On a cubic integral equation of Urysohn type with linear perturbation of second kind

• Autorzy: Awad H. K. , Darwish M. A. , Metwali M. M. A.

Identyfikatory

DOI

10.7862/rf.2018.3

• Języki publikacji: EN

Abstrakty

EN

In this paper, we concern by a very general cubic integral equation and we prove that this equation has a solution in C[0; 1]. We apply the measure of noncompactness introduced by Banaś and Olszowy and Darbo's fixed point theorem to establish the proof of our main result.

Słowa kluczowe

EN

cubic integral equation; Darbo's fixed point theorem; monotonicity; measure of noncompactness

PL

równanie całkowe; twierdzenie Darbo o punkcie stałym; monotoniczność; miara braku zgodności

• Wydawca: Oficyna Wydawnicza Politechniki Rzeszowskiej
• Czasopismo: Journal of Mathematics and Applications
• Rocznik: 2018
• Tom: Vol. 41
• Strony: 29--38
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Awad H. K.

• Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt

autor

Darwish M. A.

• Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt

autor

Metwali M. M. A.

• Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt

Bibliografia

• [1] J. Appell, Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. Math. Anal. Appl. 3 (1981) 251-263.
• [2] J. Appell, C. Chen, How to solve Hammerstein equations, J. Integral Equations Appl. 18 (2006) 287-296.
• [3] H.K. Awad, M.A. Darwish, On monotonic solutions of a cubic Urysohn Integral equation with linear modification of the argument, Adv. Dyn. Syst. Appl. 13 (2018) 91-99.
• [4] J. Banaś, K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics 60 Marcel Dekker, New York, 1980.
• [5] J. Banaś, L. Olszowy, Measures of noncompactness related to monotonicity, Comment. Math. 41 (2001) 13-23.
• [6] M. Benchohra, M.A. Darwish, On unique solvability of quadratic integral equations with linear modification of the argument, Miskolc Math. Notes 10 (2009) 3-10.
• [7] T.A. Burton, Volterra Integral and Differential Equations, Academic Press, New York, 1983.
• [8] K.M. Case, P.F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading, MA, 1967.
• [9] J. Caballero, D. O'Regan, K. Sadarangani, On nondecreasing solutions of cubic integral equations of Urysohn type, Comment. Math. (Prace Mat.) 44 (2004) 39-53.
• [10] M.A. Darwish, On integral equations of Urysohn-Volterra type, Appl. Math. Comput. 136 (2003) 93-98.
• [11] M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311 (2005) 112-119.
• [12] M.A. Darwish, On monotonic solutions of a singular quadratic integral equation with supremum, Dynam. Syst. Appl. 17 (2008) 539-549.
• [13] K. Deimling, Nonlinear Fuctional Analysis, Springer-Verlag, Berlin, 1985.
• [14] W.G. El-Sayed, A.A. El-Bary, M.A. Darwish, Solvability of Urysohn integral equation, Appl. Math. Comput. 145 (2003) 487-493.
• [15] W.G. El-Sayed, B. Rzepka, Nondecreasing solutions of a quadratic integral equation of Urysohn type, Comput. Math. Appl. 51 (2006) 1065-1074.
• [16] D. Franco, G. Infante, D. O'Regan, Positive and nontrivial solutions for the Urysohn integral equation, Acta Math. Sin. (Engl. Ser.) 22 (2006) 1745-1750.
• [17] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
• [18] S. Hu, M. Khavani, W. Zhuang, Integral equations arrising in the kinetic theory of gases, Appl. Anal. 34 (1989) 261-266.
• [19] C.T. Kelly, Approximation of solutions of some quadratic integral equations in transport theory, J. Integral Eq. 4 (1982) 221-237.
• [20] D. O'Regan, M. Meehan, Existence Theory for Nonlinear Integral and Integro-differential Equations, Kluwer Academic Publishers, Dordrecht, 1998.
• [21] M. Väth, Volterra and Integral Equations of Vector Functions, Monographs and Textbooks in Pure and Applied Mathematics 224, Marcel Dekker, Inc., New York, 2000.
• [22] P.P. Zabrejko et al., Integral Equations - a Reference Text, Noordhoff International Publishing, The Netherlands 1975 (Russian edition: Nauka, Moscow, 1968).

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).