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Existence of solutions for a class of nonlinear Volterra singular integral equations. (English) Zbl 1228.45002
Summary: We prove some results concerning the existence of solutions for a large class of nonlinear Volterra singular integral equations in the space $C[0,1]$ consisting of real functions defined and continuous on the interval $[0,1]$. The main tool used in the proof is the concept of a measure of noncompactness. We also present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results. Moreover, we compare our theory with the approach depending on the use of the theory of Volterra-Stieltjes integral equations. We also show that the results of the paper are applicable in the study of the so-called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems.

45D05Volterra integral equations
Full Text: DOI
[1] Estrada, R.; Kanwal, R. P.: Singular integral equations, (1999) · Zbl 0575.45008
[2] Lifanov, I. K.; Poltavskii, L. N.; Vainikko, G. M.: Hypersingular integral equations and their applications, (2004) · Zbl 1061.45001
[3] Muskhelishvilli, N. I.: Singular integral equations: boundary problems of function theory and their applications to mathematical physics, (1992)
[4] Agarwal, R. P.; Benchohra, M.; Seba, D.: On the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results math. 55, 221-230 (2009) · Zbl 1196.26009 · doi:10.1007/s00025-009-0434-5
[5] Agarwal, R. P.; O’regan, D.; Wong, P. I. Y.: Constant-sign solutions for systems of singular integral equations of Hammerstein type, Math. comput. Modelling 50, 999-1025 (2009) · Zbl 1193.45023 · doi:10.1016/j.mcm.2009.04.003
[6] Banaś, J.; Zaja\ogonek c, T.: A new approach to the theory of functional integral equations of fractional order, J. math. Anal. appl. 375, 375-387 (2011) · Zbl 1210.45005 · doi:10.1016/j.jmaa.2010.09.004
[7] Darwish, M. A.; Ntouyas, S. K.: Monotonic solutions of a perturbed quadratic fractional integral equation, Nonlinear anal. 71, 5513-5521 (2009) · Zbl 1177.45004 · doi:10.1016/j.na.2009.04.041
[8] Darwish, M. A.: On monotonic solutions of a quadratic integral equation with supremum, Dynam. systems appl. 17, 539-550 (2008) · Zbl 1202.45006
[9] M.A. Darwish, On a singular quadratic integral equation of Volterra type with supremum, IC/2007/071, Trieste, Italy, 2007, pp. 1--13.
[10] Diago, T.: Collocation and iterated collocation method for a class of weakly singular Volterra integral equations, J. comput. Appl. math. 229, 363-372 (2009) · Zbl 1168.65073 · doi:10.1016/j.cam.2008.04.002
[11] Liu, L.; Guo, F.; Wu, C.; Wu, Y.: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. math. Anal. appl. 309, 638-649 (2005) · Zbl 1080.45005 · doi:10.1016/j.jmaa.2004.10.069
[12] Rzepka, B.: On attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation of fractional order, Topol. methods nonlinear anal. 32, 89-102 (2008) · Zbl 1173.45003
[13] Rzepka, B.; Sadarangani, K.: On solutions of an infinite system of singular equations, Math. comput. Modelling 45, 1265-1271 (2007) · Zbl 1134.45003 · doi:10.1016/j.mcm.2006.11.006
[14] Tao, L.; Yong, H.: Extrapolation method for solving weakly singular nonlinear Volterra integral equations of the second kind, J. math. Anal. appl. 324, 225-237 (2006) · Zbl 1115.65129 · doi:10.1016/j.jmaa.2005.12.013
[15] Tao, L.; Yong, H.: A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equations of the second kind, J. math. Anal. appl. 282, 56-62 (2003) · Zbl 1030.65140 · doi:10.1016/S0022-247X(02)00369-4
[16] Banaś, J.; Rzepka, B.: Nondecreasing solutions of a quadratic singular Volterra integral equation, Math. comput. Modelling 49, 488-496 (2009) · Zbl 1165.45301 · doi:10.1016/j.mcm.2007.10.021
[17] Banaś, J.; Chlebowicz, A.: On integrable solutions of a nonlinear Volterra integral equation under Carathéodory conditions, Bull. lond. Math. soc. 41, 1073-1084 (2009) · Zbl 1191.47087 · doi:10.1112/blms/bdp088
[18] Benchohra, M.; Graef, J. R.; Hamani, S.: Existence results for boundary value problems with nonlinear fractional differential equations, Appl. anal. 87, 851-863 (2008) · Zbl 1198.26008 · doi:10.1080/00036810802307579
[19] Darwish, M. A.; Henderson, J.: Existence and asymptotic stability of solutions of a perturbed quadratic fractional integral equation, Fract. calc. Appl. anal. 12, 71-86 (2009) · Zbl 1181.45014
[20] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006) · Zbl 1092.45003
[21] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[22] Saxena, R. K.; Kalla, S. I.: On a fractional generalization of free electron laser equation, Appl. math. Comput. 143, 89-97 (2003) · Zbl 1110.45300 · doi:10.1016/S0096-3003(02)00348-X
[23] Banaś, J.; Goebel, K.: Measures of noncompactness in Banach spaces, Lect. notes pure appl. Math. 60 (1980) · Zbl 0441.47056
[24] Appell, J.; Zabrejko, P. P.: Nonlinear superposition operators, (1990)
[25] Argyros, I. K.: On a class of quadratic integral equations with perturbations, Funct. approx. Comment. math. 20, 51-63 (1992) · Zbl 0780.45005
[26] Banaś, J.; Lecko, M.; El-Sayed, W. G.: Existence theorems for some quadratic integral equations, J. math. Anal. appl. 222, 276-285 (1998) · Zbl 0913.45001 · doi:10.1006/jmaa.1998.5941
[27] Cahlon, B.; Eskin, M.: Existence theorems for an integral equation of the Chandrasekhar H-equation with perturbation, J. math. Anal. appl. 83, 159-171 (1981) · Zbl 0471.45002 · doi:10.1016/0022-247X(81)90254-7
[28] Chandrasekhar, S.: Radiative transfer, (1950) · Zbl 0037.43201
[29] Hu, S.; Khavani, M.; Zhuang, W.: Integral equations arising in the kinetic theory of gases, Appl. anal. 34, 261-266 (1989) · Zbl 0697.45004 · doi:10.1080/00036818908839899
[30] Natanson, I. P.: Theory of functions of a real variable, (1960) · Zbl 0091.05404
[31] Banaś, J.; Rodriguez, J. R.; Sadarangani, K.: On a class of Urysohn--stietjes quadratic integral equations and their applications, J. comput. Appl. math. 113, 35-50 (2000) · Zbl 0943.45002 · doi:10.1016/S0377-0427(99)00242-3