On systems of fractional differential equations with the ψ‐Caputo derivative and their applications

Abstract

Systems of fractional differential equations with a general form of fractional derivative are considered. A unique continuous solution is derived using the Banach fixed point theorem. Additionally, the dependence of the solution on the fractional order and on the initial conditions are studied. Then the stability of autonomous linear fractional differential systems with order 0 < < 1 of the -Caputo derivative is investigated. Finally, an application of the theoretical results to the problem of the leader-follower consensus for fractional multi-agent systems is presented. Sufficient conditions are given to ensure that the tracking errors asymptotically converge to zero. The results of the paper are illustrated by some examples.

Fractional differential equations and Volterra–Stieltjes integral equations of the second kind

“…subject to the initial conditions Regarding the existence and uniqueness of solutions for nonlinear fractional differential equation with the ψ-Caputo derivative of type (1), we refer the reader to Almeida et al (2018). For stability results, we suggest the work of Almeida et al (2019).…”

“…A more general unifying perspective to the subject was proposed in Agrawal (2010), Klimek and Lupa (2013), Malinowska et al (2015), by considering fractional operators depending on general kernels. In this work, we follow the special case of this approach that was developed in Almeida (2017a, b), Almeida et al (2018Almeida et al ( , 2019, Garra et al (2019), Kilbas et al (2006), Yang and Machado (2017). Namely, we focus on nonlinear fractional differential equations involving a Caputo-type fractional derivative with respect to another function, called ψ-Caputo derivative.…”

Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving ψ-Riesz-Caputo Fractional Derivative

“…In this section, we recall some notation, definitions and preliminaries about fractional calculus [6,7,21], ψ-Caputo fractional calculus [3][4][5]22,23], and Riesz or Riesz-Caputo fractional derivative [17][18][19]. Definition 1 ([6]).…”

“…Fractional order models, providing excellent description of memory and hereditary processes, are more adequate than integer order ones. Some recent contributions to fractional differential equations and inclusions have been carried out, see the monographs [1][2][3][4][5][6][7][8], and the references cited therein. The study of fractional differential equations or inclusions with anti-periodic boundary problems, which are applied in different fields, such as physics, chemical engineering, economics, populations dynamics and so on, have recently received considerable attention, see the references ( [9,10]) and papers cited therein.There are several definitions of fractional differential derivatives and integrals, such like Caputo type, Rimann-Liouville type, Hadamard type and Erdelyi-Kober type and so on.…”

On Hyers–Ulam stability of a multi-order boundary value problems via Riemann–Liouville derivatives and integrals

“…(see [1][2][3][4][5][6][7][8][9][10][11]). Many researchers helped in developments on the existence and uniqueness results of fractional differential equations [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Stability is a notion in physics, and most phenomena include the concept.…”