Existence and Stability Results on Hadamard Type Fractional Time-Delay Semilinear Differential Equations

: A delayed perturbation of the Mittag-Lefﬂer type matrix function with logarithm is proposed. This combines the classic Mittag–Lefﬂer type matrix function with a logarithm and delayed Mittag–Lefﬂer type matrix function. With the help of this introduced delayed perturbation of the Mittag–Lefﬂer type matrix function with a logarithm, we provide an explicit form for solutions to non-homogeneous Hadamard-type fractional time-delay linear differential equations. We also examine the existence, uniqueness, and Ulam–Hyers stability of Hadamard-type fractional time-delay nonlinear equations.

H D α 1 + y (t) = λy (t) + f (t) , t ∈ (1, T] , h > 0, H I 1−α 1 + y 1 + = a ∈ R, λ ∈ R, 0 < α < 1, has the form However, we find that there exists only one [33] work on the representation of explicit solutions of Hadamard type fractional order delay linear differential equations. In [33] authors studied the Hadamard type fractional linear time-delay system where B is a constant n × n square matrix. Motivated by the above researches, we investigate a new class of Hadamard-type fractional delay differential equations. We consider an explicit representation of solutions of a Hadamard type fractional time-delay differential equation of the following form by introducing a new delayed M-L type function with logarithm where H D α 1 + y (·) is the Hadamard derivative of order α ∈ (0, 1), A, B ∈ R n×n denote constant matrices, and ϕ : 1 h , 1 → R n is an arbitrary Hadamard differentiable vector function, f ∈ C ([1, T] , R n ), T = h l for a fixed natural number l.
The second purpose of this paper is to study the existence and stability of solutions for a Hadamard type fractional delay differential equation At the end of this section, we state the main contribution of the paper as follows: (i) We propose delayed perturbation of the M-L type functions Y A,B h,α,β (t, s) with logarithms, by means of the matrix Equations (6). We show that for B = Θ the function Y A,B h,α,β (t, s) coincides with the M-L type function with two parameters (ln t − ln s) β−1 E α,β A (ln t − ln s) α . For A = Θ the delayed M-L type function Y A,B h,α,β (t, s) coincides with the delayed M-L type matrix function with two parameters E B h,α,β (ln t − ln h) , introduced in (4). (ii) We explicitly write the solution of the Hadamard type fractional delay linear system (2) via delayed perturbation of the M-L type function with logarithm. Using this representation we study existence, uniqueness, and Ulam-Hyers stability of the nonlinear Equation (3).

Preliminaries
Let 0 < a < b < ∞ and C [a, b] be the Banach space of all continuous functions y : [a, b] → R n with the norm y C := max { y (t) : t ∈ [a, b]}. For 0 ≤ γ < 1, we denote the space C γ,ln (a, b] by the weighted Banach space of the continuous function y : [a, b] → R n , which is given by endowed with the norm y γ := sup ln t a γ y (t) : t ∈ (a, b] . The following definitions and lemmas will be used in this paper. Definition 1. [24] Hadamard fractional integral of order α ∈ R + of function y (t) is defined by where Γ is the Gamma function.
Next, we introduce a definition of delayed M-L type matrix function E B h,α,β (ln t) : R + → R n×n with logarithm generated by B.

Definition 4. Two parameters delayed M-L type matrix function E B
h,α,β (ln t) : R + → R n×n with logarithm generated by B is defined by Our definition of the two-parameter delayed M-L type matrix function with logarithm differs substantially from the definition given in [33].
In order to give a definition of delayed perturbation of the M-L type matrix functions with logarithm, we introduce the following matrices Y α,β,k , k = 0, 1, 2, ...
Definition 5. Let A, B ∈ R n×n be fixed matrices and k ∈ N ∪ {0}. Delayed perturbation of M-L type function Y A,B h,α,β (·, ·) : R×R→R n with logarithm generated by A, B is defined by where H (t) is a Heaviside function: 1 To prove (8), firstly using (7) we calculate it for k = 0: Similarly, for any k ∈ N, we have Lemma 3. If A and B are commutative matrices, then Proof. The proof is based on the equality (7). Using Lemma 2, for k = 1, we have Using the Mathematical Induction in a similar manner we can get (10).
According to Lemma 3 in the case AB = BA delayed M-L type function Y A,B h,α,β (t, s) has a simple form: (11) Next lemma shows some special cases of the delayed M-L type function.

Proof.
The proof is similar to that of [34] and is omitted.
It turns out that Y A,B h,α,β (t, s) is a delayed perturbation of the Cauchy matrix with logarithm of the homogeneous Equation (2) with f = 0.

Lemma 6. Y A,B
h,α,β : R×R → R n×n is a solution of Proof. According to (9) we have On the other hand for any k ∈ N: From (13) and (14) it follows that for sh k < t ≤ sh k+1 The proof is complete.
The solution y(t) of (2) with zero initial condition has the form Proof. Assume that any solution of a nonhomogeneous system y (t) has the form where h (s) , 1 ≤ s ≤ t ≤ T is an unknown continuous vector function and y(1) = 0. Having Hadamard fractional differentiation on both sides of (15) , for 1 < t ≤ h we have On the other hand, according to Lemma 2, we have Therefore, h(t) ≡ f (t). The proof is complete.

Proof.
We are looking for a solution which depends on an unknown constant c, and a vector function g (t) , of the form Moreover, y (t) satisfies initial conditions We have Thus c = a. Since 1 < t ≤ h, we obtain that Consequently, on interval 1 < t ≤ h, we can easily derive Having differentiated (16) in the Hadamard sense, we obtain Therefore, g (t) = H D α 1 + y (t) − Ay (t) = Bϕ t h and the desired formula holds.
Combining Theorems 1 and 2 together we get the following result.

Existence Uniqueness and Stability
In this section, we consider the following equivalent integral form of the nonlinear Cauchy problem for fractional time-delay differential equations with Hadamard derivative (3): Let us introduce the conditions under which existence and uniqueness of the integral Equation (17) will be investigated. (A1) f : [1, T] × R n → R n be a function such that f (t, y) ∈ C γ,ln [1, T] with γ < α for any y ∈ R n ; (A2) There exists a positive constant L f > 0 such that From (A1) and (A2), it follows that To prove existence uniqueness and stability of (17) we use the following properties of Y A,B α,β (t, s) .

Lemma 7.
We have for sh p < t ≤ sh p+1 , p = 0, 1, ..., Our first result on existence and uniqueness of (17) is based on the Banach contraction principle.

Proof.
We define an operator Θ on B r := y ∈ C γ,ln [1, T] : y γ ≤ r as follows It is obvious that Θ is well-defined due to (A1). Therefore, the existence of a solution of the Cauchy problem (3) is equivalent to that of the operator Θ has a fixed point on B r . We will use the Banach contraction principle to prove that Θ has a fixed point. The proof is divided into two steps.
Step 1. Θy ∈ B r for any y ∈ B r . Indeed, for any y ∈ B r and any δ > 0, by (A3) Firstly, we estimate the first integral: Similarly, Inserting (19) and (20) into (18) we get Step 2. Let y, z ∈ C γ,ln [1, T]. Then similar to the estimation (20) we get Hence, the operator Θ is contraction on B r and the proof is competed by using the Banach fixed point theorem.
Secondly, we discuss the Ulam-Hyers stability for the problems (3) by means of integral operator given by where Θ is defined by (17). Define the following nonlinear operator Q : For some ε > 0, we look at the following inequality: Definition 6. We say that the Equation (17) is Ulam-Hyers stable, if there exist V > 0 such that for every solution y * ∈ C γ,ln ([ 1 h , T], R n ) of the inequality (22), there exists a unique solution y ∈ C γ,ln ([ 1 h , T], R n ) of problem (17) with y − y * γ,ln ≤ Vε.
Theorem 4. Under the assumptions of Theorem 3, the problem (17) is stable in Ulam-Hyers sense.
Proof. Let y ∈ C γ,ln ([ 1 h , T], R n ) be the solution of the problem (17). Let y * be any solution satisfying (22): It follows that Therefore, we deduce by the fixed-point property (21) of the operator Θ, that and Thus, the problem (3) is Ulam-Hyers stable with

Existence Result
Our second existence result is based on the well known Schaefer's fixed point theorem. We use the following linear growth condition to replace (A 2 ): (A 3 ) There exists M f > 0 such that For the sake of convenience, we will split the proof into several steps.
Step 2. Θ maps bounded sets into bounded sets in C γ,ln [1, T]. Clearly, the function f (t, y) = L f sin y 1 t sin y 2 − y 1 cos t is jointly continuous and Lipschitz continuous with respect to y. We can choose L f > 0 so that the conditions of Theorems 3 and 4 are satisfied. Thus, the above problem has a unique solution which is Ulam-Hyers stable.

Conclusions
In this paper, we have introduced delayed perturbation of the M-L matrix exponential with logarithms, to get a representation formula for time-delay Hadamard type fractional differential equations with non-commutative linear part. Using this representation formula we have obtained several existence results for an initial value problem of time-delay Hadamard-type fractional differential equations. Furthermore, we have presented a sufficient condition for stability in the Ulam-Hyers sense. In our future work, we are planing to investigate the existence, stability and controllability of solutions to an initial value problem for time-delay fractional differential equations involving a combination of Caputo and Hadamard fractional derivatives.