Mathematical Modelling and

: This paper studies the existence of solutions for fractional dynamical systems with two damping terms in Banach space. First, we generalize the well-known Gronwall inequality. Next, according to fixed-point theorems and inequalities, the existence results for the considered system are obtained. At last, an example is used to support the main results.


Introduction
In recent years, fractional calculus has gained much attention.
Compared with the integer derivative, the fractional derivative can lead to better results in many practical problems as it serves as a powerful tool to describe the memory and genetic properties of various materials and processes. Fractional differential equations (FDEs) appear in plenty of scientific and engineering disciplines because they describe many events and processes in the domains of aerodynamics, chemistry, physics, the rheology of polymers, etc. For more details, see [1][2][3] and the references therein.
In addition, fractional damping systems based on velocity history have attracted extensive attention and been the focus of much research in the last few decades (see [4][5][6][7]). For example, problems of non-viscous damping with hysteresis have been studied in the application of magnetorheological fluids (see [8]). Similarly, this concept was used to simulate damping in a vehicle tire and plates made of composite materials (see [9,10]). The authors of [11] considered a mechanical system with viscoelastic damping, mass and a rapid jump by producing the FDEs based on Caputo fractional derivatives. For more exciting results about damped dynamic systems, see [4][5][6][7][8].
The existence of solutions for fractional dynamical systems has been widely investigated because it is a fundamental problem of fractional dynamical systems as well as a necessary condition to consider other properties such as controllability and stability (see [12][13][14]). For instance, the authors of [15] investigated the existence of solutions for fractional neutral differential equations with infinite delay. The authors of [16] obtained existence results for an impulsive fractional integro-differential equation with state-dependent delay. In addition, many authors have studied multi-term fractional systems as they have been successfully used in gas dynamics, mechanical systems, etc.
(see [17][18][19][20][21]). These systems are more complicated and interesting than one-term fractional systems. For example, Sheng and Jiang discussed the following system in [22]: where 0 < β ≤ 1 < α ≤ 2, x ∈ R n , A is an n × n matrix and f : J × R n → R n is continuous. The existence results were obtained by utilizing fixed-point theorems.
It is well known that many significant results have been achieved in the study of finite-dimensional dynamical systems. However, the research on dynamical systems is not limited to the finite dimension. From the perspective of practical problems in physics, many important dynamical system problems, such as turbulence in fluid mechanics, discrete attractors and small dissipative dynamics (see [23]), are studied in infinite dimensions.
Inspired by the above, we investigate the following fractional dynamical system with two damping terms in Banach space H: To the best of our knowledge, few people have studied this type of system. Only Zhang and Xu [24] has studied (1.1) in a finite-dimensional space. The existence and uniqueness of solutions for (1.1) have been obtained by using the Banach fixed point theorem. It is remarkable that ℏ(t, ξ) meets the Lipschitz condition, which is difficult to satisfy in practical problems. Compared with [24], this article has the following distinctive features. First, we consider (1.1) in abstract space. Second, on the basis of [22], we generalize the Gronwall inequality, which is crucial for the proof of our results. Third, ℏ(t, ξ) here is no longer required to satisfy the Lipschitz conditions. The remainder of this article is organized as follows. In Section 2, we introduce some fundamental concepts and lemmas, which will be used throughout the paper. In Section 3, first, the Gronwall inequality is extended. Second, the main results are presented and proved. Last, we give an illustrative instance to support the main results. In Section 4, the conclusion of the full text is given.

Preliminaries
In this section, we introduce some definitions and lemmas used to prove the conclusion. Throughout this paper, let PC(T, H) be the Banach space of all continuous functions from T = [0, ι] to H with the norm ∥ξ∥ c = sup{∥ξ(t)∥ : t ∈ T} for ξ ∈ PC(T, H), where ∥ · ∥ is the norm of the Banach space H. In addition, X(·) and X c (·) represent the Hausdorff measure of noncompactness of a bounded set in H and PC(T, H) respectively.
the lower zero is defined as follows: where t > 0, α > 0 and Γ(·) is a gamma function. with the lower zero is defined as follows: where n = [α] + 1, t > 0 and α > 0.
From the definition of fractional integrals and Caputo derivatives, we have the following results.
Lemma 2.6 [26] Suppose that H is a Banach space.
Let D be a closed and convex subset of H and x 0 ∈ D.
Assume that the continuous operator Q : D → D satisfies the following: Then, Q has a fixed point in D.

Generalization of Gronwall inequality
In this section, we extend the Gronwall inequality. Then, Define Then, which implies that Now, we prove that and that B r ξ(t) → 0 as r → +∞ for each t ∈ [0, ι].
Suppose that it holds for r = k.
We exchange the integral order. Then, Combining (3.7) and (3.8), we obtain k q=0 p=0 and k+1 p=1 q=k−p+1 By calculation, we can obtain can be rewritten as As a result, the inequality (3.3) is proved.

Existence of solutions
The existence results of (1.1) are given in this section.
(H3) There exist k 3 ∈ (0, 1) and a real function ω ∈ L For convenience, denote Now, we introduce the following primary results. (3.14) Then, (1.1) has at least one solution on T.
Proof. Suppose that ξ(t) is the solution of (1.1). Taking the integral of order α on both sides of (1.1), by Lemma 2.4 and Lemma 2.5, we can get that Define the operator Q on PC(T, H). Obviously, we only need to certify the existence of fixed points of Q.
Step 1. We state that Q is continuous.
In fact, for arbitrary ξ ∈ D ϱ , using (H2), we have These, together with (3.22), guarantee that Step 3. We demonstrate that S is relatively compact if S ⊂ D ϱ is countable and for some ξ 0 ∈ D ϱ .
In addition, suppose that Then (1.1) has at least one solution on T.
Proof. First, it is easy to get that Q : PC(T, H) → PC(T, H) defined as (3.15) is bounded and continuous.
Then, from Lemma 2.8, for any ε > 0 and bounded for arbitrary t ∈ T. Notice that Let ε → 0. We obtain At last, define the set From Lemma 2.9, (1.1) has at least one solution. □ Remark 3.1. Notice that l 1 ∈ L ∞ (T, H), which implies that l 1 (t) is no longer required to satisfy (3.14).

Examples
In this section, an example is used to illustrate the effectiveness of the obtained results.
In addition, A = ε 1 E and B = ε 2 E, where E is the identity operator.
We can easily conclude that the function ℏ satisfies (H1).
To verify condition (H2), let ξ = {ξ j } ∞ j=1 ∈ c 0 . Then for t ∈ T. So we choose l 1 (t) = 1 t 2 +1 , l 2 (t) = t t 2 +1 . This shows that (H2) holds. To prove (H3), we review the Hausdorff measure of noncompactness X in c 0 : where D is a bounded subset in c 0 and T n is the projection onto the linear span of the first n vectors in the standard basis [27].