On separability of a differential operator of non-classical type in an unbounded domain

Muratbekov, М.B.; Igisinov, Sabit; Shyrakbaev, Abai

doi:10.1063/1.4959688

Cited by 2 publications

(3 citation statements)

References 6 publications

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“…Using the last inequality, we obtain the following inequality:

{c}&#x0005E;{-1}{q}&#x0005E;{\frac{1}{2}}\left(x,0\right)\le {q}&#x0005E;{\ast}\left(x,0\right)\le {c}&#x0005E;{-1}{q}&#x0005E;{\frac{1}{2}}\left(x,0\right)

, where

c&gt;0

is a constant number. The last inequality is proved in the same way as Lemma 3.2 in [20]. Now, using Theorem 8.1 in [19] and repeating the computations and arguments used in the proof of Theorems 1–4 in [18], we obtain the proof of Theorems 3 and 4.…”

Section: Two‐sided Estimates Of the Approximation Numbers Of Solution...mentioning

confidence: 81%

Maximal regularity and two‐sided estimates of the approximation numbers of the nonlinear Sturm–Liouville equation solutions with rapidly oscillating coefficients in L₂(ℝ)

Muratbekov

Altynbek

2023

Math Methods in App Sciences

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A theorem on the maximum regularity of solutions of the nonlinear Sturm–Liouville equation with greatly growing and rapidly oscillating potential in the space is proved in this paper. Two‐sided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear Sturm–Liouville equation are also obtained. As is known, the obtained estimates give the opportunity to choose approximation apparatus that guarantees the minimum possible error.

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“…Using the last inequality, we obtain the following inequality:

{c}&#x0005E;{-1}{q}&#x0005E;{\frac{1}{2}}\left(x,0\right)\le {q}&#x0005E;{\ast}\left(x,0\right)\le {c}&#x0005E;{-1}{q}&#x0005E;{\frac{1}{2}}\left(x,0\right)

, where

c&gt;0

Section: Two‐sided Estimates Of the Approximation Numbers Of Solution...mentioning

confidence: 81%

Maximal regularity and two‐sided estimates of the approximation numbers of the nonlinear Sturm–Liouville equation solutions with rapidly oscillating coefficients in L₂(ℝ)

Muratbekov

Altynbek

2023

Math Methods in App Sciences

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“…An operator of mixed type has been studied by Muratbekov 15 for the case when the coefficient satisfies the condition:

y\cdotp k(y)&gt;0

for

y\ne 0

and

k(0)&#x0003D;0

.…”

Section: Introduction Statement Of Resultsmentioning

confidence: 99%

“…where 𝜆 n, k are eigenvalues of the operator (L + 𝜇 I) −1 . An operator of mixed type has been studied by Muratbekov 15 for the case when the coefficient satisfies the condition:…”

Section: D(l Nmentioning

confidence: 99%

Estimates of singular numbers (s$$ s $$‐numbers) and eigenvalues of a mixed elliptic‐hyperbolic type operator with parabolic degeneration

Muratbekov

Abylayeva

Muratbekov

2022

Math Methods in App Sciences

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This paper is concerned with a mixed type differential operator which is initially defined with , where and is a set of infinitely differentiable functions with compact support with respect to the variable and satisfying the conditions: Regarding the coefficient , with supposition that satisfies the condition: is a piecewise continuous and bounded function in . The coefficients and are continuous functions in and can be unbounded at infinity. The operator admits closure in the space , and the closure is also denoted by . Taking into consideration certain constraints on the coefficients , apart from the above‐mentioned conditions, the existence of a bounded inverse operator is proved in this paper; a condition guaranteeing compactness of the resolvent kernel is found; and we also obtained two‐sided estimates for singular numbers ( ‐numbers). Here, we note that the estimate of singular numbers ( ‐numbers) shows the rate of approximation of the resolvent of the operator by linear finite‐dimensional operators. It is given an example of how the obtained estimates for the ‐numbers enable to identify the estimates for the eigenvalues of the operator . We note that the above results are apparently obtained for the first time for a mixed‐type operator in the case of an unbounded domain with rapidly oscillating and greatly growing coefficients at infinity.

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On separability of a differential operator of non-classical type in an unbounded domain

Cited by 2 publications

References 6 publications

Maximal regularity and two‐sided estimates of the approximation numbers of the nonlinear Sturm–Liouville equation solutions with rapidly oscillating coefficients in L₂(ℝ)

Maximal regularity and two‐sided estimates of the approximation numbers of the nonlinear Sturm–Liouville equation solutions with rapidly oscillating coefficients in L₂(ℝ)

Estimates of singular numbers (s$$ s $$‐numbers) and eigenvalues of a mixed elliptic‐hyperbolic type operator with parabolic degeneration

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On separability of a differential operator of non-classical type in an unbounded domain

Cited by 2 publications

References 6 publications

Maximal regularity and two‐sided estimates of the approximation numbers of the nonlinear Sturm–Liouville equation solutions with rapidly oscillating coefficients in L2(ℝ)

Maximal regularity and two‐sided estimates of the approximation numbers of the nonlinear Sturm–Liouville equation solutions with rapidly oscillating coefficients in L2(ℝ)

Estimates of singular numbers (s$$ s $$‐numbers) and eigenvalues of a mixed elliptic‐hyperbolic type operator with parabolic degeneration

Contact Info

Product

Resources

About

Maximal regularity and two‐sided estimates of the approximation numbers of the nonlinear Sturm–Liouville equation solutions with rapidly oscillating coefficients in L₂(ℝ)

Maximal regularity and two‐sided estimates of the approximation numbers of the nonlinear Sturm–Liouville equation solutions with rapidly oscillating coefficients in L₂(ℝ)