Abstract
In this paper, we study a coupled system of generalized Sturm–Liouville problems and Langevin fractional differential equations described by Atangana–Baleanu–Caputo (ABC for short) derivatives whose formulations are based on the notable Mittag-Leffler kernel. Prior to the main results, the equivalence of the coupled system to a nonlinear system of integral equations is proved. Once that has been done, we show in detail the existence–uniqueness and Ulam stability by the aid of fixed point theorems. Further, the continuous dependence of the solutions is extensively discussed. Some examples are given to illustrate the obtained results.
Highlights
The subject of fractional calculus is a generalization of ordinary differentiation and integration to an arbitrary order, which might be noninteger
2 Preliminaries In this subsection, we introduce some notations, definitions, properties and lemmas of fractional calculus, we present briefly the so-called operators with nonsingular kernel. and present preliminary results needed in our proofs later
4 Conclusions The theory of fractional operators with nonsingular kernels is new and we need to study the qualitative properties of differential equations involving such operators
Summary
The subject of fractional calculus is a generalization of ordinary differentiation and integration to an arbitrary order, which might be noninteger. The integral solution for the linear system of fractional differential equations (3.1) is given by the pair of functions (u1, u2) ∈ X × X, with (3.2). 3.3 Existence of solutions of (3.8) In this subsection, define the following operators: E, F : Br → X × X and T : Br → X × X by E = (E1, E2), F = (F1, F2) and T = E + F, with (Eu)(t) = E1(u1, u2), E2(u1, u2) (t) and (Fu)(t) = F1(u1, u2), F2(u1, u2) (t), (3.40)
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