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doi: 10.3934/dcdsb.2021094
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## Boundary-value problems for weakly singular integral equations

 Institute of Mathematics of the National Academy of Sciences of Ukraine, 3, Tereshchenkivska Str., Kyiv, 01024, Ukraine

Received  April 2020 Revised  August 2020 Early access March 2021

Fund Project: The present work was supported by the Grant H2020-MSCA-RISE-2019, project number 873071 (SOMPATY: Spectral Optimization: From Mathematics to Physics and Advanced Technology)

We consider a perturbed linear boundary-value problem for a weakly singular integral equation. Assume that the generating boundary-value problem is unsolvable for arbitrary inhomogeneities. Efficient conditions for the coefficients guaranteeing the appearance of the family of solutions of the perturbed linear boundary-value problem in the form of Laurent series in powers of a small parameter $\varepsilon$ with singularity at the point $\varepsilon = 0$ are established.

Citation: Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021094
##### References:
 [1] P. L. Auer and C. S. Gardner, Note on singular integral equations of the Kirkwood–Riseman type, J. Chem. Phys., 23 (1955), 1545-1546.  doi: 10.1063/1.1742352.  Google Scholar [2] A. Boichuk, J. Diblík, D. Khusainov and M. Růžičková, Boundary-value problems for weakly nonlinear delay differential systems, Abstr. Appl. Anal., 2011 (2011), Art. ID 631412, 19 pp. doi: 10.1155/2011/631412.  Google Scholar [3] O. A. Boichuk and V. A. Feruk, Linear boundary-value problems for weakly singular integral equations, J. Math. Sci., 247 (2020), 248-257.  doi: 10.1007/s10958-020-04800-6.  Google Scholar [4] A. A. Boichuk, I. A. Korostil and M. Fečkan, Bifurcation conditions for a solution of an abstract wave equation, Differential Equations, 43 (2007), 495-502.  doi: 10.1134/S0012266107040076.  Google Scholar [5] O. A. Boichuk, N. O. Kozlova and V. A. Feruk, Weakly perturbed integral equations, J. Math. Sci., 223 (2017), 199-209.  doi: 10.1007/s10958-017-3348-x.  Google Scholar [6] A. A. Boichuk, N. A. Kozlova and V. A. Feruk, Weakly nonlinear integral equations of the Hammerstein type, Nonlin. Dynam. Syst. Theory, 19 (2019), 289-301.   Google Scholar [7] A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, 2$^nd$ edition, Walter de Gruyter, Berlin, Boston, 2016. doi: 10.1515/9783110378443.  Google Scholar [8] A. A. Boichuk and V. F. Zhuravlev, Solvability criterion for integro-differential equations with degenerate kernel in Banach spaces, Nonlin. Dynam. Syst. Theory, 18 (2018), 331-341.   Google Scholar [9] C. Constanda and S. Potapenko (eds.), Integral Methods in Science and Engineering. Techniques and Applications, Birkhäuser, Boston, 2008. doi: 10.1007/978-0-8176-4671-4.  Google Scholar [10] E. A. Galperin, E. J. Kansa, A. Makroglou and S. A. Nelson, Variable transformations in the numerical solution of second kind Volterra integral equations with continuous and weakly singular kernels; extensions to Fredholm integral equations, J. Comput. Appl. Math., 115 (2000), 193-211.  doi: 10.1016/S0377-0427(99)00297-6.  Google Scholar [11] I. Golovatska, Weakly perturbed boundary-value problems for systems of integro-differential equations, Tatra Mt. Math. Publ., 54 (2013), 61-71.  doi: 10.2478/tmmp-2013-0005.  Google Scholar [12] O. Gonzalez and J. Li, A convergence theorem for a class of Nystrom methods for weakly singular integral equations on surfaces in $\mathbb{R}^{3}$, Math. Comp., 84 (2015), 675-714.  doi: 10.1090/S0025-5718-2014-02869-X.  Google Scholar [13] E. Goursat, A Course in Mathematical Analysis. Vol. III. Part 2, Dover Publications, Inc., New York, 1964.  Google Scholar [14] I. G. Graham, Galerkin methods for second kind integral equations with singularities, Math. Comp., 39 (1982), 519-533.  doi: 10.1090/S0025-5718-1982-0669644-3.  Google Scholar [15] E. A. Grebenikov and Yu. A. Ryabov, Constructive Methods in the Analysis of Nonlinear Systems, Nauka, Moscow, 1979.  Google Scholar [16] D. Hilbert, Selected Works, Vol. 2, Faktorial, Moscow, 1998. Google Scholar [17] C. Huang, T. Tang and Z. Zhang, Supergeometric convergence of spectral collocation methods for weakly singular Volterra and Fredholm integral equations with smooth solutions, J. Comput. Math., 29 (2011), 698-719.  doi: 10.4208/jcm.1110-m11si06.  Google Scholar [18] S. G. Krein, Linear Equations in Banach Spaces, Nauka, Moscow, 1971.  Google Scholar [19] I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations, Gos. Izdt. Tekh.-Teor. Lit., Moscow, 1956.  Google Scholar [20] S. G. Mikhlin, Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, 2$^{nd}$ edition, Pergamon, New York, 1964.  Google Scholar [21] G. R. Richter, On weakly singular Fredholm integral equations with displacement kernels, J. Math. Anal. Appl., 55 (1976), 32-42.  doi: 10.1016/0022-247X(76)90275-4.  Google Scholar [22] A. Samoilenko, A. Boichuk and S. Chuiko, Hybrid difference-differential boundary-value problem, Miskolc Mathematical Notes, 18 (2017), 1015-1031.  doi: 10.18514/MMN.2017.2280.  Google Scholar [23] C. Schneider, Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral Equations Operator Theory, 2 (1979), 62-68.  doi: 10.1007/BF01729361.  Google Scholar [24] J. Shen, C. Sheng and Z. Wang, Generalized Jacobi spectral-Galerkin method for nonlinear Volterra integral equations with weakly singular kernels, J. Math. Study, 48 (2015), 315-329.  doi: 10.4208/jms.v48n4.15.01.  Google Scholar [25] V. I. Smirnov, A Course of Higher Mathematics. Vol. IV. Part 1, Nauka, Moscow, 1974.  Google Scholar [26] E. Vainikko and G. Vainikko, A spline product quasi-interpolation method for weakly singular Fredholm integral equations, SIAM J. Numer. Anal., 46 (2008), 1799-1820.  doi: 10.1137/070693308.  Google Scholar [27] M. I. Vishik and L. A. Lyusternik, Solution of some perturbation problems in the case of matrices and self-adjoint or nonself-adjoint differential equations. I, Russ. Math. Surveys, 15 (1960), 1-73.  doi: 10.1070/RM1960v015n03ABEH004092.  Google Scholar [28] V. F. Zhuravlev and N. P. Fomin, Weakly perturbed boundary-value problems for the Fredholm integral equations with degenerate kernel in Banach spaces, J. Math. Sci., 238 (2019), 248-262.  doi: 10.1007/s10958-019-04233-w.  Google Scholar

show all references

##### References:
 [1] P. L. Auer and C. S. Gardner, Note on singular integral equations of the Kirkwood–Riseman type, J. Chem. Phys., 23 (1955), 1545-1546.  doi: 10.1063/1.1742352.  Google Scholar [2] A. Boichuk, J. Diblík, D. Khusainov and M. Růžičková, Boundary-value problems for weakly nonlinear delay differential systems, Abstr. Appl. Anal., 2011 (2011), Art. ID 631412, 19 pp. doi: 10.1155/2011/631412.  Google Scholar [3] O. A. Boichuk and V. A. Feruk, Linear boundary-value problems for weakly singular integral equations, J. Math. Sci., 247 (2020), 248-257.  doi: 10.1007/s10958-020-04800-6.  Google Scholar [4] A. A. Boichuk, I. A. Korostil and M. Fečkan, Bifurcation conditions for a solution of an abstract wave equation, Differential Equations, 43 (2007), 495-502.  doi: 10.1134/S0012266107040076.  Google Scholar [5] O. A. Boichuk, N. O. Kozlova and V. A. Feruk, Weakly perturbed integral equations, J. Math. Sci., 223 (2017), 199-209.  doi: 10.1007/s10958-017-3348-x.  Google Scholar [6] A. A. Boichuk, N. A. Kozlova and V. A. Feruk, Weakly nonlinear integral equations of the Hammerstein type, Nonlin. Dynam. Syst. Theory, 19 (2019), 289-301.   Google Scholar [7] A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, 2$^nd$ edition, Walter de Gruyter, Berlin, Boston, 2016. doi: 10.1515/9783110378443.  Google Scholar [8] A. A. Boichuk and V. F. Zhuravlev, Solvability criterion for integro-differential equations with degenerate kernel in Banach spaces, Nonlin. Dynam. Syst. Theory, 18 (2018), 331-341.   Google Scholar [9] C. Constanda and S. Potapenko (eds.), Integral Methods in Science and Engineering. Techniques and Applications, Birkhäuser, Boston, 2008. doi: 10.1007/978-0-8176-4671-4.  Google Scholar [10] E. A. Galperin, E. J. Kansa, A. Makroglou and S. A. Nelson, Variable transformations in the numerical solution of second kind Volterra integral equations with continuous and weakly singular kernels; extensions to Fredholm integral equations, J. Comput. Appl. Math., 115 (2000), 193-211.  doi: 10.1016/S0377-0427(99)00297-6.  Google Scholar [11] I. Golovatska, Weakly perturbed boundary-value problems for systems of integro-differential equations, Tatra Mt. Math. Publ., 54 (2013), 61-71.  doi: 10.2478/tmmp-2013-0005.  Google Scholar [12] O. Gonzalez and J. Li, A convergence theorem for a class of Nystrom methods for weakly singular integral equations on surfaces in $\mathbb{R}^{3}$, Math. Comp., 84 (2015), 675-714.  doi: 10.1090/S0025-5718-2014-02869-X.  Google Scholar [13] E. Goursat, A Course in Mathematical Analysis. Vol. III. Part 2, Dover Publications, Inc., New York, 1964.  Google Scholar [14] I. G. Graham, Galerkin methods for second kind integral equations with singularities, Math. Comp., 39 (1982), 519-533.  doi: 10.1090/S0025-5718-1982-0669644-3.  Google Scholar [15] E. A. Grebenikov and Yu. A. Ryabov, Constructive Methods in the Analysis of Nonlinear Systems, Nauka, Moscow, 1979.  Google Scholar [16] D. Hilbert, Selected Works, Vol. 2, Faktorial, Moscow, 1998. Google Scholar [17] C. Huang, T. Tang and Z. Zhang, Supergeometric convergence of spectral collocation methods for weakly singular Volterra and Fredholm integral equations with smooth solutions, J. Comput. Math., 29 (2011), 698-719.  doi: 10.4208/jcm.1110-m11si06.  Google Scholar [18] S. G. Krein, Linear Equations in Banach Spaces, Nauka, Moscow, 1971.  Google Scholar [19] I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations, Gos. Izdt. Tekh.-Teor. Lit., Moscow, 1956.  Google Scholar [20] S. G. Mikhlin, Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, 2$^{nd}$ edition, Pergamon, New York, 1964.  Google Scholar [21] G. R. Richter, On weakly singular Fredholm integral equations with displacement kernels, J. Math. Anal. Appl., 55 (1976), 32-42.  doi: 10.1016/0022-247X(76)90275-4.  Google Scholar [22] A. Samoilenko, A. Boichuk and S. Chuiko, Hybrid difference-differential boundary-value problem, Miskolc Mathematical Notes, 18 (2017), 1015-1031.  doi: 10.18514/MMN.2017.2280.  Google Scholar [23] C. Schneider, Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral Equations Operator Theory, 2 (1979), 62-68.  doi: 10.1007/BF01729361.  Google Scholar [24] J. Shen, C. Sheng and Z. Wang, Generalized Jacobi spectral-Galerkin method for nonlinear Volterra integral equations with weakly singular kernels, J. Math. Study, 48 (2015), 315-329.  doi: 10.4208/jms.v48n4.15.01.  Google Scholar [25] V. I. Smirnov, A Course of Higher Mathematics. Vol. IV. Part 1, Nauka, Moscow, 1974.  Google Scholar [26] E. Vainikko and G. Vainikko, A spline product quasi-interpolation method for weakly singular Fredholm integral equations, SIAM J. Numer. Anal., 46 (2008), 1799-1820.  doi: 10.1137/070693308.  Google Scholar [27] M. I. Vishik and L. A. Lyusternik, Solution of some perturbation problems in the case of matrices and self-adjoint or nonself-adjoint differential equations. I, Russ. Math. Surveys, 15 (1960), 1-73.  doi: 10.1070/RM1960v015n03ABEH004092.  Google Scholar [28] V. F. Zhuravlev and N. P. Fomin, Weakly perturbed boundary-value problems for the Fredholm integral equations with degenerate kernel in Banach spaces, J. Math. Sci., 238 (2019), 248-262.  doi: 10.1007/s10958-019-04233-w.  Google Scholar
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