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Monotonic solutions for a quadratic integral equation of fractional order

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

*Corresponding author: Sh. M. Al-Issa shorouk.alissa@liu.edu.lb

Math2019,3,821doi:10.3934/math.2019.3.821

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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