Existence theory for $\psi$-Caputo fractional differential equations

Nadhir Bendrici; Abdellatif Boutiara; Malika Boumedien-Zidani

doi:10.3842/umzh.v76i9.7669

Nadhir Bendrici Laboratory of Dynamic Systems, University of Science and Technology Houari Boumediene, Bab ezzouar, Algeria
Abdellatif Boutiara Department of Mathematics and Computer Science, University of Ghardaia, Algeria
Malika Boumedien-Zidani Laboratory of Dynamic Systems, University of Science and Technology Houari Boumediene, Bab ezzouar, Algeria

DOI: https://doi.org/10.3842/umzh.v76i9.7669

Keywords: -caputo - Fractional derivatives equations - Boundary value problem - M¨onch fixed point theorem, Measure of noncompactness, $\psi$-Caputo fractional differential equations, Boundary value problem, Monch fixed point theorem, Measure of non-compacteness

Abstract

UDC 517.9

The target of this paper is to handle a nonlocal boundary-value problem for a specific kind of nonlinear fractional differential equations encapsuling a collective fractional derivative known as the $\psi$-Caputo fractional operator. The applied fractional operator generated by the kernel is of the following kind: $k(t,s)=\psi (t)-\psi(s).$ The existence of the solutions of the above-mentioned equations is tackled by using Mönch's fixed-point theorem combined with the technique of measures of noncompactness. In addition, we discuss the problem of stability within the scope of the Ulam–Hyers stability criteria for the main fractional system. Finally, an example is given to illustrate the viability of the reported results.

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