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Asymptotic Properties of Mild Solutions of Nonautonomous Evolution Equations with Applications to Retarded Differential Equations
, 1999
"... . We investigate asymptotic properties of mild solutions of the inhomogeneous nonautonomous evolution equation d dt u(t) = (A+B(t))u(t)+f(t); t 2 R, where (A; D(A)) is a HilleYosida operator on a Banach space X, B(t), t 2 R, is a family of operators in L(D(A); X) satisfying certain boundedness ..."
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Cited by 5 (1 self)
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. We investigate asymptotic properties of mild solutions of the inhomogeneous nonautonomous evolution equation d dt u(t) = (A+B(t))u(t)+f(t); t 2 R, where (A; D(A)) is a HilleYosida operator on a Banach space X, B(t), t 2 R, is a family of operators in L(D(A); X) satisfying certain boundedness and measurability conditions, and f 2 L 1 loc (R; X). The mild solutions of the corresponding homogeneous problem are represented by an evolution family (UB (t; s)) ts . For various function spaces F we derive conditions on (UB (t; s)) ts and f which ensure the existence of a unique mild solution u contained in F . In particular, if (UB (t; s)) ts is pperiodic and f is bounded there is a unique bounded mild solution u subject to certain spectral assumptions on UB (p; 0), f , and u. We apply our results to discuss asymptotic properties of nonautonomous retarded differential equations. For certain pperiodic retarded differential equations we derive a characteristic equation which ...
DISCRETE SPECTRUM AND ALMOST PERIODICITY
"... Abstract. The purpose of this note is to show that solutions of first and second order Cauchy problems with almost periodic inhomogeneity are almost periodic on the real line whenever the spectrum of the underlying operator is discrete. 1. ..."
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Cited by 4 (0 self)
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Abstract. The purpose of this note is to show that solutions of first and second order Cauchy problems with almost periodic inhomogeneity are almost periodic on the real line whenever the spectrum of the underlying operator is discrete. 1.
A Characteristic Equation For NonAutonomous Partial Functional Differential Equations
, 2000
"... . We characterize the exponential dichotomy of nonautonomous partial functional differential equations by means of a spectral condition extending known characteristic equations for the autonomous or time periodic case. From this we deduce robustness results. We further study the almost periodic ..."
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Cited by 3 (3 self)
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. We characterize the exponential dichotomy of nonautonomous partial functional differential equations by means of a spectral condition extending known characteristic equations for the autonomous or time periodic case. From this we deduce robustness results. We further study the almost periodicity of solutions to the inhomogeneous equation. Our approach is based on the spectral theory of evolution semigroups. 1. Introduction For the autonomous partial functional differential equation u(t) = Au(t) + Lu t ; t 0; u(t) = OE(t); \Gammar t 0; (1.1) there is a well developed semigroup approach; in particular, a powerful spectral theory is available. Here we assume that A generates a C 0 semigroup V (\Delta) on a Banach space X. Further, r 0, OE 2 E := C([\Gammar; 0]; X); L 2 L(E; X), and we let u t () := u(t + ) for 2 [\Gammar; 0], t 0, and u : [\Gammar; 1) ! X. Then the operator AL := d d ; D(AL ) := fOE 2 C 1 ([\Gammar; 0]; X) : OE(0) 2 D(A); OE 0 (0) = AOE(0) + ...
Equality Of Two Spectra Arising In Harmonic Analysis And Semigroup Theory
, 2000
"... . In [5, 6] a new notion of a spectrum of u 2 L 1 (R+ ; X) (X is a Banach space) is defined. We show that this spectrum coincides with the Arveson spectrum of some shift group, provided u is uniformly continuous. We apply this result to prove a new version of a tauberian theorem. In the last decad ..."
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Cited by 1 (0 self)
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. In [5, 6] a new notion of a spectrum of u 2 L 1 (R+ ; X) (X is a Banach space) is defined. We show that this spectrum coincides with the Arveson spectrum of some shift group, provided u is uniformly continuous. We apply this result to prove a new version of a tauberian theorem. In the last decade there has been an extensive study of the problem to determine the asymptotic behaviour of a bounded and uniformly continuous function u : R+ ! X (X a Banach space) via tauberian theorems; cf. [3, 6] for accounts of this theory and its applications to evolution equations. There are mainly two approaches to this problem: the operator theoretical approach by C 0 semigroups andgroups and the approach by harmonic analysis (Fourier transforms). In both approaches, the tauberian conditions are of spectral type. It is the purpose of this note to show that two spectra from both approaches actually coincide, and that in this sense the two approaches are equivalent. We also restate the main tauber...
EXISTENCE OF ALMOST AUTOMORPHIC SOLUTIONS TO SOME CLASSES OF NONAUTONOMOUS HIGHERORDER DIFFERENTIAL EQUATIONS
"... Abstract. In this paper, we obtain the existence of almost automorphic solutions to some classes of nonautonomous higher order abstract differential equations with Stepanov almost automorphic forcing terms. A few illustrative examples are discussed at the very end of the paper. 1. ..."
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Abstract. In this paper, we obtain the existence of almost automorphic solutions to some classes of nonautonomous higher order abstract differential equations with Stepanov almost automorphic forcing terms. A few illustrative examples are discussed at the very end of the paper. 1.
periodic equation, monodromy operator, spectrum of a function, spectral criterion. Abbreviated title: A Spectral Criterion for Almost Periodicity
"... This paper is concerned with equations of the form: u ′ = A(t)u + f(t) , where A(t) is (unbounded) periodic linear operator and f is almost periodic. We extend a central result on the spectral criteria for almost periodicity of solutions of evolution equations to some classes of periodic equations ..."
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This paper is concerned with equations of the form: u ′ = A(t)u + f(t) , where A(t) is (unbounded) periodic linear operator and f is almost periodic. We extend a central result on the spectral criteria for almost periodicity of solutions of evolution equations to some classes of periodic equations which says that if u is a bounded uniformly continuous mild solution and P is the monodromy operator, then their spectra satisfy eispAP (u) ⊂ σ(P) ∩ S1, where S1 is the unit circle. This result is then applied to find almost periodic solutions to the abovementioned equations. In particular, parabolic and functional differential equations are considered. Existence conditions for almost periodic and quasiperiodic solutions are discussed.
ftp ejde.math.txstate.edu ALMOST PERIODIC SOLUTIONS OF HIGHER ORDER DIFFERENTIAL EQUATIONS ON HILBERT SPACES
"... Abstract. We find necessary and sufficient conditions for the differential equation u (n) (t) = Au(t) + f(t), t ∈ R to have a unique almost periodic solution. Some applications are also given. 1. ..."
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Abstract. We find necessary and sufficient conditions for the differential equation u (n) (t) = Au(t) + f(t), t ∈ R to have a unique almost periodic solution. Some applications are also given. 1.
© Hindawi Publishing Corp. ASYMPTOTIC ALMOST PERIODICITY OF CSEMIGROUPS
, 2002
"... Let {T(t)}t≥0 be a Csemigroup on a Banach space X with generator A. Wewill investigate the asymptotic almost periodicity of {T(t)} via the HilleYosida space of its generator. 2000 Mathematics Subject Classification: 47D60, 47D62, 47D06. ..."
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Let {T(t)}t≥0 be a Csemigroup on a Banach space X with generator A. Wewill investigate the asymptotic almost periodicity of {T(t)} via the HilleYosida space of its generator. 2000 Mathematics Subject Classification: 47D60, 47D62, 47D06.