Well-Posedness and Stability for a Differential Problem with Hilfer-Hadamard Fractional Derivative

Motivated by the Hilfer fractional derivative (which interpolates the Riemann-Liouville derivative and the Caputo derivative), we consider a new type of fractional derivative (which interpolates the Hadamard derivative and its Caputo counterpart). We prove the well-posedness for a basic Cauchy type fractional differential equation involving this kind of derivative. This is established in an appropriate underlying space after proving the equivalence of this problem with a certain corresponding Volterra integral equation.


Introduction
In this work, we are concerned with the Hadamard derivative Its Caputo counterpart is where (J + ) ( ) := 1 Γ ( ) ∫ (log ) −1 ( ) , < < (see [1][2][3][4][5]). Here, we consider the following fractional derivative This type of fractional derivative interpolates the Hadamard fractional derivative ( = 0) and the Caputo-Hadamard fractional derivative ( = 1). It has been introduced recently in [6]. In introducing this new fractional derivative we were motivated by the Hilfer fractional derivative of order 0 < < 1 and type 0 ≤ ≤ 1 (see [7]) which interpolates the Riemann-Liouville derivative and the Caputo derivative. We study the existence and uniqueness of solutions of a basic fractional differential equation with an appropriate initial condition in a suitable underlying space after proving the equivalence of this problem with a corresponding Volterra integral equation. In addition to that, we discuss the stability of solutions for a large and important class of nonlinearities. We find that solutions decay to zero at a logarithmic rate as time goes to infinity. To this end, we prove an inequality (which is important by itself). The literature is very rich in works on well-posedness for fractional differential equations [8][9][10][11][12][13][14][15][16][17] (see also the books [1,2,4,5] and the survey paper [18]) to cite but a few. The Hadamard fractional derivative may be found in the books [1,2,[4][5][6]. Differential equations involving such a derivative and others have been treated in [2,3]. In contrast with the well-posedness, the stability issue and the long time behavior is not well studied [6,12,[19][20][21][22][23].

Abstract and Applied Analysis
The rest of the paper is organized as follows: the next section contains some material needed in our proofs. The different fractional derivatives as well as the new one are defined there. In Section 3, we present our problem and prove an existence and uniqueness result after establishing the equivalence of the differential problem with its corresponding integral equation. Section 4 is devoted to a stability result.

Preliminaries
In this section we present some definitions, lemmas, properties, and notation which will be used in our theorems later.
In the rest of the paper we shall assume ̸ = 0 when considering an interval ( , ).

Abstract and Applied Analysis
Theorem 17 (Banach fixed point theorem [2]). Let ( , ) be a non-empty complete metric space, let 0 ≤ < 1, and let : → be a map such that, for every , V ∈ , the relation holds. Then, the operator has a unique fixed point * ∈ . Furthermore, if ( ∈ N) is the sequence of operators defined by then, for any 0 ∈ , the sequence { 0 } ∞ =1 converges to the above fixed point * .
Theorem 18 (Young's inequality). If and are nonnegative real numbers and and are positive real numbers such that 1/ + 1/ = 1 then we have Equality holds if and only if = .
Finally, we refer the reader to the nice treatments of Hadamard-type fractional calculus in [25,26]

Existence and Uniqueness for an FDE with Hilfer-Hadamard Fractional Derivative
In this section we discuss the existence, uniqueness and the stability of solutions of the Cauchy type problem (46) (below) with Hilfer-Hadamard fractional derivative.
Definition 19 (Hilfer-Hadamard fractional derivative (HHFD)). The left sided fractional derivative of order , (0 < < 1) and type 0 ≤ ≤ 1 with respect to is defined by for functions for which the expression on the right hand side exists, where D + − + is the Hadamard fractional derivative (Definition 5).
This new fractional derivative (introduced for the first time in [6]) may be viewed as interpolating the Hadamard fractional derivative and the Hadamard-Caputo fractional derivative. Indeed for = 0 this derivative (46) reduces to the Hadamard fractional derivative (Definition 5) and when = 1, we recover the Hadamard-Caputo fractional derivative (Definition 7).
We will study the existence and uniqueness for the Cauchy type problem We consider the underlying spaces defined by where = + − and 0 ≤ < 1. It is clear that 0 < < 1 for 0 < , < 1. Here Our investigations are based on reducing the fractional differential problem to a Volterra integral equation of the second kind: and then using the Banach fixed point theorem.

Equivalence of the Cauchy Type Problem and the Volterra Integral Equation.
Here, we prove the equivalence of the Cauchy type problem (46) and the nonlinear Volterra integral equation (50) for some positive constant . Therefore, and by using Lemma 9 (with = 1 − > 0) we have As > , we obtain the result. Proof. First we prove the necessity. Let ∈ 1− ,log [ , ] be a solution of problem (46). We want to prove that is also a solution of the integral equation (50). By the definition of the space 1− ,log [ , ] (relation (48)) we have Moreover, by Lemma 13 . Then, by Definition 2, we have Thus, we can apply Theorem 12 (with replaced by ) to get or which is (50), and hence the necessity is proved. Now, we prove the sufficiency. Let From (64) and the fact that D + ∈ 1− ,log [ , ], we obtain that Next, applying the operator J  Now, we show that the initial condition in (46) also holds. To this end we apply the operator J 1− + to both sides of (50): and use the Lemma 8 (with replaced by 1 − and by ) and the Lemma 9 to obtain In (73), taking the limit as → , we obtain . Therefore, the sufficiency is proved, which completes the proof of Theorem 21.

Existence and Uniqueness of a Solution.
In this section we establish the existence of a unique solution to the Cauchy type problem (46) in the space Let us select 1 in ( , ) such that where > 0 is the Lipschitz constant. We start by proving that a unique solution ∈ 1− ,log [ , 1 ] to (50) exists on the interval ( , 1 ]. It is easy to see that the space 1− ,log [ , 1 ] is a complete metric space when equipped with the distance given by The integral equation (50) takes the form where We claim that maps 1− ,log [ , 1 ] into itself. Indeed, 0 given by (80)  with ∈ R (0 ≤ < 1), then, by Lemma 13(a) and (b), the integral in the right-hand side of (79) belongs to − ,log [ , ] for > and to [ , ] for ≤ . Since − < 1 − , by Lemma 16 the right-hand side of (79) belongs to 1− ,log [ , ].
Our second claim is that is a contraction; that is, This follows from (79), Lemma 13(a), and the fact that Our assumption (76) allows us to apply the Banach fixed point theorem to obtain a unique solution * ∈ 1− ,log [ , 1 ] to (50) on the interval ( , 1 ]. This solution * is the limit of a convergent sequence * 0 : where 01 ( ) is defined by and is a known function. We note that 01 ∈ [ 1 , ].
We want to prove the existence of a unique solution ∈ [ 1 , ] of (50) on the interval [ 1 , ]. For this, we also use Banach fixed point theorem for the space The space [ 1 , 2 ] is a complete metric space with the distance given by The integral equation (87) may be written shortly as where the operator (again denoted by ) is given by Moreover, using the Lipschitz condition and applying the Lemma 14, we find This, together with our assumption 0 < 2 < 1, shows that is a contraction and therefore from Theorem 17, there exists a unique solution * 1 ∈ [ 1 , 2 ] to (50) on the interval . Further, Theorem 17 guarantees that this solution is the limit of a convergent sequence * 01 : where * 01 is any function in [ 1 , 2 ], which we can pick * 01 ( ) = 01 ( ) defined by (88). Therefore, where ( ) = ( * 01 ) ( ) If 2 ̸ = , we consider the interval [ 2 , 3 ], where 3 = 2 +ℎ 2 , ℎ 2 > 0 such that 3 ≤ and Using the same arguments as above, we derive that there exists a unique solution * 2 ∈ [ 2 , 3 ] to (50) on the interval [ 2 , 3 ]. If 3 ̸ = , then we continue the process until we reach a solution to (50), ( ) = * ( ), and * ∈ [ , +1 ] ( = 1, . . . , ), where = 0 < 1 < ⋅ ⋅ ⋅ < +1 and Assume that − > Γ( + )/ Γ( ) (for otherwise, take It remains to show that such a unique solution is actually in , 1− , [ , ]. To this end we need to prove that D , + ∈ ,log [ , ]. Let us recall that our is a limit of the sequence , where = * 0 ∈ 1− ,log [ , ]; that is, with a certain choice of * 0 ( ) on each subinterval [ , 1 ], . . . , [ , ]. Indeed, this is a consequence of the construction adopted, the initial values are selected in the space (80) and (88)) and the operator maps this space into itself (see argument right after (80) and (92)). As for the convergence in that space it has been proved in (83) and (95).

Stability
In this section, we consider the weighted Cauchy-type problem where D , + is the Hilfer-Hadamard fractional derivative (HHFD) of order 0 < < 1 and type 0 ≤ ≤ 1 and ∈ R * (the set of all real numbers except 0). It is interesting to note that, using an argument similar to the one in the proof of Lemma 3.5 in [2] (see also Lemma 3.2), we can prove that the initial condition in (102) and the one in (46) are equivalent.
We will assume the following hypotheses on the function : (F * ) ( , ) is a continuous (nonlinear) function on ( , ∞) × R and is such that where is a continuous (nonnegative) function on [ , ∞). We first prove the following inequality.

Lemma 23.
If , ], > 0, then for any > , > 0 we have where is a positive constant independent of .