Asymptotic Behavior of Some Differential Inequalities with Mixed Delays and Their Applications

: In this paper, we focus on the asymptotic stability of the trajectories governed by the differential inequalities with mixed delays using the fixed-point theorem. It is interesting that the Halanay inequality is a special case of the differential inequality studied in this paper. Our results generalize and improve the existing results on Halanay inequality. Finally, three numerical examples are utilized to illustrate the effectiveness of the obtained results

The operator A in ( 2) is a neutral-type operator which has D−operator form; see [1].Neutral-type functional differential equations and dynamic systems with D−operator have been extensively studied; see [2][3][4][5][6] and related references.Hence, the study of properties of differential inequalities (1) is helpful for the study of neutral-type dynamical systems.In this paper, we need the following lemmas for neutral-type operator A and the Banach contraction mapping principle.Lemma 1 ([7]).Let the operator (Ax)(t) ∈ C(R, R) defined by (2).If |c(t)| < 1, then operator A has continuous inverse A −1 on C(R, R), satisfying where c ∞ = max t∈R |c(t)|.
We also need the following definition: is a solution of system (1) and x(t) is any solution of system (1) satisfying We call x * (t) is globally asymptotic stable.
In past decades, the theory of computational methods for neutral-type delay differential equations has been studied by many authors, and a a great many interesting results have been obtained.Since there exists a neutral-type operator in (1), the research results of this article have wide applications in neutral-type delay differential equations and other fields.
The results obtained in this article are based on the fixed-point theorem.The fixedpoint theorem has wide applications in many branches of mathematics.Jabbar [19] studied the applications of Hardy and Rogers type contractive condition and common fixed-point theorem in cone 2-metric space.Niezgoda [20] considered a companion preorder to G−majorization type fixed-point theorem.Aydi et al. [21] obtained a fixed-point theorem for set-valued quasi-contraction maps in b-metric spaces.For more applications of fixed-point theorem, see [22][23][24].
We also use the theoretical results to analyze some dynamical properties in delay dynamical systems.We list the major contributions of this paper as follows: (1) Halanay inequality and generalized Halanay inequality are the special forms of delay inequality (1).Therefore, the theoretical results of this article can be used to study more general delay dynamical systems.Specifically, the results of this paper can be used to conveniently study neutral-type differential dynamical systems.(2) Specifically, the results of this paper can be used to conveniently study neutral-type differential dynamical systems.Example 1 contains a first-order neutral-type differential equation, which can be used to model models of biological populations and lossless transmission systems.Examples 2 and 3 contain first-order nonlinear equations with mixed delays.Many neural networks, physical models, chemical models, and infectious disease models can be described by Examples 2 and 3. (3) The uniform positiveness of coefficient function is no longer required which is different from corresponding ones in [11][12][13][14][15].
The following sections are organized as follows: In Section 2, the main theoretical conclusions are given.Section 3 gives three numerical examples for the considered system.Finally, Section 4 concludes the paper.
Therefore, the results of this paper have wider applicability, particularly since we can conveniently study neutral-type differential equations using the results of this paper.
Remark 2. The theory of time scales unifies discrete systems and continuous systems.In recent years, many researchers have focused on the study of Halanay inequality on time scales.In [25], Ou, Jia, and Erbe studied the following generalized Halanay inequality on time scales: where t ∈ T, T is an arbitrary time scale, the means of other coefficients can be found in [25].Using the theory of time scales, the authors obtained dynamic behaviors for the delay system (26).Wen, Yu, and Wang [26] studied generalized Halanay inequalities for dissipativity of Volterra functional differential equations.Then, in [27], the author extends a result of [26] to Halanay inequality on time scales with unbounded coefficients.For more results about Halanay inequality on time scales, see [28][29][30].In future work, we will study neutral-type delay differential inequalities on time scales.

Examples
In this section, we give three numerical examples to test and verify our main results.
Example 1.Consider the following delay differential inequality: where Hence, all conditions of Theorem 1 hold, we know that x(t) → 0 as t → ∞.We choose different initial conditions x(0) = 9.5 and x(0) = 7.5, simulation results present in Figure 1 which verify the validity of our theoretical results.x(t) x (1) (t) x (2) (t) Example 2. Consider the following delay differential inequality: where a(t) = Hence, all conditions of Corollary 1 hold, we know that x(t) → 0 as t → ∞.We choose different initial conditions x(0) = 5.5 and x(0) = 10, simulation results present in Figure 2 which verify the validity of our theoretical results.x(t) x (1) (t) x (2) (t) Figure 2. The states' trajectory of the system (28) with different initial conditions.
Example 3. Consider the following delay differential inequality: where a(t) = Hence, all conditions of Corollary 3 hold, we know that x(t) → 0 as t → ∞.We choose different initial conditions x(0) = 6 and x(0) = 8, simulation results present in Figure 3, which verify the validity of our theoretical results.x(t) x (1) (t) x (2) (t)

Conclusions
In this paper, we used the fixed-point theorem to investigate delay inequalities with neutral-type operators and obtained some sufficient conditions of asymptotic stability.We presented some numerical examples to test and verify the theoretical results.It should be emphasized that system (1) includes the Halanay inequality and its extension.We believe that studying system (1) can help us understand the Halanay inequality in a broader context.Also, to the authors' knowledge, although there have been many results on the study of the Halanay inequality on general time scales, we have not found any research results on system (1).In future work, we are devoted to studying system (1) on time scales.We also investigate system (1) with pulse response and random disturbance in the subsequent study of this field.

Figure 1 .
Figure 1.The states' trajectory of the system (27) with different initial conditions.

Figure 3 .
Figure 3.The states' trajectory of the system (29) with different initial conditions.