Global Integrable Solution for a Nonlinear Functional Integral Inclusion

Abstract

We study the global existence of positive integrable solution for the functional integral inclusion of fractional order x(t)∈p(t)+F1(t,Iαf2(t,x(φ(t))), t∈(0,1), α≥0, where F1(t,x(t)) is a set-valued function defined on (0,1)×R+.

References

1. J. Banaś and Z. Knap, “Integrable solutions of a functional-integral equation,” Revista Matemática de la Universidad Complutense de Madrid, vol. 2, no. 1, pp. 31–38, 1989. View at: Google Scholar
2. J. Banas and W. G. El-Sayed, Solvability of Functional and Integral Equations in Some Classes of Integrable Functions, Politechnika Rzeszowska, Rzeszów, Poland, 1993.
3. G. Emmanuele, “About the existence of integrable solutions of a functional-integral equation,” Revista Matemática de la Universidad Complutense de Madrid, vol. 4, no. 1, pp. 65–69, 1991. View at: Google Scholar
4. G. Emmanuele, “Integrable solutions of a functional-integral equation,” Journal of Integral Equations and Applications, vol. 4, no. 1, pp. 89–94, 1992. View at: Publisher Site | Google Scholar | MathSciNet
5. A. M. A. El-Sayed, N. Sherif, and I. Abou El-Farag, “A nonlinear operator functional equation of Volterra type,” Applied Mathematics and Computation, vol. 148, no. 3, pp. 665–679, 2004. View at: Publisher Site | Google Scholar | MathSciNet
6. I. Podlubny and A. M. A. EL-Sayed, On Two Defintions of Fractional Calculus, Slovak Academy of Sciences, Institute of Experimental Physics, Watsonova, Slovakia, 1996.
7. I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, Calif, USA, 1999.
8. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
9. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Orders and Some of Their Applications, Nauka i Tekhnika, Minsk, Belarus, 1987.
10. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
11. J. Dungundji and A. Grance, Fixed Piont Theory, Monografie Mathematyczne, PWN, Warsaw, Poland, 1982.
12. C. Swartz, Measure, Integration and Function Spaces, World Scientific, Singapore, 1994.
13. A. Bressan and G. Colombo, “Extensions and selections of maps with decomposable values,” Studia Mathematica, vol. 90, no. 1, pp. 69–86, 1988. View at: Google Scholar
14. M. Cichoń, A. M. A. El-Sayed, and A. H. Hussien, “Existence theorems for nonlinear functional integral equations of fractional orders,” Commentationes Mathematicae, vol. 41, pp. 59–67, 2001. View at: Google Scholar
15. M. Cichoń, “Multivalued perturbations of $m$-accretive differential inclusions in a non-separable Banach space,” Commentationes Mathematicae, vol. 32, pp. 11–17, 1992. View at: Google Scholar
16. D. Repovš and P. V. Semenov, Continuous Selections of Multivalued Mappings, vol. 455 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.

Copyright

Copyright © 2010 A. M. A. El-Sayed and Sh. M. Al-Issa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.