Hyers–Ulam stability of a coupled system of fractional differential equations of Hilfer–Hadamard type

The theory of fractional differential equations (FDEs) is a growing area of research. Recently, it has been realized that FDEs can describe a large number of nonlinear phenomena in different fields of science like physics, chemistry, biology, viscoelasticity, control hypothesis, speculation, fluid dynamics, hydrodynamics, aerodynamics, information processing system networking, notable and picture processing etc. In addition, FDEs can provide marvelous tools for the depiction of memory and inherited properties of many materials and processes. Consequently, FDEs have emerged significant developments and thus important results have reported in recent years [1–17]. One of the most attractive research areas in the field of FDEs which has engrossed great consideration amongst researchers is dedicated to the existence theory of the solutions of fractional models. The aforesaid area has been extensively explored for integer order differential equations (DEs). However, for arbitrary order DEs, there are stillmany aspects that need further study and research. Differentmathematicians explore FDEs in different directions; the reader may see [18–25] and references cited therein. Another imperative and more remarkable area of researchwhich has recently attractedmore attention is committed to the stability analysis of DEs of integer and non integer order. The first effort was initiated by Ulam in 1940 and later on confirmed by Hyers in 1941 (see [26]). That’s why this type of stability is called Hyers–Ulam (HU) stability. Rassias introduced the Hyers–Ulam–Rassias (HUR) stability. Obloza was the first mathematician who introduced the HU stability for DEs; the reader can consult [27–43] for comprehensive literature. It is to be noted that, the above said areas of interest (existence and stability) have been fabulously deliberated by adapting Riemann– Liouville and Caputo derivatives.

One of the most attractive research areas in the field of FDEs which has engrossed great consideration amongst researchers is dedicated to the existence theory of the solutions of fractional models. The aforesaid area has been extensively explored for integer order differential equations (DEs). However, for arbitrary order DEs, there are still many aspects that need further study and research. Different mathematicians explore FDEs in different directions; the reader may see [18][19][20][21][22][23][24][25] and references cited therein. Another imperative and more remarkable area of research which has recently attracted more attention is committed to the stability analysis of DEs of integer and non integer order. The first effort was initiated by Ulam in 1940 and later on confirmed by Hyers in 1941 (see [26]). That's why this type of stability is called Hyers-Ulam (HU) stability. Rassias introduced the Hyers-Ulam-Rassias (HUR) stability. Obloza was the first mathematician who introduced the HU stability for DEs; the reader can consult [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] for comprehensive literature. It is to be noted that, the above said areas of interest (existence and stability) have been fabulously deliberated by adapting Riemann-Liouville and Caputo derivatives.
Recently, significant consideration has been given to the existence of solutions of boundary and initial value problems for FDEs with Hilfer-Hadamard (HH) type fractional derivative. In [44], Abbas et al. studied the existence and stability of the solution of FDEs involving HH type derivative given by , t ∈ J, 0 < α < 1, 0 < β ≤ 1, where J = (1, T] with T > 1 and H D α,β denotes HH fractional derivative of order α and type β introduced by Hilfer in [45], ϕ ∈ R, f : J × R → R is a continuous function and I 1− 1 + is the left-sided mixed Hadamard type integral of order 1 − .
The objective of this paper is to use the basic concepts mentioned in [44] combined with the methodology applied in [46], to examine the existence and uniqueness as well as different kinds of HU stability for the solutions of coupled impulsive FDEs involving HH type derivative. The proposed system is given by: where H D p,q represents the HH type derivatives for the functions u and v of order p ∈ (0, 1) and q ∈ (0, 1] and I 1− 1 + is the left-sided mixed Hadamard type integral of order 1 − . Let J = (1, T] with T > 1, then f , g : J × X × X → X are continuous and nonlinear functions on a Banach space X := R. This work is outlined as follows: In Section 2, we present some basic notions needed to prove our main results. In Section 3, we setup some adequate conditions that are used to prove the existence-uniqueness and HU stability results of solutions for system (1.2). The established results are illustrated with an example in Section 4.

Fundamental results
In this section, we introduce basic definitions and lemmas which will be used throughout this manuscript. The notations and terminologies are adopted from [1,5,8,57].
Definition 2.1. The fractional order Hadamard type derivative with order σ for a function θ : [1, ∞) → X is defined as where ⌈σ⌉ is the integer part of σ.
Definition 2.2. The fractional order Hadamard type integral with order σ for a function θ : [1, ∞) → X is given as provided that the integral on the right side exists.
, the HH type derivative of order α, β for a function θ is defined as Then the homogenous DE along with HH fractional order has solution of the form Theorem 2.5. Let S ≠ ∅ be a convex and closed subset of a Banach space E. Consider two operators G and F such that Definition 2.6. Consider a Banach space E such that Φ 1 , Φ 2 : E → E are two operators. Then the operator system is called HU stable if there exist constants C i (i = 1, 2, 3, 4) > 0 for each ϱ j (j = 1, 2) > 0 and for each solution there exists a solution (̃︀ u,̃︀ v) ∈ E of system (2.1), which satisfies the inequalities Definition 2.7. Let µ j (for j = 1, 2, . . . , m) be the eigenvalues of a matrix H ∈ C m×m . Then the spectral radius r(H) of H is defined by Furthermore, the system corresponding to H converges to zero provided that r(H) < 1.
if the spectral radius of matrix is less than one, then the fixed points corresponding to operational system (1.2) are HU stable.

Existence, uniqueness and stability results
Here, we discuss the existence, uniqueness and stability of our proposed system. Our first result is stated as follows.
if and only if u, v are the solutions of then by using Lemma 2.4, we have Applying the boundary conditions, we get b 0 = a Γ( ) and Similarly, we may have The proof is completed.
We make use of the following assumptions: (H 2 ) f , g : J × X → X are completely continuous functions ∀ u, v ∈ X and t ∈ J, there exist nondecreasing continuous linear functions µ f , µg : X → X + such that (H 3 ) Let ξ * = max{ξ 1 , ξ 2 } < 1 with It is obvious that (J, X) is a Banach space with the norm ‖u‖ = max{|u(t)|, t ∈ J} and (J, X × X) is a Banach space with norm ‖(u, v)‖ = ‖u‖ + ‖v‖. C 1− ,log(t) (J, X) denote the space of all continuous functions defined as Define the operators F = (F 1 , F 2 ), G = (G 1 , G 2 ) on Er as Proof. For any (u, v) ∈ Er, we have Similarly, Next, from (3.4), we get Hence By similar procedure, we get Also, we have Combining all these inequalities and using (3.6), we have Hence, F(u, v) + G(u, v) ∈ Er. Next, for any t ∈ J and (u, v), (u, v) ∈ X, we have Similarly, Using (3.7), we have Here β = max{β 1 , β 2 }, where , .
Hence F is a contraction mapping. Now, we show that the operator G is continuous and compact. Consider a sequence ξn = (un , vn) ∈ Er such that (un , vn) → (u, v) for n → ∞ ∈ Er. Therefore, we have

vn(s), H D p,q un(s)) − g(s, v(s), H D p,q u(s))
This implies that ‖G(un , vn)−G(u, v)‖ C 1− ,log(t) → 0 as n → ∞. Thus G is continuous. To show that the operator G is bounded on Er we have which implies that G is uniformly bounded on Er.

An example
To demonstrate our theoretical results, an example is presented as follows.
Example 4.1. Consider the following system of fractional order differential equations consisting of HH type fractional derivatives as  For any (u, v), (u, v) ∈ X, we have Here M f = M ′ f = 1 10e 2 , Mg = M ′ g = 1 20 , T = e. If we take p = 2 3 , q = 1 2 then we get = 5 6 . Upon calculations, we have ξ * = 0.0251 < 1. Therefore, system (4.1) has a unique solution. Furthermore, we observe that and if ω 1 and ω 2 are the eigenvalues, then ω 1 = 0.0149 and ω 2 = −0.0001. Since the spectral radius of H is less than one. Thus, system (4.1) converges to 0. That is, system (4.1) is HU stable.

Conclusion
We used Banach contraction principle and Krasnoselskii fixed point theorem to establish sufficient conditions for the existence and uniqueness of the solution of coupled impulsive fractional differential system of HH type given in (1.2). In addition and under particular assumptions and conditions, we have studied the UH stability results of different kinds for the solution of the proposed problem. In view of the results of this paper, we conclude that such a method is very powerful, effectual and suitable for the solution of nonlinear fractional differential equations.