On Implicit Time–Fractal–Fractional Differential Equation

: An implicit time–fractal–fractional differential equation involving the Atangana’s fractal– fractional derivative in the sense of Caputo with the Mittag–Lefﬂer law type kernel is studied. Using the Banach ﬁxed point theorem, the well-posedness of the solution is proved. We show that the solution exhibits an exponential growth bound, and, consequently, the long-time (asymptotic) property of the solution. We also give examples to illustrate our problem.


Introduction
The fractal-fractional differential equation is a link between the fractal and fractional differential equations. Fractal and fractional differential equations are known for modeling complex physical processes and phenomena, particularly irregular systems with memory. Although fractional equations are renowned for representing systems with long-term memory and long-range interactivity, fractal calculus, conversely, is immensely effective in working with occurrence in stratified or porous media. That is, fractal-fractional differential operator models physical phenomena and real-world activities that exhibit or display fractional behaviours (sponge-like media, aquifer, turbulence, etc.) namely finance, viscoelasticity, control theory, electrical networks, goundwater flow and geo-hydrology, wave propagation, plasma physics and fusion, rheology, chaotic processes, fluid mechanics and biological activities [1][2][3][4][5][6]. 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].
In 2015, Benchohra and Souid in [18] studied the existence of integrable solutions for IVP for some given implicit fractional order functional differential equations with infinite delay where C D µ is the Caputo fractional differential operator, σ : I × B × B → R is a given function and B is a phase space with its element φ τ ( ) = φ(τ + ), ∈ [−∞, 0]. In 2016, Kucche et al.,in [20], considered the following equation: where C D µ (0 < µ < 1) stands for the Caputo fractional derivative and σ : [0, T] × R × R → R is a known continuous function fulfilling some conditions. 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 where ABC 0 D µ τ denotes the ABC derivative of order 1 < µ ≤ 2 and σ : I × R × R → R is a continuous function. Here, the authors established the existence of solution, uniqueness of solution and stability of solution to the class of implicit BVPs (2) with an ABC type derivative and integral.
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: with ψ a taken to be a bounded and non-negative function, FFM 0 D µ,υ t represents Atangana's fractal-fractional derivative of orders µ, υ ∈ (0, 1] in the sense of Caputo with generalized Mittag-Leffler law type kernel, ς : [a, T] × R × R → R is Lipschitz continuous. Information within our disposal, suggests that we are the first to study this class of implicit fractalfractional differential equation. Using similar ideas in [2,3], we give the formulation of the solution to Equation (3) as follows: which follows by the definition of the operator FFM a I µ,υ t .
Next, we define the norm of the solution ψ by The organization of the paper is as follows. In Section 2, we present the preliminaries; and in Section 3, we give the statements and proofs of the main results of the paper. Section 5 contains a brief summary of the paper.

Preliminaries
In this section, one gives some concepts that will be useful for the main result.

Definition 5 ([23]
). One defines the incomplete beta function by It also has a representation in terms of a hypergeometric function given by

Proof. From Equations
and, therefore, Furthermore, one obtains
Proof. By taking absolute value on the operator A, we have Applying Condition 1 and |ϕ a | ≤ c 1 , to obtain From Lemma 2, we arrive at Evaluating the integral above, we have 1]. We observe that a υ−1 > t υ−1 since υ − 1 < 0. Thus, taking supremum over t ∈ [a, T] in (8) and recalling that µ + υ ≥ 1, we obtain T µ+υ−1 ψ , and the proof is complete.

Lemma 4.
Suppose ψ and ϕ are solutions satisfying Equation (4) and let Condition 1 be satisfied. Then, if it follows that for all µ, υ ∈ (0, 1] such that µ + υ ≥ 1, we have Proof. The proof is skipped since it follows similar steps as the proof of Lemma 3. Next, we state the existence and uniqueness theorem for Equation (3).
Then, there exists a unique solution to Equation (3).

Exponential Growth
We present an inequality needed in proving the upper growth bound: 26]). Given that f , g, h : I → R + are continuous functions. If φ : I → R + is continuous and Now, dividing byc 2 = rameter υ is to zero, the faster the rate of growth is to the bound. However, as time grows large, the growth rate is at most at t = 600 irrespective of the values of υ, as shown in the Figure 1 below.

Conclusions
Fractional order derivatives are used to represent memory formalism in modeling phenomena or processes in porous media in order to diminish the size of the pores and the permeability of the porous matrix [27]. Hence, implicit fractal-fractional differential equations are very important because they model many technical processes and systems in porous environment exhibiting long time memory property. In this paper, we estimated the higher growth bound of our solution and it is shown that the solution exhibits an exponential growth in t at a specific rate. Furthermore, the result shows a long time behaviour of the mild solution. Banach fixed point theorem was applied to prove the well-posedness of mild solution to the class of implicit time-fractal-fractional differential equation with Mittag-Leffler law. For future work, one can investigate the lower growth estimate of the solution, the stability of the solution, and the continuous dependence on the initial condition, as shown in [20].