Geodesic
Existence, compactness, estimates of eigenvalues and s-numbers of a resolvent for a linear singular operator of the Korteweg-de Vries type
Filomat, Tome 36 (2022) no. 11, p. 3689

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In this paper, we consider a linear operator of the Korteweg-de Vries type Lu = ∂u ∂y + R 2 (y) ∂ 3 u ∂x 3 + R 1 (y) ∂u ∂x + R 0 (y)u initially defined on C ∞ 0,π (Ω), where Ω = {(x, y) : −π ≤ x ≤ π, −∞ y ∞}. C ∞ 0,π (Ω) is a set of infinitely differentiable compactly supported function with respect to a variable y and satisfying the conditions: u (i) x (−π, y) = u (i) x (π, y), i = 0, 1, 2. With respect to the coefficients of the operator L , we assume that these are continuous functions in R(−∞, +∞) and strongly growing functions at infinity. In this paper, we proved that there exists a bounded inverse operator and found a condition that ensures the compactness of the resolvent under some restrictions on the coefficients in addition to the above conditions. Also, two-sided estimates of singular numbers (s-numbers) are obtained and an example is given of how these estimates allow finding estimates of the eigenvalues of the considered operator.
DOI : 10.2298/FIL2211689M
Classification : 47A10, 35L81
Keywords: Korteweg-de Vries type singular operator, separability 
M B Muratbekov; A O Suleimbekov. Existence, compactness, estimates of eigenvalues and s-numbers of a resolvent for a linear singular operator of the Korteweg-de Vries type. Filomat, Tome 36 (2022) no. 11, p. 3689 . doi: 10.2298/FIL2211689M
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     author = {M B Muratbekov and A O Suleimbekov},
     title = {Existence, compactness, estimates of eigenvalues and s-numbers of a resolvent for a linear singular operator of the {Korteweg-de} {Vries} type},
     journal = {Filomat},
     pages = {3689 },
     year = {2022},
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     doi = {10.2298/FIL2211689M},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2211689M/}
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