Equivalence Between Fractional Differential Problems and Their Corresponding Integral Forms with the Pettis Integral

Mieczysław Cichoń (), Wafa Shammakh, Kinga Cichoń and Hussein A. H. Salem
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Mieczysław Cichoń: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
Wafa Shammakh: Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
Kinga Cichoń: Institute of Mathematics, Faculty of Automatic Control, Robotics and Electrical Engineering, Poznan University of Technology, Piotrowo 3A, 60-965 Poznań, Poland
Hussein A. H. Salem: Department of Mathematics and Computer Science, Faculty of Sciences, Alexandria University, Alexandria 5424041, Egypt

Mathematics, 2024, vol. 12, issue 23, 1-29

Abstract: The problem of equivalence between differential and integral problems is absolutely crucial when applying solution methods based on operators and their properties in function spaces. In this paper, we complement the solution of this important problem by considering the case of general derivatives and integrals of fractional order for vector functions for weak topology. Even if a Caputo differential fractional order problem has a right-hand side that is weakly continuous, the equivalence between the differential and integral forms may be affected. In this paper, we present a complete solution to this problem using fractional order Pettis integrals and suitably defined pseudo-derivatives, taking care to construct appropriate Hölder-type spaces on which the operators under study are mutually inverse. In this paper, we prove, in a number of cases, the equivalence of differential and integral problems in Hölder spaces and, by means of appropriate counter-examples, investigate cases where this property of the problems is absent.

Keywords: fractional calculus; pseudo-derivatives; Pettis integral; Hölder spaces (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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