Existence and characterization of solutions of nonlinear Volterra-Stieltjes integral equations in two variables. (English) Zbl 1474.45020

Summary: The paper is devoted mainly to the study of the existence of solutions depending on two variables of a nonlinear integral equation of Volterra-Stieltjes type. The basic tool used in investigations is the technique of measures of noncompactness and Darbo’s fixed point theorem. The results obtained in the paper are applicable, in a particular case, to the nonlinear partial integral equations of fractional orders.


45D05 Volterra integral equations
Full Text: DOI


[1] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional Calculus: Models and Numerical Methods (2012), New York, NY, USA: World Scientific, New York, NY, USA · Zbl 1248.26011 · doi:10.1142/9789814355216
[2] Diethelm, K., The Analysis of Fractional Differential Equations. The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics (2010), Berlin, Germany: Springer, Berlin, Germany · Zbl 1215.34001 · doi:10.1007/978-3-642-14574-2
[3] Hilfer, R., Applications of Fractional Calculus in Physics (2000), River Edge, NJ, USA: World Scientific, River Edge, NJ, USA · Zbl 0998.26002 · doi:10.1142/9789812817747
[4] Tarasov, V. E., Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Nonlinear Physical Science (2010), Beijing, China: Springer, Heidelberg, Germany; Higher Education Press, Beijing, China · Zbl 1214.81004 · doi:10.1007/978-3-642-14003-7
[5] Dung, N. T., Fractional stochastic differential equations with applications to finance, Journal of Mathematical Analysis and Applications, 397, 1, 334-348 (2013) · Zbl 1255.60100 · doi:10.1016/j.jmaa.2012.07.062
[6] Abbas, S.; Benchohra, M.; N’Guérékata, G. M., Topic in Fractional Differential Equations. Topic in Fractional Differential Equations, Developments in Mathematics, 27 (2012), New York, NY, USA: Springer, New York, NY, USA · Zbl 1273.35001
[7] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 (2006), Amsterdam, The Netherlands: Elsevier Science B.V., Amsterdam, The Netherlands · Zbl 1092.45003
[8] Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198 (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0924.34008
[9] Banaś, J.; Zaja̧c, T., A new approach to the theory of functional integral equations of fractional order, Journal of Mathematical Analysis and Applications, 375, 2, 375-387 (2011) · Zbl 1210.45005 · doi:10.1016/j.jmaa.2010.09.004
[10] Zaja̧c, T., Solvability of fractional integral equations on an unbounded interval through the theory of Volterra-Stieltjes integral equations, Zeitschrift für Analysis und ihre Anwendungen, 33, 1, 65-85 (2014) · Zbl 1292.45006 · doi:10.4171/ZAA/1499
[11] Abbas, S.; Baleanu, D.; Benchohra, M., Global attractivity for fractional order delay partial integro-differential equations, Advances in Difference Equations, 2012, article 62 (2012) · Zbl 1302.35392 · doi:10.1186/1687-1847-2012-62
[12] Abbas, S.; Benchohra, M.; Nieto, J. J., Global attractivity of solutions for nonlinear fractional order Riemann-Liouville Volterra-Stieltjes partial integral equations, Electronic Journal of Qualitative Theory of Differential Equations, 81, 1-15 (2012) · Zbl 1340.26008
[13] Abbas, S.; Benchohra, M., Existence and stability of nonlinear, fractional order Riemann-Liouville Volterra-Stieltjes multi-delay integral equations, Journal of Integral Equations and Applications, 25, 2, 143-158 (2013) · Zbl 1288.26002 · doi:10.1216/JIE-2013-25-2-143
[14] Abbas, S.; Benchohra, M., Fractional order integral equations of two independent variables, Applied Mathematics and Computation, 227, 755-761 (2014) · Zbl 1364.45005 · doi:10.1016/j.amc.2013.10.086
[15] Vityuk, A. N.; Golushkov, A. V., Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscillations, 7, 3, 328-335 (2004) · Zbl 1092.35500 · doi:10.1007/s11072-005-0015-9
[16] Ahmad, B.; Nieto, J. J.; Alsaedi, A.; Al-Hutami, H., Existence of solutions for nonlinear fractional \(q\)-difference integral equations with two fractional orders and nonlocal four-point boundary conditions, Journal of the Franklin Institute, 351, 5, 2890-2909 (2014) · Zbl 1372.45007 · doi:10.1016/j.jfranklin.2014.01.020
[17] Appell, J.; Banaś, J.; Merentes, N., Bounded Variation and Around. Bounded Variation and Around, Series in Nonlinear Analysis and Applications, 17 (2014), Berlin, Germany: Walter de Gruyter, Berlin, Germany · Zbl 1282.26001
[18] Rudin, W., Real and Complex Analysis (1970), New York, NY, USA: McGraw-Hill, New York, NY, USA
[19] Banaś, J.; Goebel, K., Measures of Noncompactness in Banach Spaces. Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, 60 (1980), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0441.47056
[20] Akhmerov, R. R.; Kamenskiĭ, M. I.; Potapov, A. S.; Rodkina, A. E.; Sadovskiĭ, B. N., Measures of Noncompactness and Condensing Operators. Measures of Noncompactness and Condensing Operators, Operator Theory: Advances and Applications, 55 (1992), Basel, Switzerland: Birkhäuser, Basel, Switzerland · Zbl 0748.47045 · doi:10.1007/978-3-0348-5727-7
[21] Toledano, M. A.; Benavides, T. D.; Acedo, G. L., Measures of Noncompactness in Metric Fixed Point Theory. Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory: Advances and Applications, 99 (1997), Basel, Switzerland: Birkhäuser, Basel, Switzerland · Zbl 0885.47021 · doi:10.1007/978-3-0348-8920-9
[22] Appell, J.; Zabrejko, P. P., Nonlinear Superposition Operators. Nonlinear Superposition Operators, Cambridge Tracts in Mathematics, 95 (1990), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0701.47041 · doi:10.1017/CBO9780511897450
[23] Srivastava, H. M.; Saxena, R. K., Operators of fractional integration and their applications, Applied Mathematics and Computation, 118, 1, 1-52 (2001) · Zbl 1022.26012 · doi:10.1016/S0096-3003(99)00208-8
[24] Darwish, M. A., Nondecreasing solutions of a fractional quadratic integral equation of Urysohn-Volterra type, Dynamic Systems and Applications, 20, 4, 423-438 (2011) · Zbl 1243.45007
[25] Darwish, M. A.; Ntouyas, S. K., On a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument, Nonlinear Analysis: Theory, Methods & Applications, 74, 11, 3510-3517 (2011) · Zbl 1228.45006 · doi:10.1016/j.na.2011.02.035
[26] Abbas, S.; Benchohra, M.; Henderson, J., Global asymptotic stability of solutions of nonlinear quadratic Volterra integral equations of fractional order, Communications on Applied Nonlinear Analysis, 19, 1, 79-89 (2012) · Zbl 1269.26003
[27] Caballero, J.; Darwish, M. A.; Sadarangani, K., Solvability of a fractional hybrid initial value problem with supremum by using measures of noncompactness in Banach algebras, Applied Mathematics and Computation, 224, 553-563 (2013) · Zbl 1339.34008 · doi:10.1016/j.amc.2013.08.060
[28] Fichtenholz, G. M., Differential and Integral Calculus, 2, Warsaw, Poland: Wydawnictwo Naukowe PWN, Warsaw, Poland · Zbl 0900.26001
[29] Argyros, I. K., On a class of quadratic integral equations with perturbation, Functiones et Approximatio Commentarii Mathematici, 20, 51-63 (1992) · Zbl 0780.45005
[30] Chandrasekhar, S., Radiative Transfer (1950), London, UK: Oxford University Press, London, UK · Zbl 0037.43201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.