Invertibility and dichotomy of differential operators on a half-line

Ben-Artzi, Asher; Gohberg, Israel; Kaashoek, M. A.

doi:10.1007/bf01063733

Cited by 44 publications

(32 citation statements)

References 9 publications

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“…Recently there has been an increasing interest in the asymptotic behavior of solutions of differential equations in Banach spaces, in particular, in the unbounded case (see, e.g., [2], [3], [7], [8], [9], [13], [16], [22]). In this direction, we would like to mention a recent paper [14] in which a new characterization of exponential dichotomy was given in Hilbert spaces using only conditions on d dt − A(t) and A(t) − d dt (more precisely, its closure).…”

Section: Huy Ieotmentioning

confidence: 99%

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Exponentially Dichotomous Operators and Exponential Dichotomy of Evolution Equations on the Half-Line

Nguyen

2004

Integral Equations and Operator Theory

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To an evolution family on the half-line U = (U (t, s)) t≥s≥0 of bounded operators on a Banach space X we associate operators I X and I Z related to the integral equationdξ and a closed subspace Z of X. We characterize the exponential dichotomy of U by the exponential dichotomy and the quasi-exponential dichotomy of the operators I X and I Z , respectively. (2000). Primary 34G10, Secondary 47H20. Mathematics Subject Classification

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Section: Huy Ieotmentioning

confidence: 99%

“…In this paper we shall use the concept of exponentially dichotomous operators (which is introduced in [1,3]) and introduce the concept of quasi-exponentially dichotomous operators to characterize the exponential dichotomy of evolution family U. Our main results are contained in Theorems 3.4, 3.5 and 4.1.…”

Section: Huy Ieotmentioning

confidence: 99%

Exponentially Dichotomous Operators and Exponential Dichotomy of Evolution Equations on the Half-Line

Nguyen

2004

Integral Equations and Operator Theory

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“…t %=%+t on 3=R is well understood by now, see [4,5,20,22,34,36,37,39]. Here, the cocycle is given by 8 t (%)=U(%+t, %) for a strongly continuous evolution family [U(t, s)] t s on a Banach space X which can be thought as solution operator for the evolution equation u* (t)=A(t) u(t) with unbounded operators A(t).…”

Section: And1mentioning

confidence: 97%

Evolution Semigroups, Translation Algebras, and Exponential Dichotomy of Cocycles

Latushkin

Schnaubelt

1999

Journal of Differential Equations

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We study the exponential dichotomy of an exponentially bounded, strongly continuous cocycle over a continuous flow on a locally compact metric space 3 acting on a Banach space X. Our main tool is the associated evolution semigroup on C 0 (3; X ). We prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup. To show the latter equivalence, we establish the spectral mappingÂannular hull theorem for the evolution semigroup. In addition, dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c 0 (Z; X ). Here we develop Banach algebra techniques and study weighted translation algebras that contain the evolution operators. These results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space. Also, exponential dichotomy follows from pointwise discrete dichotomies with uniform constants. Finally, we extend to our situation the classical Perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded, continuous, mild solutions to the inhomogeneous equation. Academic Press

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“…, in the study of expo-nential dichotomy, have been also considered by Ben-Artzi, Gohberg and Kaashoek in [1].…”

mentioning

confidence: 99%

“…In the last few years, outstanding results concerning the asymptotic behaviour of evolution equations have been obtained using discrete-time methods (see [1]- [3], [6], [7], [14]). In [3], Chow and Leiva gave discrete and continuous characterizations for exponential dichotomy of linear skew-product semiflows.…”

Section: Introductionmentioning

confidence: 99%