An Initial Value Problem Involving Caputo-Hadamard Fractional Derivative: The Extremal Solutions and Stabilization

In this paper, the existence of extremal solutions of Caputo-Hadamard-type fractional di erential equations (CHFDEs) with order α ∈ (1, 2) is established by employing the method of lower and upper solutions. Moreover, su cient condition that ensure the stability of a class of CHFDE is also provided. Some examples are given to illustrate our main results.


Introduction
Fractional calculus and fractional dierential equation models have been studied in a variety of elds such as physics, mathematics, engineering, bioengineering, and other applied sciences. For a general overview of the theory and applications of fractional dierential equations involving the Riemann-Liouville fractional derivative and the Caputo derivative, we refer the reader to the monograph of Kilbas [7]. Recently, fractional dierential equations with the Hadamard derivative and the Caputo-Hadamard derivative have attracted the attention of a large number of researchers (see [1,2,4,8,13] and the references therein). In particular, Caputo-Hadamard fractional derivatives were introduced by Jarad et al. [5], and it was shown that there are many advantages over the usual Hadamard fractional derivative. Moreover, Gambo et al. [4] presented the fundamental theorem of fractional calculus in the Caputo-Hadamard setting based on the concept in [5], and recently Almeida [2] proposed three types of Caputo-Hadamard derivatives of variable fractional order, and studied the relation between them. Adjabi et al. [1] investigated the existence of solutions to fractional dierential equations with the Caputo-Hadamard derivative using Banach's xed point theorem. Yukunthorn et al. [13] studied the existence of solutions for impulsive hybrid systems of Caputo-Hadamard fractional dierential equations equipped with integral boundary conditions using xed point theorems.
In this work, we present some existence results for the initial value problem for fractional dierential equations with order α ∈ (1, 2) involving Caputo-Hadamard fractional derivative using the method of upper and lower solutions coupled with its associated monotone iteration scheme. This technique is a powerful tool for proving the existence of solutions of nonlinear fractional dierential equations; see [3,9,10,14,15,12]. To the best of our knowledge this technique has not been applied to the initial value problem for CHFDE, when α ∈ (1, 2), to investigate the existence of minimal and maximal solutions. In this paper, using this technique we discuss the existence of extremal solutions and the uniqueness of the solution of the following fractional dierential equation with the Caputo-Hadamard-type fractional derivative for α ∈ (1, 2) where t ∈ (a, b], and a ≥ 1. Section 2 contains some basic denitions and the related lemma that will be used. In Section 3, using the method of upper and lower solutions, we prove the existence of extremal solutions of problem (1). Finally, the stabilization of a class of fractional dierential equations is established in Section 4.

Preliminaries
In this section, some basic denitions, propositions, remarks and lemma are introduced (see [5,4] for more detail), which will be used in the following discussions. Denote by C[a, b], AC[a, b] the space of continuous functions and the space of absolutely continuous functions from [a, b] into R, respectively. In this paper, we denote by AC n [a, b], C n [a, b] and C γ [a, b], where n ∈ N, the spaces dened by where γ ∈ (0, 1]. If n = 1, the space AC 1 [a, b] coincides with AC[a, b]. Now, we provide some denitions and properties of fractional calculus. The Hadamard fractional integral of the function ψ is dened by (see [7]): The Hadamard fractional derivative of order α ∈ (1, 2) for the function ψ is dened as follows (see [7]): is dened by (see [5,4]) Then, one has (see [4,5]) On the other hand, the denition of the Caputo-Hadamard fractional derivative C−H D α a + ψ of order α ∈ (1, 2) is dened by (see [4,5]) 150 c 2020 Journal of Advanced Engineering and Computation (JAEC) VOLUME: 4 | ISSUE: 2 | 2020 | June Furthermore, we observe that: where t ∈ [a, b] and ψ 0 = sup t∈[a,b] |ψ(t)|.

The existence of extremal solutions
Consider the following fractional dierential equation with the order α ∈ (1, 2): where t ∈ (a, b], and a ≥ 1. A function ψ : Then, a function ψ is a solution of the problem (10) if and only if ψ satises ] be a solution of the problem (10). Then by (10) and Remark 3 one has that Because f (t, u) ∈ C γ ([a, b], R) and from problem (10), we have that This yields the necessity condition of the proof. On the other hand, because of the continuity of the function f , the function t → f α (t, u) is continuous on (a, b] and f α (a, u(a)) = lim t→a + f α (t, u(t)) = 0. Then, ψ(a) = ψ 0 and ψ (a) = ξ 0 . By taking the operator H D α a + on two sides of (11) and by Proposition 2. .1, one gets To show the main results of this paper, we need the formula of solution of the problem (10) in the linear form as the below. Remark 4. The formula of the solution of the following linear Caputo-Hadamard fractional dierential equation is expressed by Indeed, to get the explicit formula of the solution of (14), we shall employ the method of successive approximations. First of all, based on Lemma 2 we observe that a function ψ is a solution of the problem ( 14) if it satises Next, we set ψ 0 (t) = ψ 0 +ψ (1) (a)(ln t − ln a) and for n = 1, 2, 3, ... For n = 2, we also see that .
If one proceeds inductively and let n → ∞, one gets the solution Then, by using the denition of the Mittag- , α, β > 0, the solution of the problem (14) is given (15).
Then, we get the formula of the solution as follows: The graph of the solution ψ(t) is shown in Fig.  1.
is a lower solution for the initial value problem (10) if is an upper solution for (10) if it satises the reverse inequalities of (16), i.e., As in the proof of Lemma 2.2 in [11], we also get the remark below.
Step 1: The initial value problem (10) has at least one solution if and only if the operator P has a xed point ψ satisfying We now show that the integral operator P is welldened, that is, Pψ ∈ C[a, b] for ψ ∈ C[a, b]. Let us consider ψ n , ψ ∈ C[a, b], where ψ 0 (t) = ψ 0 + ψ (1) (a)(ln t − ln a), such that ψ n → ψ as n → ∞, and then from Remark 6 and hypothesis (26) we have that for n ∈ N Pψ n − Pψ 0 = max Therefore, this yields This allows to conclude that the operator P is continuous on [a, b]. Therefore, P is well-dened.
Step 2: We now show that the operator P has a xed point, and this is done using Schauder's xed point theorem. In the previous step, we have Pψ ∈ C[a, b] if ψ ∈ C[a, b], i.e. P maps the set C[a, b] into itself. Next, let S ⊂ C[a, b] be a bounded set, and then we shall show that P(S) = {(Pψ)(t) : ψ ∈ S} is a relatively compact set, and this is done using the Arzela-Ascoli Theorem (see Theorem 1.8 in [7]). First of all we shall verify that the set P(S) is uniformly bounded. Let W (t) ∈ P(S). Then from Remark 6 and the condition (26) we have that for all t ∈ [a, b], α ∈ (1, 2), This argument shows that P(S) is uniformly bounded. Next, we show that P(S) is equicontinuous. For every ψ ∈ C[a, b], from the continuity of the function f , from Remark 6 and by letting K f = sup t∈[a,b] |f (t, ψ) + λψ|, we get for a ≤ t 1 ≤ t 2 ≤ b, Since α ∈ (1, 2), λ < 0, the functions E i (t), i = {1, 2} are uniformly continuous and bounded on [a, b]. Therefore, as t 2 → t 1 , the right-hand side of the above inequality tends to zero. Thus P(S) is equicontinuous. So, by Schauder's xed point theorem (see Theorem 1.7 in [7]), we assert that P has at least one xed point ψ * such that Pψ * = ψ * . This xed point is the required solution of the initial value problem (10).
Step 3: In order to prove that the operator P has a extremal solutions, we show that the conditions of Lemma 1 are satised. Indeed, let ψ L , ψ U be lower and upper solutions of the initial value problem (10). Then by the denition of the lower solution, there exist g(t) ≥ 0 and ε 1 , ε 2 ≥ 0 such that where t ∈ (a, b]. Using Remark 4 and the denition (28), we have that Similarly, ψ U ≥ Pψ U is satised. Set D := [ψ L , ψ U ]. Based on Step 2, we have that the operator P : D → C[a, b] is relatively compact, and so it follows that P is completely continuous. In addition, the condition (27) yields f is monotone nondecreasing in ψ if M = 0. Thus, it follows that the operator P is nondecreasing on D. Using Lemma 1, the existence of the extremal solutions of the initial value problem (10) is obtained. The proof is complete.
As in the proof of Theorem 1, we also obtain the below corollary. In this section, we discuss the stability of the solution of the following problem where t ≥ a ≥ 1, α ∈ (1, 2), A is a constant, and the continuous and bounded function g : [a, b] × R → R is the nonlinear term of state ψ(t) and it satises g(t, 0) ≡ 0. We observe that the state ψ = 0 is the equilibrium of the problem (30), which can be taken as a target orbit. In this section, to force the state of the problem into the target orbit, we shall choose a linear feedback controller u(t) = Bψ(t) to the problem (30). Therefore, we consider the following the controlled problem where t ≥ a, and the feedback gain B needs to be determined, α ∈ (1, 2).
Remark 7. Based on the result of Section 3, by putting f (t, ψ) = Aψ(t) + g(t, ψ(t)), then from the continuous hypothesis of the function f we observe that the problem (30) has at least solution on [a, b].
Proof. From the result of Remark 4, one can obtain the following solution of system (31): Now, because lim Furthermore, we have for all α ∈ (1, 2) and i ∈ {0, 1, 2, ..., ∞} where B(·, ·) is the Beta function. Therefore, using condition (32) we have the following estimate for all α ∈ (1, 2) and t > a Then, substituting (32), (34) and (37) into (33), we obtain that for all t ∈ [a, δ) On the other hand, using Leibniz's rule for dierentiation under the integral sign of the function I(t) with respect to t and by (38), we get and by using Gronwall's inequality and since a ≥ 1, we also get the estimate · exp (−µ(ln t − ln a)) .
Suppose that T is nite. Then, from (41) we have that |ψ(t)| < M 2 on [a, T). According to the existence theory in Section 3, it follows that the solution ψ(t) can be further extended so that T is contained in the maximal existent interval [a, T * ), where T * > T. Then, since ψ(t) is continuous, one has |ψ(t)| < M on [a, T * ). Therefore, this leads to the contradiction of the denition of T. Thus we conclude that if |ψ 0 | < θ, then |ψ(t)| < M for all t ≥ a. It further follows that the inequality (39) is valid for all t ≥ a and lim t→∞ |ψ(t)| = 0 provided that ε < µ ρ . This implies that the solution ψ(t) of system (31) can be forced to the equilibrium ψ ≡ 0. The proof is complete.
Based on the assertion of Theorem 2, we get the following corollary.
The following example is given to illustrate the assertion of Theorem 2.  In this example, if we take α = 1.9, then problem (43) is unstable at the equilibrium point ψ(t) = 0. The unstable orbit of the problem (43) is numerically shown in Fig. 8. Now, we assume that the equilibrium point 0 is the target orbit.
To force the trajectory of the problem (43) into the target orbit, we shall add a feedback controller u(t) = Bψ(t) to the problem (43), where the constant B is a control gain. Based on Theorem 2, if the control gain is taken as B = 100 and the parameters are chosen ε = 1, θ = 1, ρ = 1.5, µ = 2, then the condition (32) is valid for the above-mentioned control gain and parameters (see Fig. 9). According to Theorem 2, this implies that the orbit of the problem (43) can be controlled to equilibrium ψ(t) = 0 via the feedback controller u(t) = Bψ(t). The graph of the controlled orbit of the problem (43) is shown in Fig. 10.

Conclusion
In the paper, by using the well-known method of lower and upper solutions, the existence theory of the extremal solutions for a class of fractional dierential equations under the Caputo-Hadamard derivative with the case of α ∈ (1, 2) has been established. We also provide sucient conditions that ensure the stability of a class of fractional dierential equations. Finally, an example was implemented to demonstrate the feasibility and validity of the proposed method, which were consistent with the theoretical results. The approach proposed in this paper may be extended to other fractional dierential systems and in the future, it will be applied to investigate the stabilization of non-linear systems with the fractional derivative.