On the existence of continuous solutions of a nonlinear quadratic fractional integral equation

Authors

  • Wagdy G. El-Sayed Faculty of Science, Alexandria University, Alexandria, Egypt https://orcid.org/0000-0003-3527-0206
  • Mahmoud M. El-Borai Faculty of Science, Alexandria University, Alexandria, Egypt
  • Mohamed M.A. metwali Department of mathematics, Faculty of Science , Damanhour Universty, Egypt
  • Nagwa I. Shemais Department of mathematics, Faculty of Science , Damanhour Universty, Egypt

DOI:

https://doi.org/10.24297/jam.v19i.8802

Keywords:

fractional calculus, Darbo’s fixed point theorem, measure of noncompactnes, Quadratic integral equation

Abstract

We prove an existence theorem for a nonlinear quadratic integral equation of fractional order, in the Banach space of real functions defined and continuous on a closed interval. This equation contains as a special case numerous integral equation studied by other authors. Finally, we give an example for indicating the natural realizations of our abstract result presented in this paper.

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References

A. Aghajani, J. Banas, N. Sabzali, Some generalizations of Darbo fixed point therom and applications, Bull. Belg. Math. Soc. simon Stevin. 20(2)(2013), 345-358. https://doi.org/10.36045/bbms/1369316549

A. Aghajani, J. Banas, Y. Jalilian, Existence of solutions for a class of nonlinear Volterra singular integral equations, Comput. Math. Appl. 62(2011), 1215-1227. https://doi.org/10.1016/j.camwa.2011.03.049

A. Aghajani, R. Allahyari, M. Mursaleen, Ageneralization of Darbo’s theorm with application to the solvability of systems of integral equations, J. Comput. Appl. Math. 260(2014), 68-77.https://doi.org/10.1016/j.cam.2013.09.039

D. Beglov, B. Roux, An integral equation to describe the solvation of polar molecules in liquid water, J. Phys. Chem. B. 101(1997), 7821-7826. https://doi.org/10.1021/jp971083h

G. Darbo, Punti untiti in transformazioni a condominio noncompatto, Rend. Sem. Mat. Univ. Padora 24 (1955), 84-92.http://www. numdam. org

H.W. Hethcote, D.W. Tudor, Integral equation models for endemic infectious diseases, J. Math. Biology. 9(1980), 37-47, doi:10.1007/BF00276034

HA. Schenck, Improved integral formulation for acoustic radiation problem, J Acoust Soc Am. 44 (1968), 41-58. https://doi.org/10.1121/1.1911085

I.M. Olaru, Generalization of an integral equation related to some epidemic models, Carpathian J. Math. 26(1)(2010), 92-96.http://www.jstor.org/stable143999436

J. Banas and A. Martinon, Monotonic solutions of quadratic integral equation of voltera type, Comput. Math. Appl. 47(2004), 271-279. https://doi.org/10.1016/S0898-1221(04)90024-7

J. Banas and B. Rzepka, Monotonic solutions of quadratic integral equation of fractional order, J. Math. Anal. Appl. 332(2007), 1370-1378.

J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Lect. Notes in Math. 60, M. Dekker, New york and Basel, 1980.

J. Banas, M. Lecko and w. G. El-sayed, Existence theorem of some quadratic integral equations, J. Math. Anal. Appl. 222(1998), 276-285. https://doi.org/10.1006/jmaa.1998.5941

J. Appell, P.P. Zabrejko, Nonlinear Superposition operator, Cambridge Tracts in Mathematics.95,Cambridge University Press, 1990. https://doi.org/10.1017/CBO9780511897450

J. Sladek, V. Sladek, S.N. Atluri, Local boundary integral equation(LBIE)method for solving problem of elasticity with nonhomogeneous material properties, Computational Mechanics. 24(2000), 456-462. https://doi.org/10.1007/s004660050005

K. Sadarangani, B. Samet, Solvability of a fractional integral equation with the concept of measure of noncompactness, Bull. Belg. Math. Soc. Simon Stevin. 24 (2017), 1-17. https://doi.org/10.36045/bbms/1489888812

M.A. Darwish, On monotonic solutions of a singular quadratic integral equation with supremum,Dynam. Systems appl. 17(20008), 539-550.

M.A. Darwish, On quadratic integral equation of fractional orders,J. Math. Anal. Appl. 311(2005), 112-119. https://doi.org/10.1016/j.jmaa.2005.02.012

M.A. Darwish, On Erdélyi -Kober fractional Urysohn-Volterra quadratic integral equation,Appl. Math. Comput. 273(2016), 562-569. https://doi.org/10.1016/j.amc.2015.10.040

M.A. Darwish, K. Sadarangani, On quadratic integral equation with supremum involving Erdélyi -Kober fractional order,Mathematische Nachrichten. 288(5-6)(2015),566-576.

M. Benchohra, D. Seba, Integral equations of fractional order with multiple time delays in banach spaces,Electron. J. differential Equations. 65(2012), 1-8.

M. Cichon, M.A. Metwali, On quadratic integral equations in Orlicz spaces,j. Math. Anal. Appl. 387(2012), 419-432. https://doi.org/10.1016/j.jmaa.2011.09.013

R. Agarwal, M. Meehan, D. O’Regan, Fixed point Theory and Applications, Cambridge University Press, 2004.

R.P. Agarwal, B. Samet, An existence result for a class of nonlinear integral equations of fractional orders, Nonlinear Analysis. Modelling and Control. 21(5)(2016), 616- 629. https://doi.org/10.15388/NA.2016.5.10

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific. Singapore, 2000.

https://doi.org/10.1142/3779

S. Chandraseker, Relative transfer, Dover publications, Newyork, 1960.

S.Hu, M. Khavani and W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Anal. 34(1989), 261-266. https://doi.org/10.1080/00036818908839899

U. Çakan, I. Ozdemir, An application of Darbo fixed-point theorem to a class of functional integral equations,Numer. Funct. Anal. Optim. 36(2015), 29-40. https://doi.org/10.1080/01630563.2014.951771

V. Muresan, Volterra integral equations with iterations of linear modification of argument, Novi Sad J. Math. 33(2003), 1-10.

W.G. El-Sayed, B. Rzepka, Nondecreasing solutions of a quadratic integral equation of Urysohn type, Comput. Math. Appl. 51 (2006), 1065-1074. https://doi.org/10.1016/j.camwa.2005.08.033

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Published

2020-07-28

How to Cite

Wagdy G. El-Sayed, Mahmoud M. El-Borai, Mohamed M.A. metwali, & Nagwa I. Shemais. (2020). On the existence of continuous solutions of a nonlinear quadratic fractional integral equation. JOURNAL OF ADVANCES IN MATHEMATICS, 19, 14-25. https://doi.org/10.24297/jam.v19i.8802

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