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On a discussion of Volterra-Fredholm integral equation with discontinuous kernel. (English) Zbl 1450.45011

Summary: The purpose of this paper is to establish the general solution of a Volterra-Fredholm integral equation with discontinuous kernel in a Banach space. Banach’s fixed point theorem is used to prove the existence and uniqueness of the solution. By using separation of variables method, the problem is reduced to Volterra integral equations of the second kind with continuous kernel. Normality and continuity of the integral operator are also discussed.

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45B05 Fredholm integral equations
45D05 Volterra integral equations
65R20 Numerical methods for integral equations
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[1] Bazm, S., Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations, J. Comput. Appl. Math., 275, 44-60 (2015) · Zbl 1297.65202 · doi:10.1016/j.cam.2014.07.018
[2] Brauer, F.; Castillo-Chvez, C., Mathematical Models in Population Biology and Epidemiology (2012), New York: Springer, New York · Zbl 1302.92001
[3] Craciun, C.; Serban, M. A., A nonlinear integral equation via picard operators, Fixed Point Theory, 12, 1, 57-70 (2011) · Zbl 1233.47062
[4] Hamoud, A. A.; Ghadle, K. P., Approximate solutions of fourth-order fractional integro-differential equations, Acta Univ. Apulensis, 55, 49-61 (2018) · Zbl 1424.35008
[5] Hamoud, A. A.; Ghadle, K. P., On the numerical solution of nonlinear Volterra-Fredholm integral equations by variational iteration method, Int. J. Adv. Sci. Tech. Res., 3, 45-51 (2016)
[6] Ana Ilea, Veronica; Otrocol, Diana, Some Properties of Solutions of a Functional-Differential Equation of Second Order with Delay, The Scientific World Journal, 2014, 1-8 (2014) · Zbl 1324.47096 · doi:10.1155/2014/878395
[7] Otrocol, D.; Ilea, V., Ulam stability for a delay differential equation, Cent. Eur. J. Math., 11, 7, 1296-1303 (2013) · Zbl 1275.34098
[8] Rus, I. A., Results and problems in Ulam stability of operatorial equations and inclusions, Handbook of Functional Equations (2014), New York: Springer Optim. Appl, New York · Zbl 1311.39052
[9] Abdou, M. A.; Nasr, M. E.; Abdel-Aty, M. A., A study of normality and continuity for mixed integral equations, J. Fixed Point Theory Appl., 20, 1, 5 (2018) · Zbl 1390.45024 · doi:10.1007/s11784-018-0490-0
[10] Abdou, M. A.; Nasr, M. E.; Abdel-Aty, M. A., Study of the normality and continuity for the mixed integral equations with phase-lag term, Int. J. Math. Anal., 11, 16, 787-799 (2017) · doi:10.12988/ijma.2017.7798
[11] Karapinar, E.; Kumari, P.; Lateef, D., A new approach to the solution of the Fredholm integral equation via a fixed point on extended b-metric spaces, Symmetry, 10, 10, 512 (2018) · doi:10.3390/sym10100512
[12] Hammad, H. A.; De La Sen, M., A solution of fredholm integral equation by using the cyclic \(\eta^s_q\)-rational contractive mappings technique in b-metric-like spaces, Symmetry, 11, 9, 1184 (2019) · doi:10.3390/sym11091184
[13] Tunç, C.; Tunç, O., On behaviours of functional volterra integro-differential equations with multiple time lags, J. Taibah Univ. Sci., 12, 2, 173-179 (2018) · doi:10.1080/16583655.2018.1451117
[14] Tunç, C.; Tunç, O., A note on the qualitative analysis of volterra integro-differential equations, J. Taibah Univ. Sci., 13, 1, 490-496 (2019) · doi:10.1080/16583655.2019.1596629
[15] Basseem, M., Degenerate method in mixed nonlinear three dimensions integral equation, Alex. Eng. J., 58, 1, 387-392 (2019) · doi:10.1016/j.aej.2017.10.010
[16] Al-Jawary, M.; Radhi, G.; Ravnik, J., Two efficient methods for solving Schlömilch’s integral equation, Int. J. Intell. Comput. Cybernet., 10, 3, 287-309 (2017) · doi:10.1108/IJICC-11-2016-0042
[17] Cattani, C.; Kudreyko, A., Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind, Appl. Math. Comput., 215, 12, 4164-4171 (2010) · Zbl 1186.65160
[18] Gu, Z.; Guo, X.; Sun, D., Series expansion method for weakly singular Volterra integral equations, Appl. Numer. Math., 105, 112-123 (2016) · Zbl 1416.65535 · doi:10.1016/j.apnum.2016.03.001
[19] Hashemizadeh, E.; Rostami, M., Numerical solution of Hammerstein integral equations of mixed type using the Sinc-collocation method, J. Comput. Appl. Math., 279, 31-39 (2015) · Zbl 1306.65291 · doi:10.1016/j.cam.2014.10.019
[20] Maleknejad, K.; Almasieh, H.; Roodaki, M., Triangular functions (TF) method for the solution of nonlinear Volterra-Fredholm integral equations, Commun. Nonlinear Sci. Numer. Simul., 15, 11, 3293-3298 (2010) · Zbl 1222.65147 · doi:10.1016/j.cnsns.2009.12.015
[21] Nasr, M. E.; Abdel-Aty, M. A., Analytical discussion for the mixed integral equations, J. Fixed Point Theory Appl., 20, 3, 115 (2018) · Zbl 1401.45001 · doi:10.1007/s11784-018-0589-3
[22] Sizikov, V.; Sidorov, D., Generalized quadrature for solving singular integral equations of abel type in application to infrared tomography, Appl. Numer. Math., 106, 69-78 (2016) · Zbl 1416.65552 · doi:10.1016/j.apnum.2016.03.004
[23] Abdou, M. A.; Al-Kader, G. M., Mixed type of integral equation with potential kernel, Turk. J. Math., 32, 1, 83-101 (2008) · Zbl 1153.45001
[24] Haagerup, U.: Lp – spaces associated with an arbitrary von neumann algebra. In: Algebres D’opérateurs et leurs applications en physique mathématique (Colloques internationaux C.N.R. S., Paris, 1948), vol. 274, pp. 175-184 (1979). · Zbl 0426.46045
[25] Muscat, Joseph, Hilbert Spaces, Functional Analysis, 171-219 (2014), Cham: Springer International Publishing, Cham · Zbl 1312.46002
[26] Abdou, M. A., Fredholm-Volterra integral equation of the first kind and contact problem, Appl. Math. Comput., 125, 2-3, 177-193 (2002) · Zbl 1028.45003
[27] Popov, G. Y., Contact Problems for a Linearly Deformable Base (1982), Kiev: Odessa, Kiev · Zbl 0563.73098
[28] Linz, P., Analytical and Numerical Methods for Volterra Equations, vol. 7 (1985), Siam: Society for Industrial and Applied Mathematics, Siam · Zbl 0566.65094
[29] Burton, T. A., Volterra Integral and Differential Equations, vol. 202 (2005), USA: Elsevier, USA · Zbl 1075.45001
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