Journal of Mathematics and Applications
03/41, DOI: 10.7862/rf.2018.3
On a Cubic Integral Equation of Urysohn Type with Linear Perturbation of Second Kind
Hamed Kamal Awad, Mohamed Abdalla Darwish, Mohamed M.A. Metwali
DOI: 10.7862/rf.2018.3
Abstract
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.
References
- J. Appell, Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. Math. Anal. Appl. 3 (1981) 251-263.
- J. Appell, C. Chen, How to solve Hammerstein equations, J. Integral Equations Appl. 18 (2006) 287-296.
- 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.
- J. Banaś, K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics 60 Marcel Dekker, New York, 1980.
- J. Banaś, L. Olszowy, Measures of noncompactness related to monotonicity, Comment. Math. 41 (2001) 13-23.
- 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.
- T.A. Burton, Volterra Integral and Differential Equations, Academic Press, New York, 1983.
- K.M. Case, P.F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading, MA, 1967.
- 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.
- M.A. Darwish, On integral equations of Urysohn-Volterra type, Appl. Math. Comput. 136 (2003) 93-98.
- M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311 (2005) 112-119.
- M.A. Darwish, On monotonic solutions of a singular quadratic integral equation with supremum, Dynam. Syst. Appl. 17 (2008) 539-549.
- K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
- W.G. El-Sayed, A.A. El-Bary, M.A. Darwish, Solvability of Urysohn integral equation, Appl. Math. Comput. 145 (2003) 487-493.
- W.G. El-Sayed, B. Rzepka, Nondecreasing solutions of a quadratic integral equation of Urysohn type, Comput. Math. Appl. 51 (2006) 1065-1074.
- 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.
- A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
- S. Hu, M. Khavani, W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Anal. 34 (1989) 261-266.
- C.T. Kelly, Approximation of solutions of some quadratic integral equations in transport theory, J. Integral Eq. 4 (1982) 221-237.
- D. O'Regan, M. Meehan, Existence Theory for Nonlinear Integral and Integro-differential Equations, Kluwer Academic Publishers, Dordrecht, 1998.
- M. Väth, Volterra and Integral Equations of Vector Functions, Monographs and Textbooks in Pure and Applied Mathematics 224, Marcel Dekker, Inc., New York, 2000.
- P.P. Zabrejko et al., Integral Equations - a Reference Text, Noordhoff International Publishing, The Netherlands 1975 (Russian edition: Nauka, Moscow, 1968).
About this Article
TITLE:
On a Cubic Integral Equation of Urysohn Type with Linear Perturbation of Second Kind
AUTHORS:
Hamed Kamal Awad (1)
Mohamed Abdalla Darwish (2)
Mohamed M.A. Metwali (3)
AUTHORS AFFILIATIONS:
(1) Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, EGYPT
(2) Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, EGYPT
(3) Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, EGYPT
JOURNAL:
Journal of Mathematics and Applications
03/41
KEY WORDS AND PHRASES:
Cubic integral equation; Darbo's fixed point theorem; Monotonicity measure of noncompactness
FULL TEXT:
http://doi.prz.edu.pl/pl/pdf/jma/66
DOI:
10.7862/rf.2018.3
URL:
http://dx.doi.org/10.7862/rf.2018.3
RECEIVED:
2018-11-09
COPYRIGHT:
Publishing House of Rzeszow University of Technology Powstańców Warszawy 12, 35-959 Rzeszow