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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Keywords fractional calculus; quadratic integral equation; fixed point theory; measure of noncompactness

Citation: A. M. A. El-Sayed, Sh. M. Al-Issa. Monotonic solutions for a quadratic integral equation of fractional order. AIMS Mathematics, 2019, 4(3): 821-830. doi: 10.3934/math.2019.3.821


  • 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).


This article has been cited by

  • 1. Ahmed Mohamed El-Sayed, Shorouk Mahmoud Al-Issa, On the existence of solutions of a set-valued functional integral equation of Volterra–Stieltjes type and some applications, Advances in Difference Equations, 2020, 2020, 1, 10.1186/s13662-020-2531-4

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