Existence, Compactness, Estimates of Eigenvalues and s-Numbers of a Resolvent for a Linear Singular Operator of the Korteweg-de Vries Type

Muratbekov M.B. Suleimbekova A.O.
2022 University of Nis

Filomat
2022 #36 Issue 11 3689 - 3700 pp.

In this paper, we consider a linear operator of the Korteweg-de Vries type Lu =^∂u ∂y^+R2(y)^∂3u ∂x^{3 +R}1(y)^∂u ∂x^+R0(y)u initially defined on C^∞0,π^{(Ω), where Ω = {(x,y): −π ≤x≤ π, −∞ 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.

Korteweg-de Vries type singular operator , resolvent , separability

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Taraz Regional University named after M.Kh.Dulaty, Tole bi 60, Taraz, 080000, Kazakhstan
L.N. Gumilyov Eurasian National University, Satpayev str. 2, Nur-Sultan, 010008, Kazakhstan

Taraz Regional University named after M.Kh.Dulaty
L.N. Gumilyov Eurasian National University

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