Solvability of a class of mixed type second order equations and nonlocal estimates

Muratbekov, М.B.; Ospanov, Kordan N.; Igisinov, Sabit

doi:10.1016/j.aml.2012.01.033

Cited by 5 publications

(5 citation statements)

References 4 publications

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“…We give the sufficient conditions for a discreteness of spectrum and the bilateral estimates of s-numbers of the resolvent of operator corresponding to this system. We note that in [8][9][10][11][12][13], using the semiboundedness of Schrödinger operators the authors obtained similar results for second-order Schrödinger-type equations. But the operator corresponding to the system (1.3) is not semibounded and its spectrum, in general, is continuous.…”

Section: Introductionsupporting

confidence: 58%

Qualitative and approximate characteristics of solutions of Beltrami-type systems

Ospanov

2015

Complex Variables and Elliptic Equations

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Section: Introductionsupporting

confidence: 58%

Qualitative and approximate characteristics of solutions of Beltrami-type systems

Ospanov

2015

Complex Variables and Elliptic Equations

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“…To prove the following statements below, we use computations and arguments that have been used in [5].…”

Section: Auxiliary Lemmas and Inequalitiesmentioning

confidence: 99%

On Existence of the Resolvent and Discreteness of the Spectrum of a Class of Differential Operators of Hyperbolic Type

Muratbekov¹,

Muratbekov

2015

Trends in Mathematics

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The existence and compactness of the resolvent are studied in this paper. One of the main results is the criterion of discreteness of the spectrum of a hyperbolic singular differential operator. Singular differential operators, for example operators defined in an unbounded domain, in general may have not only a discrete but also a continuous spectrum. Therefore in general an arbitrary function cannot be decomposed into a series of eigenfunctions. For this reason the most important problem in the study of the spectrum in dependence of the behavior of the coefficients in the case of an unbounded domain is the discreteness of the spectrum. Spectral characteristics of singular elliptic differential operators are well-studied and the typical difficulties encountered in connection with bad behaving coefficients clarified. An extensive literature is devoted to their study and we mention [1-3]. Review of the literature shows that such questions as: 1) the existence and compactness of the resolvent, 2) the discreteness of the spectrum of hyperbolic differential operators defined in an unbounded domain are not well studied. We consider in the space L 2 (Ω) the differential operator of hyperbolic type A 0 u = u xx − u yy + a (y) u x + c (y) u with the domain D(A 0) of infinitely differentiable functions satisfying the conditions u (−π; y) = u (π; y) , u x (−π; y) = u x (π; y) and compactly supported with respect to the variable y, where Ω = {(x, y) : −π < x < π, −∞ < y < ∞}. Further, we assume that the coefficients a (y) , c (y) satisfy the conditions: i) |a(y)| ≥ δ 0 > 0, c(y) ≥ δ > 0 are continuous functions in R = (−∞; ∞). It is easy to verify that the operator A 0 admits closure in the space L 2 (Ω), which is denoted by A. We note that the operator A corresponds to the problem of propagation of the boundary regime (see [1], p. 106), i.e. the problem without initial conditions. Here the term au x describes the friction force. The question of the existence of solutions of the problem without initial conditions, in general, depends on the behavior of the coefficients a and c. For example, when a = 0, the solution does not always exist. The main results of this paper are the following theorems. 1

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“…It is known that under assumptions (i)-(iii), there exists a resolvent of the operator l t and the estimate [6][7][8] …”

Section: Proof Of Main Resultsmentioning

confidence: 99%

On spectral properties of a class of differential equations with operator coefficients

Muratbekov¹,

Musilimov²,

Suleimbekova³

2014

AIP Conference Proceedings

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Solvability of a class of mixed type second order equations and nonlocal estimates

Cited by 5 publications

References 4 publications

Qualitative and approximate characteristics of solutions of Beltrami-type systems

Qualitative and approximate characteristics of solutions of Beltrami-type systems

On Existence of the Resolvent and Discreteness of the Spectrum of a Class of Differential Operators of Hyperbolic Type

On spectral properties of a class of differential equations with operator coefficients

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