Fundamentals of Functional Analysis

Кутателадзе, Семeн Самсонович

doi:10.1007/978-94-015-8755-6

Cited by 58 publications

(28 citation statements)

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“…This follows from the compactness of the operator ( ) 2 c P K − Λ + and the Fredholm Theorem (see. [20], p.146), that…”

Section: Proof Of Propositionmentioning

confidence: 99%

The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice

V.N.¹,

V.²,

Dushkin³

2014

AJAMS

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In this article the method for numerical solution of boundary integral equations of the original problem is proposed. This method is one of the modifications of Nystrom-type methods; particularly the method of discrete vortices. The convergence of the numerical solutions to the exact solution of the problem is guaranteed by propositions proved in this article. Also, the rate of convergence of the approximate solutions to the exact solution had been found. Keywords: singular integral equation, modification of method of discrete vortices, existence of approximate solution, the rate of convergence of the approximate solutionsCite This Article: Maurya V.N., Gandel Yu. V., and Dushkin V.D., "The Approximate Method for Solving the Boundary Integral

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“…This follows from the compactness of the operator ( ) 2 c P K − Λ + and the Fredholm Theorem (see. [20], p.146), that…”

Section: Proof Of Propositionmentioning

confidence: 99%

The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice

V.N.¹,

V.²,

Dushkin³

2014

AJAMS

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“…Using the bounded index stability theorem [17], the operator I + qΠ 1 is invertible, and hence the Ferdholm alternative can be applied to the singular integral equation (3.13). Its solution can be given by the following Neumann series, for which (3.16) is a sufficient condition to converge (see [22]),…”

Section: Write (31) and (32) In The Formmentioning

confidence: 99%

Neumann boundary value problems in fan-shaped domains

Akel¹,

Aldawsari²

2018

Turk J Math

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In this article we give the solvability conditions and the integral representations of the solutions of the Neumann boundary value problem for the Cauchy-Riemann operator and the Beltrami operator with constant coefficient in a disc sector with angle ϑ = π n , n ∈ N . Moreover, the Neumann problem for second-order operators with the Bitsadze/Laplace operator as the main part is studied. Classical results of complex analysis are used to obtain the expressions of the solvability conditions and the integral representations for the solutions explicitly.

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“…It is well known that if a convex functional defined on a topological linear space (say, on a Banach space) is bounded in a neighborhood of some point, then it is continuous (see, e.g., Kutateladze [20]). Thus, if the functional in Theorem 2.1 is defined, say, on L m ([0,n],λ), where λ is a finite measure, and satisfies the local boundedness condition, then the continuity condition connected with the step functions x (m) (t) can be omitted.…”

Section: Applications To Empirical Processesmentioning

confidence: 99%

“…Averaging both sides of this identity with respect to the other distributions, we then obtain (with more convenient representation of the remainder in (3.19)) the equality 20) where ζ is a random variable with the density 2(1 − t) on the unit interval, which is defined on the main probability space and independent of the sequence {X k } (we may assume here that this space is rich enough). It is worth noting that, because of integrability of the left-hand side of (3.19), the expectation on the right-hand side of (3.20) is well defined due to Fubini's theorem.…”

Section: Lemma 32 For Eachmentioning

confidence: 99%