# Existence of integrable solutions for integro-differential inclusions of fractional order; coupled system approach

Volume 13, Issue 4, pp 180--186
Publication Date: January 08, 2020 Submission Date: August 26, 2019 Revision Date: October 31, 2019 Accteptance Date: November 27, 2019
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### Authors

A. M. A. El-Sayed - Faculty of Science, Alexandria University, Alexandria, Egypt. Sh. M. Al-Issa - Faculty of Science, Lebanes International University, Beirut, Lebanon. - Faculty of Science, The International University of Beirut, Saida, Lebanon.

### Abstract

In this article, we establish the existence of solutions for a functional integral equation of fractional order. The study upholds the case when the set-valued function has $L^1$-Caratheodory selections, we reformulate the functional integral inclusion according to these selections via a classical fixed point theorem of Schauder and present theorem for the existence of integrable solutions. As an application, the existence of solutions of nonlinear functional integro-differential inclusion with an initial value, and the initial value problem for the arbitrary-order differential inclusion will be studied.

### Share and Cite

##### ISRP Style

A. M. A. El-Sayed, Sh. M. Al-Issa, Existence of integrable solutions for integro-differential inclusions of fractional order; coupled system approach, Journal of Nonlinear Sciences and Applications, 13 (2020), no. 4, 180--186

##### AMA Style

El-Sayed A. M. A., Al-Issa Sh. M., Existence of integrable solutions for integro-differential inclusions of fractional order; coupled system approach. J. Nonlinear Sci. Appl. (2020); 13(4):180--186

##### Chicago/Turabian Style

El-Sayed, A. M. A., Al-Issa, Sh. M.. "Existence of integrable solutions for integro-differential inclusions of fractional order; coupled system approach." Journal of Nonlinear Sciences and Applications, 13, no. 4 (2020): 180--186

### Keywords

• Fractional calculus
• integro-differential inclusion
• $L^1$-Caratheodory selections
• Schauder fixed point principle
• Kolmogorov compactness criterion

•  26A33
•  47H30
•  47G10

### References

• [1] S. Al-Issa, A. M. A. El-Sayed, Positive integrable solutions for nonlinear integral and differential inclusions of fractional-orders, Comment. Math., 49 (2009), 171--177

• [2] J.-P. Aubin A. Cellina, Differential Inclusion, Springer-Verlag, Berlin (1984)

• [3] J. Banaś, On the superposition operator and integrable solutions of some functional equations, Nonlinear Anal., 12 (1988), 777--784

• [4] J. Banaś, Integrable solutions of Hammerstein and Urysohn integral equations, J. Austral. Math. Soc. Ser. A, 46 (1989), 61--68

• [5] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl., 20 (1967), 197--228

• [6] A. Cellina, S. Solimini, Continuous extensions of selections, Bull. Polish Acad. Sci. Math., 35 (1989), 573--581

• [7] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin (1985)

• [8] B. C. Dhage, A functional integral inclusion involving discontinuities, Fixed Point Theory, 5 (2004), 53--64

• [9] J. Dugundji,, A. Granas, Fixed Point Theory, Państwowe Wydawnictwo Naukowe (PWN), Warsaw (1982)

• [10] A. M. A. El-Sayed, A.-G. Ibrahim, Multivalued fractional differential equations, Appl. Math. Comput., 68 (1995), 15--25

• [11] A. M. A. El-Sayed, A.-G. Ibrahim, Set-valued integral equations of fractional-orders, Appl. Math. Comput., 118 (2001), 113--121

• [12] A. M. A. El-Sayed, S. M. Al-Issa, Monotonic continuous solution for a mixed type integral inclusion of fractional order, J. Math. Appl., 33 (2010), 27--34

• [13] A. M. A. El-Sayed, S. M. Al-Issa, Existence of continuous solutions for nonlinear functional differential and integral inclusions, Malaya J. Mat., 7 (2019), 541--544

• [14] A. M. A. El-Sayed, S. M. Al-Issa, Monotonic integrable solution for a mixed type integral and differential inclusion of fractional orders, Int. J. Differ. Equations Appl., 18 (2019), 10 pages

• [15] A. M. A. El-Sayed, S. M. Al-Issa, Monotonic solutions for a quadratic integral equation of fractional order, AIMS Mathematics, 4 (2019), 821--830

• [16] G. Emmanuele, Integrable solutions of Hammerstein integral equations, Appl. Anal., 50 (1993), 277--284

• [17] A. Fyszkowski, Continuous selection for a class of non-convex multivalued maps, Studia Math., 76 (1983), 163--174

• [18] A.-G. Ibrahim, A. M. A. El-Sayed, Definite integral of fractional order for set-valued function, J. Fract. Calc., 11 (1997), 81--87

• [19] K. Kuratowski, C. Ryll-Nardzewski, Ageneral theorem on selectors, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 13 (1965), 397--403

• [20] K. S. Miller, B. Ross, An Introduction to the fractional calculus and fractional differential equations, John Wiley & Sons, New York (1993)

• [21] D. O'Regan, Integral inclusions of upper semi-continuous or lower semi-continuous type, Proc. Amer. Math. Soc., 124 (1996), 2391--2399

• [22] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999)

• [23] I. Podlubny, A. M. A. EL-Sayed, On two defintions of fractional calculus, Solvak Academy Sci.-Ins. Eyperimental Phys., 1996 (1996), 3--96

• [24] S. G. Samko, A. A. Kilbas, O. I. Marichev, Integrals and Derivatives of Fractional Orders and Some of their Applications, Nauka i Teknika, Minsk (1987)

• [25] C. Swartz, Measure, integration and function spaces, World Scientific Publishing Co., River Edge (1994)