Coercive solvability of odd-order differential equations and its applications

Muratbekov, М.B.; Muratbekov, Madi; Ospanov, Kordan N.

doi:10.1134/s1064562410060189

Cited by 10 publications

(7 citation statements)

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“…The theorem is proved. If r ¼ 0 and q x ð Þ ≥ δ > 0 in (6.1), then N λ,M ð Þsatisfies the following estimates (see [8]):…”

Section: Some Spectral Applicationsmentioning

confidence: 99%

“…If

r = 0

and

q (x) \geq δ > 0

in (), then

N (λ, M)

satisfies the following estimates (see [8]):

C_{1}^{- 1} λ^{- \frac{1}{3}} μ \{x \in R : q (x) \leq C_{2} λ^{- 2}\} \leq N (λ, M) \leq C_{1} λ^{- \frac{1}{3}} μ \{x \in R : q (x) \leq C_{2} λ^{- 2}\} .

…”

Section: Some Spectral Applicationsmentioning

confidence: 99%

“…We note that the coercive estimate problem for a solution of the two‐term equation

ly = - y^{'''} + qy = f (x),

where

q (x)

is continuous and

∣ q (x) ∣ \geq δ > 0

was considered in numerous works (see previous works [8, 9] and references therein).…”

Section: Introductionmentioning

confidence: 99%

“…If the operator

r (x) \frac{d}{italicdx}

is a small perturbation of

A = - \frac{d^{3}}{{dx}^{3}} + q (x) E

(

E

is the identity operator), then properties of () are same as those of equation () and the methods of [8, 9] can be extended to (). In the other cases, Equation () has rarely studied.…”

Section: Introductionmentioning

confidence: 99%

“…where q x ð Þ is continuous and j q x ð Þ j ≥ δ > 0 was considered in numerous works (see previous works [8,9] and references therein).…”

mentioning

confidence: 99%

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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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“…The theorem is proved. If r ¼ 0 and q x ð Þ ≥ δ > 0 in (6.1), then N λ,M ð Þsatisfies the following estimates (see [8]):…”

Section: Some Spectral Applicationsmentioning

confidence: 99%

“…If

r = 0

and

q (x) \geq δ > 0

in (), then

N (λ, M)

satisfies the following estimates (see [8]):

C_{1}^{- 1} λ^{- \frac{1}{3}} μ \{x \in R : q (x) \leq C_{2} λ^{- 2}\} \leq N (λ, M) \leq C_{1} λ^{- \frac{1}{3}} μ \{x \in R : q (x) \leq C_{2} λ^{- 2}\} .

…”

Section: Some Spectral Applicationsmentioning

confidence: 99%

“…We note that the coercive estimate problem for a solution of the two‐term equation

ly = - y^{'''} + qy = f (x),

where

q (x)

is continuous and

∣ q (x) ∣ \geq δ > 0

was considered in numerous works (see previous works [8, 9] and references therein).…”

Section: Introductionmentioning

confidence: 99%

“…If the operator

r (x) \frac{d}{italicdx}

is a small perturbation of

A = - \frac{d^{3}}{{dx}^{3}} + q (x) E

(

E

is the identity operator), then properties of () are same as those of equation () and the methods of [8, 9] can be extended to (). In the other cases, Equation () has rarely studied.…”

Section: Introductionmentioning

confidence: 99%

“…where q x ð Þ is continuous and j q x ð Þ j ≥ δ > 0 was considered in numerous works (see previous works [8,9] and references therein).…”

mentioning

confidence: 99%

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