×

Nondecreasing solutions of a quadratic singular Volterra integral equation. (English) Zbl 1165.45301

Summary: We study the existence of nondecreasing solutions of a quadratic singular Volterra integral equation in the space of continuous functions on bounded interval. The main tool utilized in our considerations is the technique associated with certain measure of noncompactness related to monotonicity. The results obtained in the paper may be applied to a wide class of singular Volterra integral equations.

MSC:

45D05 Volterra integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agarwal, R. P.; O’Regan, D.; Wong, P. J.Y., Positive Solutions of Differential, Difference and Integral Equations (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0923.39002
[2] Angelov, V. G.; Bainov, D. D., On the functional differential equations with “maximums”, Appl. Anal., 16, 177-194 (1983) · Zbl 0542.45003
[3] Appell, J.; Zabrejko, P. P., (Nonlinear Superposition Operators. Nonlinear Superposition Operators, Cambridge Tracts in Mathematics, vol. 95 (1990), Cambridge Univ. Press) · Zbl 0701.47041
[4] Banaś, J.; Goebel, K., (Measures of Noncompactness in Banach Spaces. Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 60 (1980), Marcel Dekker: Marcel Dekker New York and Basel) · Zbl 0441.47056
[5] Banaś, J.; Martinon, A., On monotone solutions of a quadratic integral equation of Volterra type, Comput. Math. Appl., 47, 271-279 (2004) · Zbl 1059.45002
[6] Banaś, J.; Olszowy, L., Measures of noncompactness related to monotonicity, Comment. Math., 41, 13-23 (2001) · Zbl 0999.47041
[7] Banaś, J.; Sadarangani, K., Monotonicity properties of the superposition operator and their applications, J. Math. Anal. Appl., 340, 1385-1394 (2008) · Zbl 1137.47046
[8] Burton, T. A., Volterra Integral and Differential Equations (1983), Academic Press: Academic Press New York · Zbl 0515.45001
[9] Caballero, J.; Lopez, B.; Sadarangani, K., On monotonic solutions of an integral equation of Volterra type with supremum, J. Math. Anal. Appl., 305, 304-315 (2005) · Zbl 1076.45002
[10] Corduneanu, C., Integral Equations and Applications (1991), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0714.45002
[11] Darwish, M. A., On quadratic integral equation of fractional orders, J. Math. Anal. Appl., 311, 112-119 (2005) · Zbl 1080.45004
[12] Deimling, K., Nonlinear Functional Analysis (1985), Springer Verlag: Springer Verlag Berlin · Zbl 0559.47040
[13] Fichtenholz, G. M., Differential and Integral Calculus, vol. II (1980), PWN: PWN Warsaw, (in Polish) · Zbl 0900.26001
[14] Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific: World Scientific Singapore · Zbl 0998.26002
[15] Hristova, S. G.; Bainov, D. D., Monotone-iterative techniques of V. Lakshmikantham for a boundary value problem for systems of impulsive differential equations with “supremum”, J. Math. Anal. Appl., 172, 339-352 (1993) · Zbl 0772.34047
[16] Lin, Z.; Kang, S. M., Existence of monotone solutions for a nonlinear quadratic integral equation of Volterra type, Rocky Mount. J. Math., 37, 1971-1980 (2007) · Zbl 1147.45005
[17] O’Regan, D.; Meehan, M., Existence Theory for Nonlinear Integral and Integrodifferential Equations (1998), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0932.45010
[18] I. Podlubny, Fractional Differential Equations, San Diego, New York, London, 1999; I. Podlubny, Fractional Differential Equations, San Diego, New York, London, 1999 · Zbl 0924.34008
[19] I. Podlubny, A.M.A. El-Sayed, On Two Definitions of Fractional Calculus, Slovak Academy of Science, Institute of Experimental Physics, 1996; I. Podlubny, A.M.A. El-Sayed, On Two Definitions of Fractional Calculus, Slovak Academy of Science, Institute of Experimental Physics, 1996
[20] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach: Gordon and Breach Amsterdam · Zbl 0818.26003
[21] Väth, M., Volterra and Integral Equations of Vector Functions, Pure and Applied Mathematics (2000), Marcel Dekker: Marcel Dekker New York
[22] Zabrejko, P. P.; Koshelev, A. I.; Krasnosel’skii, M. A.; Mikhlin, S. G.; Rakovschik, L. S.; Stetsenko, V. J., Integral Equations (1975), Nordhoff: Nordhoff Leyden
[23] Zima, M., Applications of the spectral radius to some integral equations, Comment. Math. Univ. Carolinae, 36, 695-703 (1995) · Zbl 0845.47047
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.