Monotonic solutions for a quadratic integral equation of fractional order

A. M. A. El-Sayed; Sh. M. Al-Issa; A. M. A. El-Sayed; Sh. M. Al-Issa

doi:10.3934/math.2019.3.821

AIMS Mathematics

2019, Volume 4, Issue 3: 821-830. doi: 10.3934/math.2019.3.821

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Research article

Monotonic solutions for a quadratic integral equation of fractional order

A. M. A. El-Sayed ¹,
Sh. M. Al-Issa ^{2,3
,
,}

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

Received: 18 April 2019 Accepted: 30 June 2019 Published: 16 July 2019

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 \gt 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.
- 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[J]. AIMS Mathematics, 2019, 4(3): 821-830. doi: 10.3934/math.2019.3.821

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Abstract

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 \gt 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.

References

[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. doi: 10.1016/0362-546X(88)90038-7
[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. doi: 10.1016/S0898-1221(04)90024-7
[7]	J. Banaś and B. Rzepka, Monotonic solutions of a quadratic integral equation of fractional order, J. Math. Anal. Appl., 332 (2007), 1371-1379. doi: 10.1016/j.jmaa.2006.11.008
[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. doi: 10.1006/jmaa.1998.5941
[10]	M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. Int., 13 (1967), 529-539. doi: 10.1111/j.1365-246X.1967.tb02303.x
[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).

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