Stability analysis for a class of implicit fractional differential equations involving Atangana–Baleanu fractional derivative

Abstract

Some fundamental conditions and hypotheses are established to ensure the existence, uniqueness, and stability to a class of implicit boundary value problems (BVPs) with Atangana–Baleanu–Caputo type derivative and integral. The required results are established by utilizing the Banach contraction mapping principle and fixed point theorem of Krasnoselskii. In addition, various types of stability results including Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam–Rassias, and generalized Hyers–Ulam–Rassias stability are formulated for the problem under consideration. Pertinent examples are given to justify the results we obtain.

On Implicit Time–Fractal–Fractional Differential Equation

“…For more applications of fractal-fractional differential equations, see [7][8][9][10] and for recent results on fractional differential equations and their applications, see [11][12][13]. To explore more results on implicit fractional differential equations and their applications, see [14][15][16][17]. There are many results relating to implicit fractional differential equations in literature involving Caputo fractional derivatives both for initial value problems (IVP) and boundary value problems (BVP) [15,[18][19][20][21].…”

“…The authors investigated the well-posedness, interval of existence, and continuous dependence on the initial condition of solutions to Equation (1). Recently, in 2021, Shabbir et al [17] worked on an implicit boundary value problem (BVP) involving an Atangana-Baleanu-Caputo (ABC) derivative of the form…”

“…Motivated by some applications of the implicit fractal-fractional differential equation in modeling complex phonemena and systems in porous media with memory, and the result in [17], where the authors used the ABC derivative operator to study Equation (2); therefore, we generalize (2) for a class of fractal-fractional derivative operator known as the Mittag-Leffler kernel law Fractal-Fractional (FFM), to study the well-posedness, exponential growth bound, and long-time behaviour of a solution to the class of implicit time-fractal-fractional differential equation:…”