A discussion concerning the existence results for the Sobolev-type Hilfer fractional delay integro-differential systems

The goal of this study is to propose the existence results for the Sobolev-type Hilfer fractional integro-differential systems with infinite delay. We intend to implement the outcomes and realities of fractional theory to obtain the main results by Monch’s fixed point technique. Moreover, we show the existence and controllability of the thought about the fractional system with the nonlocal condition. In addition, an application to illustrate the outcomes is also included.

The differential system with Sobolev-type is frequently evident in the mathematical structure of several physical events similar to the flowing of fluids through fractured rocks, thermodynamics. The readers may refer to [50][51][52][53][54][55][56]. Many authors discussed the relations between the asymptotic stability of the zero solution for retarded differential equations and real parts of all characteristic roots of characteristic equations. In [57] the author investigated the asymptotic stability of the zero solution for Caputo-Hadamard fractional dynamic equations on a time scale. These equations guarantee the effectiveness of the zero solution, and several authors reported interesting fixed point results in the framework of complete b-metric spaces, recently, Lazreg et al. [58] established some impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces.
Control hypothesis is a significant region of usage arranged in mathematics which deals with the design and assessment of control structures. The development of modern mathematical control theory is heavily influenced by controllability. The problem of controllability of dynamical systems is commonly employed in control system analysis and design. Fractional-order control systems defined by fractional-order differential equations have gotten a lot of interest in recent years, a wide list of these distributions can be found in [25,26,28,29,40,43,48,51,56,[59][60][61][62]. The controllability of impulsive fractional evolution inclusions with state-dependent delay is demonstrated in [63], which employs a fixed point theorem for condensing maps.
From the above literature survey, to our knowledge the existence and exact controllability of the fractional system have not been studied fully. Motivated by this fact, we consider the Sobolev-type Hilfer fractional integro-differential system of the form (1.1) 2) and assume that the system with control has the following form: 3) where D α,β 0 + stands for Hilfer fractional derivative of type 1 2 < β < 1, order 0 ≤ α ≤ 1. The state z(·) takes values in a Banach space along with the norm · , A is the infinitesimal generator of a C 0 -semigroup. The control function u(·) ∈ L 2 (N, U). The histories We organize the remaining part of our article as follows: Some new notations, important facts, lemmas, vital definitions, and theoretical results are recalled in Sect. 2. Section 3 provides the existence of fractional system (1.1)-(1.2) which is proven by Monch's fixed point theorem. We extended the study to deal with the exact controllability for (1.3)-(1.4) in Sect. 4. In Sect. 5, we discuss the system with nonlocal conditions. Finally, we end with Sect. 6, which presents our conclusions.

Preliminaries
We review the essential hypothesis which is utilized all through the work in request to acquire new outcomes. [53].
(F 1 ) The linear operators A and J are closed.
is continuous. In addition, from (F 1 ), (F 2 ), J -1 is closed. Applying the closed graph theorem and (F 3 ), we obtain the boundedness of AJ -1 : Z → Z. Designate J -1 = J m and J = J m .

Definition 2.3
The Hilfer fractional derivative of order 0 ≤ α ≤ 1 and type 0 < β < 1 for f (t) of the form

Remark 2.4
(i) In case α = 0, b = 0, the Hilfer fractional differential is identical to the classical Riemann-Liouville fractional derivative for f of the form (ii) In case α = 1, 0 < β < 1, and b = 0, the Hilfer fractional derivative is identical to the classical Caputo derivative for f of the form As of now, we characterize the abstract phase space B l , which is introduced in [51]. Let g : (-∞, 0] → (0, +∞) be a continuous function with j = Define fix · b in B l , and it is characterized by Remark 2.7 We define the mild solution of (1.1)-(1.2) as follows: where M β (θ ) is a Wright function and satisfies

Lemma 2.8 ([65])
The operators S α,β (t) and Q β (t) satisfy the following conditions: • For t ≥ 0, the operators S α,β (t) and Q β (t) are linearly bounded, i.e., for every z ∈ Z, The measure of noncompactness of φ is called: The measure of noncompactness of Hausdorff μ is defined by To know more information about the properties of MNC, the readers can refer to [66].
here Y is a Banach space and k is any constant.

Existence
In this section, we mainly focus on the existence of (1.1)-(1.2), and in order to prove the main theorem, we have the following assumptions.
for almost all t ∈ N, Proof We define the operator ϒ : B l → B l by For φ ∈ B l , we defineη as follows: Clearly, z satisfies (2.1) if and only if g satisfies g 0 = 0 and Hence (B l , · b ) is a Banach space. Now > 0, we fix F = {g ∈ B l : g b ≤ }, then F ⊆ B l is uniformly bounded, g ∈ F , and referring to Lemma 2.5, We define the operatorΥ : B l → B l as follows: To prove that ϒ has a fixed point. Now we divide the proof into a few steps for our benefit.

Controllability
In this section, we mainly focus on the controllability of (1. (4.1) Controllability results are proved in relation to the following hypotheses: For our convenience, we introduce Proof By using (A 3 ), we define u z (t) by Let ϒ : B l → B l be defined by For φ ∈ B l , we havê . Now, we identified that z satisfies (4.1) if and only if g satisfies g 0 = 0 and We define the operatorΥ : Now, to show ϒ has a fixed point. We divide the proof into the following steps for our convenience.
Step 1: To prove that there exists a constant > 0 such that ϒ(F ) ⊆ F . If it fails, then g (·) ∈ F and t ∈ N such that ϒ(g )(t) > .
Take > 0 and consider {F = z ∈ C 1-v : z v ≤ }. Apparently, F is a closed, bounded, and convex set of C.
Step 2: Similar to Step 2 of Theorem 3.1.
Hereafter, we continue our proof as per Step 3 of Theorem 3.1, and hence ϒ(F ) is equicontinuous.

Nonlocal conditions
The nonlocal Cauchy problem for differential equation was first studied by Byszewski [69].
Their research is driven by imaginative enthusiasm and the manner in which these types of problems usually occur when proving practical applications. For example, material science and life sciences can be depicted by techniques for the differential framework subject to nonlocal limit conditions, the readers can refer to [48,60,62,69,70]. We presently expect that the nonlocal Sobolev-type Hilfer fractional integro-differential equations with control are as follows: t 0 e(t, s, z s ) ds + Bu(t), t ∈ N = (0, b], (5.1) The result is proved in relation to the following hypothesis: is satisfied.
is satisfied.

Conclusion
In this article, we have fundamentally focused on a class of Sobolev-type Hilfer fractional integro-differential framework with infinite delay, which generalized the Riemann-Liouville fractional derivative. At first, we dealt with the new existence result of a mild solution with the assumptions that the framework satisfies the initial condition and noncompactness measure condition. Later, we have presented the controllability results of the thought about the fractional framework. In the end, we introduced an example to show the procured hypothetical results. We will try to investigate the neutral differential equation and controllability of a similar problem in our future research work.