On the Solutions of a Quadratic Integral Equation of the Urysohn Type of Fractional Variable Order

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Date
2022-06-27Author
Benkerrouche, Amar
Souid, Mohammed Said
Stamov, Gani
Stamova, Ivanka
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In this manuscript we introduce a quadratic integral equation of the Urysohn type of
fractional variable order. The existence and uniqueness of solutions of the proposed fractional model
are studied by transforming it into an integral equation of fractional constant order. The obtained
new results are based on the Schauder’s fixed-point theorem and the Banach contraction principle
with the help of piece-wise constant functions. Although the used methods are very powerful, they
are not applied to the quadratic integral equation of the Urysohn type of fractional variable order.
With this research we extend the applicability of these techniques to the introduced the Urysohn
type model of fractional variable order. The applicability of the new results are demonstrated by
providing Ulam–Hyers stability criteria and an example. Moreover, the presented results lead to
future progress and expansion of the theory of fractional-order models, as well as of the concept of
entropy in the framework of fractional calculus. Further, an example is constructed to demonstrate
the reasonableness and effectiveness of the observed results.
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Mathematics
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