Existence and uniqueness of solutions for nonlinear impulsive differential equations with three-point boundary conditions

Abstract This paper is devoted to a system of nonlinear impulsive differential equations with three-point boundary conditions. The Green function is constructed and considered original problem is reduced to the equivalent impulsive integral equations. Sufficient conditions are found for the existence and uniqueness of solutions for the boundary value problems for the first order nonlinear system of the impulsive ordinary differential equations with three-point boundary conditions. The Banach fixed point theorem is used to prove the existence and uniqueness of a solution of the problem and Schaefer’s fixed point theorem is used to prove the existence of a solution of the problem under consideration. We illustrate the application of the main results by two examples.


Introduction
The theory of impulsive differential equations is a relatively new and crucial branch of differential equations. The theory has found applications in many areas where evolutionary processes undergo rapid changes at certain times of their development. There has been a great deal of development in impulsive theory, especially in the area of impulsive differential equations with fixed point moments (see, for instance, the monographs [1][2][3][4] and the references therein). We must note here that there are applications of impulsive differential equations to intervention models and interrupted time series analysis, percussive systems with vibrations to population dynamics, and relaxation oscillations of the electromechanical systems ( [6][7][8] and the references therein). Some authors have produced an extensive portfolio of results on differential equations undergoing impulse effects. The existence questions for impulsive differential equations have been studied in [10][11][12][13][14][15][16][17][18][19] and references therein. Three-point boundary value problems for ordinary differential equations are also studied in recent years. They arise in the modeling and analyzing of many physical systems, such as some problems in the theory of elastic stability, the vibrations of a guy-wire of a uniform cross-section and composed on N parts of different densities [5]. Moreover, boundary value problems have been studied by a number of authors in [9,[20][21][22][23][24][25][26][27][28][29] and references therein.

Problem statement
This paper deals with the existence and uniqueness of solution for the system of nonlinear impulsive differential equations of the typė subject to impulsive conditions and three-point boundary conditions where A, B, C are constant square matrices of order n such that det N = 0, N = (A + B + C); f : [0, T ] × R n → R n and I i : R n → R n are given functions; are the right-and left-hand limits of x(t) at t = t i , respectively. In the following, we first introduce necessary background. Next, theorems related to existence and uniqueness of a solution of problem (1)-(3) are proved under some sufficient conditions on the nonlinear terms. At last, an example of application of the main result of this paper is given.

Preliminaries
In this section, we give notations, preliminary facts and basic definitions which are used throughout this paper. We denote by C([0, T ]; R n ), the Banach space of all vector continuous functions x(t) from [0, T ] into R n with the norm where |·| is the norm in the space R n .
We consider the linear space Now we study the problem (1)- (3). We give first the definition of solution of the problem(1)-(3).
is given by Proof. Assume that x(t) is a solution of boundary value problem (4)- (6). Then integrating equation (4) for t ∈ (0, t j+1 ), we obtain Using the above formula and condition (5), we can write In order for (7) to satisfy the condition (6), we get Thus, we obtain Differential equations with three-point boundary conditions 25 So putting (9) in (7), we get Now consider t ∈ (t j , t j+1 ], t j+1 < τ . Then we can rewrite equality (10) as follows: Grouping the terms and then simplifying, we get Let us define the new function as follows: Using this function in (10), we obtain the following result For the case t ∈ (t j , t j+1 ], t j > τ we can write equality (10) as follows Here we introduce the new function Thus, for each t ∈ (t j , t j+1 ], we have Differential equations with three-point boundary conditions 27 Consequently, we deduce that the solution of boundary value problem (4)-(6) is in the form The proof is completed.

., p. Then the function x(t) is a solution of impulsive boundary value problem (1)-(3) if and only if x(t) is a solution of the impulsive integral equation
for t ∈ (t i , t i+1 ], i = 0, 1, ..., p. Proof. Let x(t) be a solution of boundary-value problem (1)-(3). Analogously, as in Lemma 3.2 we can prove that the impulsive integral equation (13) is also a solution of boundary value problem (1)

Main results
Now we present the existence uniqueness result for problem (1)-(3). Our considerations are based on the Banach fixed point theorem.
then the boundary value problem (1) be defined as for t ∈ (t i , t i+1 ], i = 0, 1, ..., p. We will show that F is a contraction. Consider x, y ∈ P C([0, T ], R n ). Then for each t ∈ (t i , t i+1 ], l k x − y P C . showing that F is a contraction, and thus it has a unique fixed point. So, the operator F is a solution of problem (1)  Proof. We will show that under the assumptions of the theorem, the operator F has a fixed point. The proof will be given in several steps.
Step 1. Here we prove that F is continuous. Let {x n } be a sequence such that x n → x in P C([0, T ], R n ). Then for any t ∈ (t i , t i+1 ], i = 0, 1, ..., p, Since f and I k , k = 1, 2, ..., p are continuous functions, we have as n →∝. So, F is continuous.
Step 2. In this step, we prove that the operator F maps bounded sets in It is enough to show for any η > 0, there exists a positive constant l such that for any {x ∈ B η = P C([0, T ], R n ) : x ≤ η} , we have F (x(·)) ≤ l. Applying triangle inequality, assumptions (H4) and (H5), for t ∈ (t i , t i+1 ] we get So, Step 3. The purpose of this step is to prove that the operator F maps bounded sets into equicontinuous sets of P C([0, T ], R n ).
We prove Case 3 in the similar way as the Case 1. So, The right-hand side of the above inequalities for all three cases tends to zero as ξ 1 → ξ 2 . As a consequence of Steps 1, 2 and 3 together with the Arzela-Ascoli theorem, we can conclude that the operator F : P C([0, T ], R n ) → P C([0, T ], R n ) is completely continuous.
Step 4. Now it remains to show that the set is bounded. Assume that x = λ(F x) for some 0 < λ < 1. Thus, for any t ∈ (t i , t i+1 ], i = 0, 1, ..., p we have This fact in combination with (H4) and (H5) shows that for any t ∈ [0, T ] we have So, This shows that the set ∆ is bounded and F has a fixed point by Schaefer's fixed point theorem, that is a solution of (1)-(3). Theorem 4.2 is proved.

Examples
Example 5.1. Let us consider the following there-point boundary value problem for a system of impulsive differential equations with three-point boundary conditions: and It is clear that Here S ≤ 1, 5, M = 0.2, l = 0.1. Thus, the conditions (H1) and (H2) hold with M = 0.2, l 1 = 0.1. We can easily see that condition (14) is satisfied:

Conclusion
In this research, the existence and uniqueness of solutions for nonlinear impulsive differential equations with three-point boundary conditions have been studied by using some well-known fixed point theorems. The technique used in this research can be applied to the similar problems for impulsive differential equations subject to multi-point nonlocal conditions where E is a unit matrix, B j ∈ R n ×n are given matrices and J j=1 B j < 1, 0 < λ 1 < ... < λ j ≤ T.