Langevin equation with nonlocal boundary conditions involving a $ \psi $-Caputo fractional operators of different orders

ψ$$ \psi $$‐Caputo type time‐delay Langevin equations with two general fractional orders

“…However, to the best of our knowledge, few results can be found on the existence and the Ulam–Hyers stability of solutions for fractional Langevin equations with the $$$ \psi $$$‐Caputo fractional derivative except that of previous works 35,36 . Seemab et al 35 discuss the solvability of Langevin equations with two Hadamard fractional derivatives by using Schauder's fixed point theorem and Banach's fixed point theorem. Baitiche et al 36 study the Ulam–Hyers stability of solutions for a new form of nonlinear fractional Langevin differential equations involving two fractional orders in the $$$ \psi $$$‐Caputo sense.…”

“…While it is involved in many sorts of real‐world problems, simultaneously it has been developed as in previous studies 11–28 . However, to the best of our knowledge, few results can be found on the existence and the Ulam–Hyers stability of solutions for fractional Langevin equations with the $$$ \psi $$$‐Caputo fractional derivative except that of previous works 35,36 . Seemab et al 35 discuss the solvability of Langevin equations with two Hadamard fractional derivatives by using Schauder's fixed point theorem and Banach's fixed point theorem.…”

Euler–Lagrange-Type Equations for Functionals Involving Fractional Operators and Antiderivatives

“…Later, several definitions were presented, most of them motivated by the works of Riemann and Liouville. More recently, with the concepts of fractional operators with respect to another function, we can generalize some of those operators into a single one and present results valid for a wide class of fractional operators [5,[7][8][9][10]. We remark that these fractional operators depend on an arbitrary function g, and for particular choices of such function, we can recover some well-known fractional operators like the Riemann-Liouvile, the Caputo, the Hadamard, or the Erdelyi-Kober fractional operators.…”

Qualitative Analysis for Multiterm Langevin Systems with Generalized Caputo Fractional Operators of Different Orders

“…is recently defined fractional operator could model more precisely the process utilizing differential kernel. In order to evolve these definitions, special kernels and some kinds of operators are selected to apply on FDEs; for more details, we refer to some recent results associated with this development (see [20][21][22][23][24][25][26][27][28][29]).…”