April 2009

Raúl Naulin
Raúl Naulin
Affiliation:
Departamento de Matemática, Universidad de Oriente, Venezuela
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Abstract

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In this work we are concerned with the problem of the existence of an exponential dichotomy for the linear singularly perturbed system εx′ = A(t)x, where the matrix A(t) is piecewise uniformly continuous, that is, A(t) admits points of discontinuity but is uniformly continuous in any interval where it is continuous. We shall prove that the classical result regarding the existence of an exponential dichotomy extends to this case, when there is a constant γ > 0 such that |Reλ(t)| ≥ γ > 0 for any eigenvalue λ(t) of A(t). The proofs are obtained by means of the quasidiagonalisation of a non-constant matrix: For A(t), a piecewise uniformly continuous matrix and σ > 0 there exists a bounded, piecewise constant function L(t): J → ℂn×n, and a bounded matrix Δ(t, σ) such that L-1(t)A(t)L(t) = Λ(t) + Δ(t, σ), |Δ(t, σ)| ≤ σ, where Λ(t) is the diagonal matrix consisting of eigenvalues of A(t).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 45 , Issue 1 , February 1992 , pp. 135 - 141
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Ascher, U.M., Matheij, R.M.M. and Russel, R.D., Numerical solution of boundary value problems for ordinary differential equations (Prentice Hall, New Jersey, 1988).Google Scholar
[2]Bellman, R., Stability theory of differential equations (Dover Publications, New York, 1953).Google Scholar
[3]Chang, K.W., ‘Almost periodic solutions of singularly perturbed systems of differential equations’, J. Differential Equations 4 (1968), 300–307.CrossRefGoogle Scholar
[4]Coppel, W.A., Dichotomies in stability theory: Lecture notes in mathematics 629 (Springer Verlag, Berlin, 1978).CrossRefGoogle Scholar
[5]Javid, S.H., ‘Uniform asymptotic stability of linear time varying singularly perturbed systems’, J. Franklin Inst. 305 (1978), 27–37.CrossRefGoogle Scholar
[6]Coddington, E.A. and Levinson, N., Theory of ordinary differential equations (McGraw-Hill, New York, 1955).Google Scholar
[7]Lin, X.B., ‘Shadowing lemma and singularly perturbed boundary value problems’, SIAM J. Appl. Math. 49 (1989), 26–54.CrossRefGoogle Scholar
[8]Rožkov, V.I., ‘Periodic solutions of linear systems with a small parameter in the derivative’, Soviet Math. Dokl. 16 (1975).Google Scholar
[9]Palmer, K.J., ‘Exponential dichotomies and transversal homoclinic points’, J. Differential Equations 55 (1984), 225–256.CrossRefGoogle Scholar