Abstract: The paper continues research of the criteria of applicability to complete singular integral operators of approximate methods using families of strongly approximating them operators with the "cut out'' singularity of the Cauchy kernel. The case of a complete singular integral operator with continuous coefficients acting on \(L_{p}\)-space on a closed contour is considered. It is assumed that the contour is piecewise Lyapunov and has no cusps. The task is reduced to a criterion of invertibility of an element in some Banach algebra. The study is performed using the local principle of Gokhberg and Krupnik. The focus is on the local analysis at the corner points. For this purpose, an analogue of the method of quasi-equivalent operators proposed by I. B. Simonenko is used. The criterion is formulated in terms of invertibility of some integral operators associated with the corner points acting on \(L_{p}\)-space on the real axis, and strong ellipticity conditions at the contour points with the Lyapunov condition.
Keywords: Lyapunov condition, piecewise-Lyapunov contour, complete singular integral operator, convergence of approximation method, uniform invertibility, local principle.
For citation: Abramyan, A. V. and Pilidi, V. S. Criterion of Uniform Invertibility of Regular Approximations of One-Dimensional Singular Integral Operators on
a Piecewise-Lyapunov Contour, Vladikavkaz Math. J., 2019, vol. 21, no. 1, pp. 5-15 (in Russian). DOI 10.23671/VNC.2019.1.27645
1. Probdorf, S. and Schmidt G. A finite element collocation method
for singular integral equations, Math. Nachr., 1981, vol. 100, pp. 33-60.
2. Silbermann, B. Lokale Theorie des Reduktionsverfahrens
fur Toeplitzoperatoren, Math. Nachr., 1981, vol. 104, pp. 137-146.
3. Hagen, R., Roch, S. and Silbermann, B. \(C_{\ast}\)-algebras
and numerical analysis, New York, Marcel Dekker, 2001, 376 p.
4. Pilidi, V. S. On uniform invertibility of regular approximations
of one-dimensional singular integral operators with piecewise continuous
coefficients, Dokl. Akad. Nauk SSSR, 1989, vol. 307, no. 2, pp. 280-283 (in Russian).
5. Pilidi, V. S. A method for excision of singularity for bisingular
integral operators with continuous coefficients, Funct. Anal. Appl.,
1989, vol. 23, no. 1, pp. 82-83 (in Russian).
6. Pilidi, V. S. A criterion for uniform invertibility of
regular approximations of one-dimensional singular integral
operators with piecewise continuous coefficients,
Izv. Akad. Nauk SSSR, Ser. Mat., 1990, vol. 54, no. 6, pp. 1270-1294 (in Russian).
7. Pilidi, V. S. On uniform invertibility of regular approximations
of one-dimensional singular integral operators
in variable exponent spaces, Izvestiya Vuzov. Severo-Kavkazskii Region.
Natural Science, 2011, no. 1, pp. 12-17 (in Russian).
8. Abramyan, A. V. and Pilidi, V. S. On uniform invertibility of regular approximations of one-dimensional singular integral operators with piecewise continuous coefficients in variable exponent spaces,
Izvestiya Vuzov. Severo-Kavkazskii Region. Natural Science,
2013, no. 5, pp. 5-10 (in Russian).
9. Gokhberg, I. Ts. and Fel'dman, I. À. Convolution Equations and
Projection Methods for Their Solution, Moskow, Nauka, 1971, 432 p. (in Russian).
10. Muskhelishvili, N. I. Singular Integral Equations. 3rd ed., Moskow, Nauka, 1968, 513 p. (in Russian).
11. Gokhberg, I. Ts. and Krupnik, N. Y. Introduction to the Theory of One-Dimensional Singular Integral Operators,
Kishinev, SHtiintsa, 1973, 426 p. (in Russian).
12. Edvards, R. E. Functional Analysis, Moskow, Mir, 1969, 1072 p. (in Russian).
13. Simonenko, I. B. A new general method of investigating linear
operator equations of singular integral equation type. I, Izv.
Akad. Nauk SSSR Ser. Mat., 1965, vol. 29, no. 3, pp. 567-586 (in Russian).
