Finite-dimensional perturbations of Volterra operators

Хромов, А. П.

doi:10.1007/s10958-006-0346-9

Cited by 11 publications

(7 citation statements)

References 39 publications

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“…Consequently, one can apply Proposition 2.1 to conclude that A := J n−α A BC generates an EXS analytic semigroup on H = L 2 (0, 1), provided that σ(A) satisfies (2.1). Operators given by the RHS of (2.5) have been applied to another problemssee [22] and references therein.…”

Section: A Class Of Sturm-liouville Problems With Caputo Derivative Letmentioning

confidence: 99%

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On analytic semigroup generators involving Caputo fractional derivative

Grabowski

2022

EECT

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<p style='text-indent:20px;'>Our investigations are motivated by the well - posedness problem of some dynamical models with anomalous diffusion described by the Caputo spatial fractional derivative of order <inline-formula><tex-math id="M1">\begin{document}$ \alpha \in (1, 2) $\end{document}</tex-math></inline-formula>. We propose a characterization of an exponentially stable analytic semigroup generator using the inverse operator. This characterization enables us to establish the form of a generator involving the Caputo fractional derivative, under various boundary conditions. In particular, the results simplify those known from literature obtained by means of the fractional Sobolev spaces and some perturbation results. Going further, we show how to construct a control system in factor form, having such a generator as the state operator.</p>

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Section: A Class Of Sturm-liouville Problems With Caputo Derivative Letmentioning

confidence: 99%

“…here J 0 is the Bessel function of the first kind and zero index [4, vol. I, p. 132, formula (22) and vol. II, p. 5].…”

Section: A Class Of Sturm-liouville Problems With Caputo Derivative Letmentioning

confidence: 99%

On analytic semigroup generators involving Caputo fractional derivative

Grabowski

2022

EECT

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“…The resolvent of the generator D of that semigroup is a finitedimensional perturbation of the Volterra operator. Such operators attracted attention of many authors (see, e.g., [7] and the references therein).…”

Section: Proposition 12mentioning

confidence: 99%

Sharp estimates for solutions of systems with aftereffect

Власов¹,

Иванов²

2009

St. Petersburg Math. J.

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Abstract. Sharp estimates are established for strong solutions of systems of differential-difference equations of both neutral and retarded type.The approach is based on the study of the resolvent corresponding to the generator of the semigroup of shifts along the trajectories of a dynamical system. In the case of neutral type equations, the Riesz basis property of the subsystem of exponential solutions is used. §1. Introduction. Statement of the problem and main resultsIn this paper, we consider the initial value problem for a differential-difference equation of the formHere, A kj is a matrix of size r × r with constant complex entries, and the numbers h k are shifts, 0We denote by L(λ) the characteristic matrix of equation (1.1), The following statement about quasipolynomials is well known. Proposition 1.2. If det A 0m = 0, then the quantity κ + := sup λ q ∈Λ Re λ q is finite. Now we state the main result of the paper.2000 Mathematics Subject Classification. Primary 34K12.

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“…Investigation of operators (0.3) was initiated by A.P. Hromov [7]. Question of unconditional basisness plays an important role in the spectral theory.…”

Section: Preliminariesmentioning

confidence: 99%

Resolvent growth and Birkhoff-regularity

Minkin¹

2006

Journal of Mathematical Analysis and Applications

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We prove a long standing conjecture in the theory of two-point boundary value problems that unconditional basisness implies Birkhoff-regularity. It is a corollary of our two main results: minimal resolvent growth along a sequence of points implies nonvanishing of a regularity determinant, and sparseness of nthorder roots of eigenvalues in small sectors provided that eigen and associated functions of the boundary value problem form an unconditional basis.Considerations are based on a new direct method, exploiting almost orthogonality of Birkhoff's solutions of the equation l(y) = λy. This property was discovered earlier by the author.

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Finite-dimensional perturbations of Volterra operators

Cited by 11 publications

References 39 publications

On analytic semigroup generators involving Caputo fractional derivative

On analytic semigroup generators involving Caputo fractional derivative

Sharp estimates for solutions of systems with aftereffect

Resolvent growth and Birkhoff-regularity

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