On The Solvability of Nonlinear Integral
Functional Equation

Mahmoud M. El-Borai; Wagdy G. El-Sayed; Faez N. Ghaffoori

doi:https://doi.org/10.14445/22315373/IJMTT-V34P509

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 34 | Number 1 | Year 2016 | Article Id. IJMTT-V34P509 | DOI : https://doi.org/10.14445/22315373/IJMTT-V34P509

On The Solvability of Nonlinear Integral Functional Equation

Mahmoud M. El-Borai, Wagdy G. El-Sayed, Faez N. Ghaffoori

Abstract

We study the existence solution of a functional Volterra integral equation in the space of Lebesgueintegrable functions on an unbounded interval by using the Schauder fixed point theorem and weak measure of noncompactness.

Keywords

Superposition operator- Caratheodory conditions- Nonlinear integral functional equationMeasure of weak noncompactness- Schauder fixed point theorem.

References

[1] J. Appell and P. P. Zabroejko, Continuity properties of the superposition operator, No. 131, Univ. Augsburg, (1986).
[2] J. Banas, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Caratheodory, conditions, Nonlinear Anal. 70 ( 2009), 3172-3179.
[3] J. Banas and W. G. El-Sayed, Monotonic solutions of a nonlinear Volterra integral equation, Aportaciones Mathematics en Memoria V. M. Onieva, Sanander, Spain, (1991), 19-26.
[4] J. Bana`s and, W. G. El-Sayed , Solvability of functional and Integral Equations in some classes of integrable functions, 1993.
[5] J. Banas and K. Goebel, Measures of noncompactness in Banach spaces, Lectures notes in Pure and Appl. Math., 60 Dekker, New York- Basel (1980).
[6] J. Banas and Z. Knab, Integrable solutions of a functional integral equation Revista Mathematica de la Univ. Complutense de Madrid, 2(1989), 31-38.
[7] J. Banas and Z. Knab, Measures of weak noncompactness and nonlinear integral equations of convolution type, J. Math.Anal. Appl., 146(1990), 353-362.
[8] J. Banas, T. Zajac, Solvability of a functional integral equation of functional order in the class of functions having limits at infinity, Nonlinear A 71 (2009), 5491-5500.
[9] M. Cichon, M. Metwali, Monotonic solutions for quadratic integral equations, Discuss. Math. Diff. Incl. 31 (2011), 157-171.
[10] F. S. De-Blasi, On a property of the unit sphere in a Banach spaces, Bull. Math. Soc. Sci. Math R.S. Roumania, 21 (1977), 259-262.
[11] J. Dieudonne, Sur les espace de K the, J. Anal.Math. (1951), 81-115.
[12] D. Dinculeanu, Vector Measures, Pergamon Press, 1967.
[13] M. M. El-Borai, The Fundamental Solution for fractional evolution equations of parapolic type, J. of Appl. Math.stochastic Analysis (JAMSA) (2004) 199-211.
[14] M. M. El-Borai, On some fractional differential equations in the Hilbert space. Journal of Discrete and ContiuousDynemicalsystem , series A, (2005) . 233-241.
[15] M. M. El-Borai, Khairia El-Said El-Nadi, and Iman G. El-Akabawi, On some integro-differential equations of fractional orders, The International J.of Contemporary Mathematics, Vol 1. No 15, (2006), pp 719-726.
[16] M. M. EL-Borai, Exact solution for some nonlinear fractional parapolic fractional partial differntial equations, Journal of Applied Mathematics and Computation 206. (2008), 141-153.
[17] M. M. El-Borai, Khairia El-Said El-Nadi, An inverse fractional abstract Cauchy problems with nonlocal conditions , Life Science Journal , (2013),10(3), 1705-1709.
[18] Mahmoud M. El-Borai, Wagdy G. El-Sayed, and Faez N. Ghaffoori, On the Cauchy problem for some parabolic fractional partial differential equations with time delays J. of Math. And Sys. Sci. 6 (2016) 194-199.
[19] M. M. El-Borai, Khairia El-Said El-Nadi, Integrated semi groups and Cauchy problems for some fractional abstract differential equations, Life Science Journal, 2013, 10(3), 793-795.
[20] Mahmoud M. El-Borai, Wagdy G. El-Sayed, and Amany M. Moter, Continuous solutions of a quadratic integral equation, Inter. J. Life Science and Math. (IJLSM), Vol. 2(5)-4, 21-30 (2015).
[21] W. G. El-Sayed, and E.I. Deebs, Monotonic solutions of a functional integral equati-on P. U. M. A. Bud.Unin. Hungary Vol. 6, No. 1, (1995), 41-50.
[22] W. G. El-Sayed, B. Rzepka, Nondecreasing solutions of quadratic integral equationof Urysohn type, Comp. Math. Appl. 51 (2006) 1065- 1074.
[23] W. G. El-Sayed, Monotonic solutions of a functional integral equation, J. Egypt. Math. Soc. Vol. 14 (2), (2006) pp 235-241.
[24] W. G. El-Sayed, On the Solvability of a Functional Integral Equation, East-West J. Math. Vol. 10, No.2 (2008) 145-152.
[25] G. Emmanuele, About the existence of integrable solutions of a functional integral equation, RevistaMathematica de la Univ. ComplutensedeMadrid, 4 (1991), 65-69.
[26] K. Goebel and W. A.Kirk, Topics in metric fixed point theory, Gambridge Univ. Press (1990).
[27] J. K. Hale, Theory of functional differential equations, Springer- verlag, Berlin (1977).
[28] Mohamed M. A. Metwali, Solvability of functional quadratic integral equations with perturbation, Opuscula Math. 33, No. 4 (2013), 725-739.
[29] B. Ricceri and A. Villani, Separability and ScorzaDragoni'sproperty, Le Math., 37 (1982), 156-161.
[30] D.Saveljeva, Quadratic and cubic spline collection for Volterra integral equations, University of Tartu (2006).
[31] P. P. Zarejko, A.I. Koshlev, M. A. Krasnoselskii, S. G. Mikhlin, L. S. Rakovshchik, V. J. Stecenko, Integral Equations, Noordhoff, Leyden, 1975.

Citation :

Mahmoud M. El-Borai, Wagdy G. El-Sayed, Faez N. Ghaffoori, "On The Solvability of Nonlinear Integral Functional Equation," International Journal of Mathematics Trends and Technology (IJMTT), vol. 34, no. 1, pp. 39-44, 2016. Crossref, https://doi.org/10.14445/22315373/IJMTT-V34P509