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Existence of analytic solutions for some classes of singular integral equations of non-normal type with convolution kernel. (English) Zbl 1505.45001

The article is devoted to the study of the existence and uniqueness of solutions of singular integral equations of dual type, \[ c_1x(x)+\frac{d_1}{\pi i}\int_{-\infty}^\infty\frac{x(\tau)}{\tau-t}d\tau+\int_{-\infty}^\infty R_1(t-\tau)x(\tau)d\tau=f(t), \qquad 0<t<\infty,\\ c_2x(x)+\frac{d_2}{\pi i}\int_{-\infty}^\infty\frac{x(\tau)}{\tau-t}d\tau+\int_{-\infty}^\infty R_2(t-\tau)x(\tau)d\tau=f(t), \qquad -\infty<t<0, \] where \(x\) is the unknown function, \(c_i, d_i\in \mathbb{R}\), \(i=1,2 \) (\(d_1\) and \(d_2\) do not vanish simultaneously), and a singular integral Wiener-Hopf equation, \[ ax(t)+\frac{b}{\pi i}\int_{0}^\infty\frac{x(\tau)}{\tau-t}d\tau+\int_{0}^\infty h(t-\tau)x(\tau)d\tau=g(t). \]
The author uses the theory of Fourier analysis and the regularity theory of Fredholm integral equations, as well as methods of complex analysis and the principle of analytic continuation.
As a result the author establishes the Noether solvability of the above equations and generalizes the Sokhotski-Plemelj formula for the Fourier transform. In addition, by means of integral transforms, analytical methods of boundary value problems, and principles of analytic continuation, he transforms the above equations into boundary value problems for analytical functions. Analytic solutions and conditions of Noether solvability are obtained in the non-normal type case. Finally, he discusses the asymptotic property of solutions for the equations at nodal points.

MSC:

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45E05 Integral equations with kernels of Cauchy type
45M05 Asymptotics of solutions to integral equations
45P05 Integral operators
30E25 Boundary value problems in the complex plane
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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[1] Gahov, F. D.; Cherskiy, U. I., Integral Equations of Convolution Type (1980), Moscow: NauKa, Moscow
[2] Duduchava, R. V., Integral equations of convolution type with discontinuous coefficients, Math. Nachr., 79, 75-78 (1977) · Zbl 0362.45004 · doi:10.1002/mana.19770790108
[3] Muskhelishvilli, N. I., Singular Integral Equations (2002), Moscow: NauKa, Moscow
[4] Litvinchuk, G. S., Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift (2004), London: Kluwer Academic, London
[5] Lu, J. K., Boundary Value Problems for Analytic Functions (2004), Singapore: World Sci, Singapore
[6] Duduchava, R. V., Wiener-Hopf integral operators, Math. Nachr., 65, 59-82 (1975) · Zbl 0317.47019 · doi:10.1002/mana.19750650106
[7] Li, P. R., Generalized convolution-type singular integral equations, Appl. Math. Comput., 311, 314-323 (2017) · Zbl 1416.65539 · doi:10.1016/j.cam.2016.07.027
[8] Li, P. R.; Ren, G. B., Solvability of singular integro-differential equations via Riemann-Hilbert problem, J. Differ. Equ., 265, 5455-5471 (2018) · Zbl 1406.35215 · doi:10.1016/j.jde.2018.07.056
[9] Li, P. R., Non-normal type singular integral-differential equations by Riemann-Hilbert approach, J. Math. Anal. Appl., 483, 2 (2020) · Zbl 1443.45004 · doi:10.1016/j.jmaa.2019.123643
[10] Li, P. R., Solvability theory of convolution singular integral equations via Riemann-Hilbert approach, J. Comput. Appl. Math., 370, 2 (2020) · Zbl 1443.45005 · doi:10.1016/j.cam.2019.112601
[11] Li, P. R., Two classes of linear equations of discrete convolution type with harmonic singular operators, Complex Var. Elliptic Equ., 61, 1, 67-73 (2016) · Zbl 1332.45003 · doi:10.1080/17476933.2015.1057712
[12] Li, P. R., On solvability of singular integral-differential equations with convolution, J. Appl. Anal. Comput., 9, 3, 1071-1082 (2019) · Zbl 1458.45001
[13] Li, P. R., Singular integral equations of convolution type with reflection and translation shifts, Numer. Funct. Anal. Optim., 40, 9, 1023-1038 (2019) · Zbl 1412.45009 · doi:10.1080/01630563.2019.1586721
[14] Li, P. R.; Ren, G. B., Some classes of equations of discrete type with harmonic singular operator and convolution, Appl. Math. Comput., 284, 185-194 (2016) · Zbl 1410.44003
[15] Li, P. R., Generalized boundary value problems for analytic functions with convolutions and its applications, Math. Methods Appl. Sci., 42, 2631-2643 (2019) · Zbl 1417.30036 · doi:10.1002/mma.5538
[16] Li, P. R., Singular integral equations of convolution type with Cauchy kernel in the class of exponentially increasing functions, Appl. Math. Comput., 344-345, 116-127 (2019) · Zbl 1438.65330 · doi:10.1007/s40314-019-0892-7
[17] Hörmander, L., The Analysis of Linear Partial Differential Operators. I (2003), Berlin: Springer, Berlin · Zbl 1028.35001 · doi:10.1007/978-3-642-61497-2
[18] Du, H.; Shen, J. H., Reproducing kernel method of solving singular integral equation with cosecant kernel, J. Math. Anal. Appl., 348, 1, 308-314 (2008) · Zbl 1152.45007 · doi:10.1016/j.jmaa.2008.07.037
[19] De-Bonis, M. C.; Laurita, C., Numerical solution of systems of Cauchy singular integral equations with constant coefficients, Appl. Math. Comput., 219, 1391-1410 (2012) · Zbl 1291.65378
[20] Begehr, H.; Vaitekhovich, T., Harmonic boundary value problems in half disc and half ring, Funct. Approx., 40, 2, 251-282 (2009) · Zbl 1183.30039
[21] Nakazi, T.; Yamamoto, T., Normal singular integral operators with Cauchy kernel, Integral Equ. Oper. Theory, 78, 233-248 (2014) · Zbl 1339.47066 · doi:10.1007/s00020-013-2104-y
[22] Chuan, L. H.; Mau, N. V.; Tuan, N. M., On a class of singular integral equations with the linear fractional Carleman shift and the degenerate kernel, Complex Var. Elliptic Equ., 53, 2, 117-137 (2008) · Zbl 1221.45003 · doi:10.1080/17476930701619782
[23] Praha, E. K.; Valencia, V. M., Solving singular convolution equations using inverse Fast Fourier Transform, Appl. Math., 57, 5, 543-550 (2012) · Zbl 1265.42020 · doi:10.1007/s10492-012-0032-9
[24] Tuan, N. M.; Thu-Huyen, N. T., The solvability and explicit solutions of two integral equations via generalized convolutions, J. Math. Anal. Appl., 369, 712-718 (2010) · Zbl 1197.45004 · doi:10.1016/j.jmaa.2010.04.019
[25] Li, P. R., The solvability and explicit solutions of singular integral-differential equations of non-normal type via Riemann-Hilbert problem, J. Comput. Appl. Math., 374, 2 (2020) · Zbl 1444.45006 · doi:10.1016/j.cam.2020.112759
[26] Wöjcik, P.; Sheshko etc, M. A., Application of Faber polynomials to the approximate solution of singular integral equations with the Cauchy kernel, Differ. Equ., 49, 2, 198-209 (2013) · Zbl 1266.45006 · doi:10.1134/S0012266113020067
[27] Gomez, C.; Prado, H.; Trofimchuk, S., Separation dichotomy and wavefronts for a nonlinear convolution equation, J. Math. Anal. Appl., 420, 1-19 (2014) · Zbl 1296.45005 · doi:10.1016/j.jmaa.2014.05.064
[28] Gong, Y. F.; Leong, L. T.; Qiao, T., Two integral operators in Clifford analysis, J. Math. Anal. Appl., 354, 435-444 (2009) · Zbl 1182.30082 · doi:10.1016/j.jmaa.2008.12.021
[29] Ren, G. B.; Kaehler, U.; Shi, J. H.; Liu, C. W., Hardy-Littlewood inequalities for fractional derivatives of invariant harmonic functions, Complex Anal. Oper. Theory, 6, 2, 373-396 (2012) · Zbl 1277.26044 · doi:10.1007/s11785-010-0123-0
[30] Colliander, J.; Keel, M.; Staffilani etc, G., Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation, Invent. Math., 181, 1, 39-113 (2010) · Zbl 1197.35265 · doi:10.1007/s00222-010-0242-2
[31] Abreu-Blaya, R.; Bory-Reyes, J.; Brackx, F.; De-Schepper, H.; Sommen, F., Cauchy integral formulae in Hermitian Quaternionic Clifford Analysis, Complex Anal. Oper. Theory, 6, 971-983 (2012) · Zbl 1275.30017 · doi:10.1007/s11785-011-0168-8
[32] Blocki, Z., Suita conjecture and Ohsawa-Takegoshi extension theorem, Invent. Math., 193, 149-158 (2013) · Zbl 1282.32014 · doi:10.1007/s00222-012-0423-2
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