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

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