Boundedness and square integrability of solutions of certain third-order differential equations

Beldjerd, Djamila; Remili, Moussadek

doi:10.21136/mb.2018.0029-17

Cited by 3 publications

(2 citation statements)

References 8 publications

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“…A very comprehensive bibliography for third-order differential equations is contained in [1][2][3][4][5]. In these works, in the space of continuous functions, the properties of oscillatory and non-oscillating solutions were studied, as well as the boundedness and asymptotic stability of solutions to third-order differential equations.…”

Section: Introduction Formulation Of Results Examplementioning

confidence: 99%

CMMSE:Estimates of approximation numbers, nuclearity of the resolvent of a third order singular differential operator and the completeness of its root vectors

Muratbekov,

Muratbekov

2023

Preprint

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In this paper, we consider a third order singular differential operator L w + μ w = - w ′′′ + q ( x ) w + μ w in space L 2 ( R ) originally defined on the set C 0 ∞ ( R ) , where C 0 ∞ ( R ) is the set of infinitely differentiable compactly supported functions, μ≥0. Regarding the coefficient q ( x ) , we assume that it is a continuous function in R ( - ∞ , ∞ ) and can be a growing function at infinity. The operator L allows closure in the space L 2 ( R ) and the closure also be denoted by L. In the paper, under certain restrictions on q ( x ) , in addition to the above condition, the existence of the resolvent of the operator L and the existence of the estimate ‖ - w ′′′ ‖ L 2 ( R ) + ‖ q ( x ) w ‖ L 2 ( R ) ≤ c ( ‖ L w ‖ L 2 ( R ) + ‖ w ‖ L 2 ( R ) ) (0.1) have been proved, where c>0 is a constant. Example. Let q ( x ) = e 100 | x | , then the estimate (0.1) holds. The compactness of the resolvent is proved and two-sided estimates for singular numbers ( s-numbers) are obtained. Here we note that the estimates of singular numbers ( s-numbers) show the rate of approximation of the resolvent of the operator L by linear finite-dimensional operators. In the present paper, apparently for the first time, the nuclearity of the resolvent of the third-order differential operator and completeness of its root vectors are proved in the case of an unbounded domain with a greatly growing coefficient q ( x ) at infinity.

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Section: Introduction Formulation Of Results Examplementioning

confidence: 99%

CMMSE:Estimates of approximation numbers, nuclearity of the resolvent of a third order singular differential operator and the completeness of its root vectors

Muratbekov,

Muratbekov

2023

Preprint

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“…The questions of solvability and regularity of solutions of third-order differential equations given in an infinite domain were considered in a number of papers (see in previous works [2,3] and references therein). The third-order differential equations study is being actively continued at the present time (e.g., see previous studies [4,5]). However, in these works, the coefficients of the differential equation are assumed to be bounded and smooth.…”

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confidence: 99%

The maximal regularity of the third‐order differential equation and its applications

Ospanov,

Ospanov

2024

Math Methods in App Sciences

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In this work, we study the solvability and maximal regularity questions for the singular third‐order differential equation with unbounded coefficients and some applications. Unlike previously studied cases, the leading and intermediate coefficients of this equation can grow independently. We obtain sufficient coefficient conditions for correctness of this equation and compactness for inverse of the corresponding differential operator. We also prove the maximal regularity estimate for a generalized solution. Using these results, we obtain upper and lower estimates for the number of Kolmogorov ‐diameters of the set associated with the linear Korteweg–de Vries equation's solutions. We give an example and show that from the above inequalities follow two‐sided estimates of the Kolmogorov ‐diameters.

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Maximal Regularity Estimates and the Solvability of Nonlinear Differential Equations

Ospanov

2022

Mathematics

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We study a type of third-order linear differential equations with variable and unbounded coefficients, which are defined in an infinite interval. We also consider a non-linear generalization with coefficients that depends on an unknown function. We establish sufficient conditions for the correctness of this linear equation and the maximal regularity estimate for their solution. Using these results, we prove the solvability of a nonlinear differential equation and estimate the norms of its terms.

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Boundedness and square integrability of solutions of certain third-order differential equations

Abstract: We establish some new sufficient conditions which guarantee the boundedness and square integrability of solutions of certain third order differential equation. Example is included to illustrate the results. By this work, we extend and improve some results in the literature.

Cited by 3 publications

References 8 publications

CMMSE:Estimates of approximation numbers, nuclearity of the resolvent of a third order singular differential operator and the completeness of its root vectors

CMMSE:Estimates of approximation numbers, nuclearity of the resolvent of a third order singular differential operator and the completeness of its root vectors

The maximal regularity of the third‐order differential equation and its applications

Maximal Regularity Estimates and the Solvability of Nonlinear Differential Equations

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