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Open AccessArticle
Fractional Bagley-Torvik Problem Analysis with Hilfer Fractional Derivatives and Pettis Function Space
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Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
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Faculty of Science, Derna University, Derna 417230, Libya
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Department of Mathematics and Computer Science, Faculty of Sciences, Alexandria University, Alexandria 5424041, Egypt
4
Faculty of Automatic Control, Robotics and Electrical Engineering, Poznan University of Technology, Piotrowo 3A, 60-965 Poznań, Poland
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(11), 743; https://doi.org/10.3390/fractalfract9110743 (registering DOI)
Submission received: 17 October 2025
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Revised: 2 November 2025
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Accepted: 13 November 2025
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Published: 17 November 2025
Abstract
This paper analyzes the Bagley–Torvik fractional-order equation with generalized fractional Hilfer derivatives of two orders for functions in Banach spaces under conditions expressed in the language of weak topology. We develop a comprehensive theory of fractional-order differential equations of various orders. Our focus is on the equivalence results (or the lack thereof) of this new class of fractional-order Hilfer operators and on maximizing the regularity of the solution. To this end, we examine the equivalence of differential problems involving pseudo-derivatives and integral problems involving Pettis integrals. Our results are novel, even within the context of integer-order differential equations. Another objective is to incorporate fractional-order problems into the growing research field that uses weak topology and function spaces to study vector-valued functions. The auxiliary results obtained in this article are general and applicable beyond its scope.
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MDPI and ACS Style
Cichoń, M.; Al-Fadel, M.M.A.; Salem, H.A.H.; Cichoń, K.
Fractional Bagley-Torvik Problem Analysis with Hilfer Fractional Derivatives and Pettis Function Space. Fractal Fract. 2025, 9, 743.
https://doi.org/10.3390/fractalfract9110743
AMA Style
Cichoń M, Al-Fadel MMA, Salem HAH, Cichoń K.
Fractional Bagley-Torvik Problem Analysis with Hilfer Fractional Derivatives and Pettis Function Space. Fractal and Fractional. 2025; 9(11):743.
https://doi.org/10.3390/fractalfract9110743
Chicago/Turabian Style
Cichoń, Mieczysław, Masouda M. A. Al-Fadel, Hussein A. H. Salem, and Kinga Cichoń.
2025. "Fractional Bagley-Torvik Problem Analysis with Hilfer Fractional Derivatives and Pettis Function Space" Fractal and Fractional 9, no. 11: 743.
https://doi.org/10.3390/fractalfract9110743
APA Style
Cichoń, M., Al-Fadel, M. M. A., Salem, H. A. H., & Cichoń, K.
(2025). Fractional Bagley-Torvik Problem Analysis with Hilfer Fractional Derivatives and Pettis Function Space. Fractal and Fractional, 9(11), 743.
https://doi.org/10.3390/fractalfract9110743
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