Estimates of singular numbers (s$$ s $$-numbers) and eigenvalues of a mixed elliptic-hyperbolic type operator with parabolic degeneration
Muratbekov M. Abylayeva A. Muratbekov M.
April 2023John Wiley and Sons Ltd
Mathematical Methods in the Applied Sciences
2023#46Issue 66368 - 6380 pp.
This paper is concerned with a mixed type differential operator (Formula presented.) which is initially defined with (Formula presented.), where (Formula presented.) and (Formula presented.) is a set of infinitely differentiable functions with compact support with respect to the variable (Formula presented.) and satisfying the conditions: (Formula presented.). Regarding the coefficient (Formula presented.), with supposition that (Formula presented.) satisfies the condition: (Formula presented.) is a piecewise continuous and bounded function in (Formula presented.). The coefficients (Formula presented.) and (Formula presented.) are continuous functions in (Formula presented.) and can be unbounded at infinity. The operator (Formula presented.) admits closure in the space (Formula presented.), and the closure is also denoted by (Formula presented.). Taking into consideration certain constraints on the coefficients (Formula presented.) (Formula presented.), 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 ((Formula presented.) -numbers). Here, we note that the estimate of singular numbers ((Formula presented.) -numbers) shows the rate of approximation of the resolvent of the operator (Formula presented.) by linear finite-dimensional operators. It is given an example of how the obtained estimates for the (Formula presented.) -numbers enable to identify the estimates for the eigenvalues of the operator (Formula presented.). 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.
compactness , eigenvalues , mixed type operator , resolvent , separability , singular numbers (s$$ s $$-numbers)
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Mathematics in Education, Taraz Regional University named after M.Kh. Dulaty (Dulaty University), Taraz, Kazakhstan
Fundamental Mathematics, L.N. Gumilyov Eurasian National University, Astana, Kazakhstan
Distance Learning Center, Esil Univesity, Astana, Kazakhstan
Mathematics in Education
Fundamental Mathematics
Distance Learning Center
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