Study on the Multi-Point Boundary Value Problem for Second-Order Nonlinear Impulsive Integro-Differential Equation

. In the feld of biological control, there are a large number of systems that gradually evolve over a certain period of time. However, due to some natural or human intervention behavior, the system state will be subjected to some relatively short time interference, so that the system state changes in an instant. Tis sudden change of state makes the system not simply described by continuous or discrete dynamical systems, but by means of impulse dynamical systems. In this paper, the multi-point boundary value problem of a class of second-order nonlinear impulsive integral diferential equations is studied on the basis of impulsive diferential equation theory. Te main results of this kind of equation are obtained by using the fxed point theorem of strict set contraction operator. Under certain assumptions, the existence of the solution of the equation is proved by constructing the operator on the special cone. Finally, combined with the practical application, the theory was applied to the stability prediction of biological ecosystem, and the correctness of the conclusion was verifed.


Introduction
Impulsive diferential equation is a fundamental tool for studying process state transitions and has important applications in the life sciences. Compared with the diferential equation without pulse, impulsive diferential equation can refect the motion processes in nature and science feld more truly and accurately. Terefore, it is widely used in the population dynamic system, infectious disease dynamic model, microbial model, chemotherapy in medicine, and neural network system [1][2][3][4]. Especially in recent years, infectious diseases are rampant, so a large number of mathematical models have been used to analyze various infectious diseases. To get closer to practical problems and truly refect certain biological states, instantaneous changes of the system can be expressed in the form of pulses in mathematical modeling. Terefore, diferential equation models with impulse efects have been widely studied by scholars. A large number of studies have shown that it is of great practical signifcance to study the biological control of such systems and predict the future development trend of biological systems by using impulsive diferential equation. For example, Baleanu et al. [2] used the Picard-Lindelöf method and the fxed point theory to discuss a new fractional human liver model with exponential kernel Caputo-Fabrizio derivative and obtained the known uniqueness result. Compared with the actual clinical data, the new fractional-order model is superior to the existing integerorder model with ordinary time derivatives. Mohammadi et al. [3] studied the Caputo-Fabrizio fractional-order model of mumps virus-induced hearing loss by the Picard-Lindelöf technique, proved that the fractional-order system of the model has a unique solution, and studied the stability of the iterative method by using the fxed point theory. Te results identify optimal control of the system using treatment as a control strategy to reduce the number of infections and take into account the importance of reproductive numbers in the continuation of disease transmission. Gul et al. [4] analyzed the coupling system of a nonlinear three-point boundary value problem, deduced the necessary conditions for the uniqueness and stability of the mixed sequence BVP coupling system, extended the results to a practical fourchamber typhoid therapy mathematical model, and carried out numerical simulation to analyze the infuence of different order changes on the dynamic behavior of the typhoid model mixed system.
On the other hand, as an important branch of diferential equation research, impulsive diferential equation boundary value problem has been widely used in various felds of mathematics and physics. Especially in the last 20 years, impulsive diferential equation boundary value problem has been studied extensively by many scholars at home and abroad, and they obtained more abundant results than diferential equation [5][6][7][8]. However, it is a pity that the current research results mainly focus on the exploration of two-point, three-point, or periodic boundary value problems, while there is little research on the multi-point boundary value problems [8], especially the research on multi-point boundary value problem for impulsive integrodiferential equation in Banach space has not been reported. Terefore, it is of great theoretical and practical value to focus on this kind of problems.
In paper [5], Guo et al. studied the existence of solutions for the following m-point boundary value problem: where b, k i > 0, 0 < ξ 1 < ξ 2 , · · · , < ξ m− 2 < 1. By using the Schauder fxed point theorem, the authors obtained the existence solutions of the above equation.
Recently, Jiang in [9] studied the existence of positive solutions for multi-point BVPs: under the following conditions: Te author investigated the existence of positive solutions by applying the fxed point index theory.
In paper [6], Guo and Liu used the fxed point index theory to study the following impulsive diferential equation: Zhang et al. in [10] were concerned with the multiple positive solutions for a class of m-point boundary value problems: where λ > 0, a, b, c, and d are all nonnegative numbers, 1]. By using the fxed point index theory, the authors showed the existence of multiple positive solutions for the above problem.
In [11], Wang et al. used the cone theory and the monotone iterative technique to investigate the existence of maximal and minimal solutions of the initial value problem for second-order impulsive integro-diferential equations: Very recently, Guo in [12] discussed the existence of two positive solutions for second-order impulsive singular integro-diferential equations: In recent years, impulsive integro-diferential equation has been rarely studied, and most of the related literature focuses on two-point, three-point, or periodic boundary value problems, and most of the methods adopted are upper and lower solution method, monotone iteration technique, cone theory, or fxed point index theory [9][10][11][12][13][14][15][16][17][18][19][20][21]. Based on this situation and inspired by the literature [6,12,15], this paper utilizes the fxed point theorem of strict set compression operator to study multi-point boundary value problem for impulsive integro-diferential equation. Te equations studied are combined with the literature [6,10,11], and the problem is extended to two-point, three-point, and multi-point boundary value problems. By constructing operators on special cones, the existence results of solutions of this kind of equations are studied under certain assumptions. Te main results of this paper extend the conclusions of existing literature to a large extent and enrich the research results of multi-point boundary value problem for impulsive integro-diferential equation in Banach space. Te method adopted in this paper is innovative and extensible to some extent.
Tis paper is organized as follows. In Section 2, we shall introduce some lemmas and provide some background. In Section 3, we obtain the main results and prove them. Finally, an example is given to illustrate the efectiveness of our conclusion.
can be expressed as the union of a fnite number of sets such that the diameter of each set does not exceed ε, i.e.,

Lemma 2 (see [13]). Suppose H is a bounded, closed, and convex set and H ⊂ E. If operator T is a strict set contraction from H to H, then T has a fxed point in H.
For convenience, we list the following assumptions: Journal of Mathematics where where

Journal of Mathematics
Proof. First assume that y ∈ BC 1 [J, E] ∩ C 2 (J ′ ) is a solution of BVP (1), and by integration of BVP (1),

Journal of Mathematics
From the above two equations, we obtain s g s, y(s), y ′ (s), (Ay)(s), (By)(s) ds Hence, we have G(t, s)g s, y(s), y ′ (s), (Ay)(s), (By)(s) ds + n k�1 G t, t k I * k y t k , y ′ t k Conversely, assume that y ∈ BC 1 [J, E] is a solution of BVP (1). It is easy to see that ∆y| t�t k � I k y t k , (k � 1, 2, · · · , n).
From (28), we ca get For convenience, let Lemma 5 (see [13]). If W is bounded set in BC 1 [J, E] and the elements of W′ are equicontinuous on each J k (k � 1, 2, · · · , n), then is a strict set contraction operator.

Application
In the past diferential equation theory, people always assume that the state of dynamic system is continuously changing with time. However, in the real world, there are many natural systems continuously changing in certain time intervals. For some reasons, the state of the system is often subjected to some temporary interference, which makes the state of the system change greatly in a short moment. For example, when fsh farmers release fry or catch fsh, the number of fsh in the pond will suddenly increase or decrease. When spraying pesticides in agriculture and forestry, the number of pests will be greatly reduced. In the control of infectious diseases such as the novel coronavirus pneumonia, the number of vulnerable people in the region has been rapidly reduced through containment, quarantine, and vaccination. Te disturbance that makes the system change dramatically instantaneously can be expressed in the form of pulse in mathematical modeling. Obviously, the evolution process of pulsed disturbance is widely used in various biological physiological systems, so it is of great practical signifcance to study the biological control of these systems and predict the future development trend of biological population by using impulsive diferential equations.
represented by mathematical modeling in the form of pulses. Based on this, the paper discusses the multi-point boundary value problem of a class of integro-diferential equation with impulse efects by using the fxed point theorem of the strict set construction operators under the interference of many external conditions. Te main result that the system has at least one solution is obtained, and the results are extended to the study of the stability of biological systems. Finally, the correctness of the mathematical model is verifed by the application examples of two biological population models under multiple external interferences. Tis model provides a new way to study the future development trend of population.

Data Availability
All the datasets are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.