AIMS Mathematics, 2019, 4(3): 821-830. doi: 10.3934/math.2019.3.821

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Monotonic solutions for a quadratic integral equation of fractional order

1 Faculty of Science, Alexandria University, Alexandria, Egypt
2 Faculty of Science, Lebanese International University, Lebanon
3 Faculty of Science, The International University of Beirut, Lebanon

In this paper we present a global existence theorem of a positive monotonic integrable solution for the mixed type nonlinear quadratic integral equation of fractional order\[x(t) = p(t) + h(t,x(t)) \int_{0}^{t} k(t,s)( f_1( s, I^\alpha f_2(s, x(s)))+g_1( s, I^\beta g_2(s, x(s))))ds,~t\in [0,1],\alpha,\beta>0\]by applying the technique of measures of weak noncompactness. As an application, we consider an initial value problem of arbitrary (fractional) order differential equations.
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1.J. A. Alamo and J. Rodriguez, Operational calculus for modified ErdélyiKober operators, Serdica Bulgaricae Math. Publ., 20 (1994), 351-363.

2.Sh. M. Al-Issa and A. M. A. El-Sayed, Positive integrable solutions for nonlinear integral and differential inclusions of fractional-orders, Commentat. Math., 49 (2009), 171-177.

3.J. Appell and E. De Pascale, Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. Un. Mat. Ital., 6 (1984), 497-515.

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

5.J. Banaś, Integrable solutions of Hammerstein and Urysohn integral equations, J. Aust. Math. Soc., 46 (1989), 61-68.

6.J. Banaś and A. Martinon, Monotonic solutions of a quadratic integral equation of Volterra type, Comput. Math. Appl., 47 (2004), 271-279.    

7.J. Banaś and B. Rzepka, Monotonic solutions of a quadratic integral equation of fractional order, J. Math. Anal. Appl., 332 (2007), 1371-1379.    

8.J. Banaś and K.Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA.60 (1980)

9.J. Banaś, M. Lecko and W. G. El-Sayed, Existence theorems of some quadratic integral equation, J. Math. Anal. Appl., 222 (1998), 276-285.    

10.M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. Int., 13 (1967), 529-539.    

11.F. S. De Blasi, On a property of the unit sphere in a Banach space, Bull. math. Soc. Sci. Math. R. S. Roumanie, 21 (1977), 259-262.

12.A. M. A. El-Sayed and H. H. G. Hashem, Integrable and continuous solutions of nonlinear quadratic integral equation, Electron. J. Qual. Theory Differ. Equations, 25 (2008), 1-10.

13.A. M. A. El-Sayed and Sh. M. Al-Issa, Monotonic continuous solution for a mixed type integral inclusion of fractional order, J. Math. Appl., 33 (2010), 27-34.

14.A. M. A. El-Sayed and Sh. M. Al-Issa, Global integrable solution for a nonlinear functional integral inclusion, SRX Mathematics, 2010 (2010).

15.A. M. A. El-Sayed and Sh. M. Al-Issa, Monotonic integrable solution for a mixed type integral and differential inclusion of fractional orders, Int. J. Differ. Equations Appl., 18 (2019).

16.A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland, 204 (2006).

17.A. C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions, Research Notes in Mathematics, 31 (1979).

18.K. S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiely and Sons Inc, (1993).

19.I. Podlubny, Fractional Differential Equations, Academic Press, San Diego-New York-london, (1999).

20.I. Podlubny and A. M. A. EL-Sayed, On two defintions of fractional calculus, Preprint UEF, Solvak Academy of science-Institute of Experimental Phys, (1996), 03-69.

21.B. Ross and K. S. Miller, An introduction to the fractional calculus and fractional differential equations, John Wiley, New York, (1993).

22.S. G. Samko, A. A. Kilbasa and O. Marichev, Integrals and derivatives of fractional order and some of their applications, Nauka i tekhnika, Minsk, (1987).

23.P. P. Zabrejko, A. I. Koshelev, M. A. Krasnoselskii, et al. Integral Equations, Noordhoff, Leyden, (1975).

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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