Asymptotic Stability ofNeutral Set-ValuedFunctionalDifferential Equation by Fixed Point Method

It is well known that Lyapunov’s direct method is an important technique to consider the stability of various differential equations. However, this method is not always valid for stability analysis in the functional differential equation when the delay is unbounded or when the equation has unbounded terms [1–3]. Burton and many researchers found a way to these difficulties by using various fixed point theorems; we can refer to the literature studies [4–14]. Recently, the study of qualitative analysis of the setvalued differential equation has attracted much attention. )e stability results of various set-valued differential equations were obtained by applying Lyapunov’s direct method.)e results can be found in the monograph [15], the papers for the set-valued differential equation [16–24], setvalued functional differential equation [25–29], and other equations [6, 9, 30–32]. However, according to what we know so far, there are few stability results for the set-valued differential equation via the fixed point method. Inspired by the application of the fixed point method mentioned above, in this paper, we study a class of nonlinear neutral set-valued functional differential equations:


Introduction
It is well known that Lyapunov's direct method is an important technique to consider the stability of various differential equations. However, this method is not always valid for stability analysis in the functional differential equation when the delay is unbounded or when the equation has unbounded terms [1][2][3]. Burton and many researchers found a way to these difficulties by using various fixed point theorems; we can refer to the literature studies [4][5][6][7][8][9][10][11][12][13][14].
Recently, the study of qualitative analysis of the setvalued differential equation has attracted much attention.
e stability results of various set-valued differential equations were obtained by applying Lyapunov's direct method. e results can be found in the monograph [15], the papers for the set-valued differential equation [16][17][18][19][20][21][22][23][24], setvalued functional differential equation [25][26][27][28][29], and other equations [6,9,[30][31][32]. However, according to what we know so far, there are few stability results for the set-valued differential equation via the fixed point method. Inspired by the application of the fixed point method mentioned above, in this paper, we study a class of nonlinear neutral set-valued functional differential equations: denote the collection of all nonempty, compact convex subsets of Banach space E; θ denotes the null set-valued function θ : I ⟶ K c (E), and θ(t) � 0 { } for t ∈ I, τ > 0 is a constant. e aim of this paper is to obtain an asymptotic stability theorem with a necessary and sufficient condition via the fixed point method. In addition, an application of the main result is presented.

Preliminaries
To get the desired result, we first give some notations, definitions, and propositions briefly; for the details, see the literature [15].
Let K c (E) denote the collection of all nonempty, compact convex subsets of Banach space E, given A, B ∈ K c (E), defining the Hausdorff metric between A and B as follows: where e Hausdorff metric satisfies the properties as follows: for all A, B, C ∈ K c (E) and λ ∈ R.
In the sense of the above metric D, the set K c (E) is a complete metric space.
Definition 1 (see [15]). e set-valued function F: By embedding K c (E) as a complete cone in a corresponding Banach space and taking into account the differentiation of the Bochner integral, we can find that if Proposition 1 (see [15] Proposition 2 (see [15]). If X, Y: respectively, and ‖X‖ 0 � max ‖X‖, ‖D H X‖ . By the properties of Hausdorff metric D and the definition of ‖ · ‖ 0 , we can get the following: e trivial solution of (1) is said to be For A, B ∈ K c (E) and λ ∈ R, we can define the following addition and scalar multiplication as follows: en, with the algebraic operations of addition and nonnegative scalar multiplication, (K c (E), D) becomes a semilinear metric space.
In order to apply the fixed point method, we first need to prove the following lemma.
Proof. Firstly, from the previous discussion, we know that (X 0 , ‖·‖ 0 ) is a semilinear metric space. Next, we prove the space is complete. Assume that X m (t) is a Cauchy sequence; then, for any ε > 0, there exists positive integer N such that, for all m, n > N, en, X n (t) and D H X n (t) are Cauchy sequences in K c (E), respectively. Since (K c (E), D) is a complete metric space, there exist set-valued functions X(t) and Z(t) such that First, we prove that X(t) is a continuous function on Using the same way, we can also prove that Z(t) is a continuous function.

Main Result
In this section, we establish the necessary and sufficient conditions for the global asymptotic stability of trivial solution of equation (1) by using the fixed point method.

Theorem 1. Assume that the following conditions hold:
en, the trivial solution of equation (1) is globally asymptotically stable in C 1 if and only if +∞ t 0 a(s)ds � +∞. (16) Proof. We will prove this conclusion in two steps.

Proof of Sufficiency.
Let D � X ∈ X 0 : X(t) � Φ(t), t ∈ I 0 }. en, D is a nonempty, closed subset of X 0 . At the same time, we can get equation (1) with initial condition X(t 0 ) � Φ(t) which is equivalent to the following integral equation: We define the mapping Q: D ⟶ C 1 (I 0 ∪I, K c (E)) as follows: Firstly, we prove that Q is a mapping of D ⟶ D. In fact, we just need to prove that (QX)(t) ⟶ 0 us, from (18), (19), (A 2 ), and (A 3 ), we have en, by condition (16), there exists T 1 > T such that, for t > T 1 implies us, we have ‖(QX)(t)‖ < (1 + α)ε for t > T 1 , and we can obtain (QX)(t) ⟶ 0 { } as t ⟶ +∞. In addition, we can also get Furthermore, from (22), we can get From (A 1 ) and lim Secondly, we prove that the mapping Q has a unique fixed point.

Conclusion
Stability is one of the main problems encountered in applications and has recently attracted considerable attention. e fixed point method is an effective method to discuss the stability for the differential equation with unbounded delay or the differential equation with unbounded terms. In this paper, we investigate a class of neutral set-valued functional differential equations and obtain a criterion for the globally asymptotic stability theorem with necessary and sufficient conditions by the fixed point method. Finally, we verify the validity of the result by an example.

Data Availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.