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Article

Analytical Study of a ϕ− Fractional Order Quadratic Functional Integral Equation

1
Faculty of Science, Alexandria University, Alexandria 21544, Egypt
2
Department of Mathematics, Faculty of Arts and Sciences, The International University of Beirut, Beirut 1107, Lebanon
3
Department of Mathematics, Faculty of Arts and Sciences, Lebanese International University, Saida 1600, Lebanon
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Academic Editor: Sotiris K. Ntouyas
Foundations 2022, 2(1), 167-183; https://doi.org/10.3390/foundations2010010
Received: 9 December 2021 / Revised: 11 January 2022 / Accepted: 12 January 2022 / Published: 25 January 2022
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Quadratic integral equations of fractional order have been studied from different views. Here we shall study the existence of continuous solutions of a ϕ fractional-orders quadratic functional integral equation, establish some properties of these solutions and prove the existence of maximal and minimal solutions of that quadratic integral equation. Moreover, we introduce some particular cases to illustrate our results. View Full-Text
Keywords: Carathéodory theorem; ϕ− fractional integration; quadratic integral equation; continuous solution; maximal and minimal solutions Carathéodory theorem; ϕ− fractional integration; quadratic integral equation; continuous solution; maximal and minimal solutions
MDPI and ACS Style

El-Sayed, A.M.A.; Hashem, H.H.G.; Al-Issa, S.M. Analytical Study of a ϕ− Fractional Order Quadratic Functional Integral Equation. Foundations 2022, 2, 167-183. https://doi.org/10.3390/foundations2010010

AMA Style

El-Sayed AMA, Hashem HHG, Al-Issa SM. Analytical Study of a ϕ− Fractional Order Quadratic Functional Integral Equation. Foundations. 2022; 2(1):167-183. https://doi.org/10.3390/foundations2010010

Chicago/Turabian Style

El-Sayed, Ahmed M. A., Hind H. G. Hashem, and Shorouk M. Al-Issa. 2022. "Analytical Study of a ϕ− Fractional Order Quadratic Functional Integral Equation" Foundations 2, no. 1: 167-183. https://doi.org/10.3390/foundations2010010

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