Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line (original) (raw)
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Journal of Mathematical Analysis and Applications, 2001
In this paper we investigate the characterization of dichotomies of an evolution family = U t s t≥s≥0 of bounded linear operators on Banach space X. We introduce operators I 0 and I X on subspaces of L p R + X using the integral equation u t = U t s u s + t s U t ξ f ξ dξ. The exponential and ordinary dichotomies of are characterized by properties of I 0 I X .
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Journal of Dynamics and Differential Equations, 1998
We prove that the exponential dichotomy of a strongly continuous evolution family on a Banach space is equivalent to the existence and uniqueness of continuous bounded mild solutions of the corresponding inhomogeneous equation. This result addresses nonautonomous abstract Cauchy problems with unbounded coefficients. The technique used involves evolution semigroups. Some applications are given to evolution families on scales of Banach
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This dissertation entitled ON EVOLUTION EQUATIONS IN BANACH SPACES AND COMMUTING SEMIGROUPS by SAUD M. A. ALSULAMI has been approved for the Department of Mathematics and the College of Arts and Sciences by Quoc-Phong Vu Professor of Mathematics Leslie A. Flemming Dean, College of Arts and Sciences ALSULAMI, SAUD M. A. Ph.D. June 2005. Mathematics On Evolution Equations In Banach Spaces And Commuting Semigroups (102 pp.) Director of Dissertation: Quoc-Phong Vu This dissertation is concerned with several questions about the qualitative behavior of mild solutions of differential equations with multi-dimensional time on Banach spaces. For the differential equations ∂u(s,t) ∂s
Journal of Mathematical Analysis and Applications, 2009
Consider an evolution family U = (U (t, s)) t s 0 on a half-line R + and a semi-linear integral equation u(t) = U (t, s)u(s) + t s U (t, ξ) f (ξ, u(ξ)) dξ. We prove the existence of stable manifolds of solutions to this equation in the case that (U (t, s)) t s 0 has an exponential dichotomy and the nonlinear forcing term f (t, x) satisfies the non-uniform Lipschitz conditions: f (t, x 1) − f (t, x 2) ϕ(t) x 1 − x 2 for ϕ being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of L p type, the Lorentz spaces L p,q and many other function spaces occurring in interpolation theory.
On Uniform Exponential Dichotomy of Evolution Operators
Annals of West University of Timisoara - Mathematics, 2012
The present paper presents three distinct concepts of uniform exponential dichotomy for evolution operators in Banach spaces. Characterizations and relationships between them and some illustrative counterexamples are given.