Impulsive differential equations describe processes which have a sudden change of their state at certain moments. Impulse effects are important in many real world applications, such as physics, medicine, biology, control theory, population dynamics, etc. (see, for example 1 2 3 ). In this article, we consider the following boundary value problem for first order impulsive integro-differential equations (BVP):

x ′ t = f t , x t , F x t , S x t , t ∈ J = 0 , T , t ≠ t k , Δ x t k = I k ∑ l = 1 c k ρ l k x η l k , k = 1 , 2 , … m , x 0 + μ ∑ k = 1 m ∑ l = 1 c k τ l k x η l k = x T ,

where f ∈ C(J × R ³, R), 0 = t ₀ < t ₁ < t ₂ < · · · < t _m < t _m+1 = T,

F x t = ∫ 0 t k t , s x s d s , S x t = ∫ 0 T h t , s x s d s ,

k ∈ C(D, R ⁺), D = {(t, s) ∈ J × J: t ≥ s}, h ∈ C(J × J, R ⁺). I_k ∈ C(R, R), Δ x t k = x t k + - x t k - , t k - 1 < η 1 k < η 2 k < … < η c k k ≤ t k , τ l k , ρ l k ≥ 0 , l = 1, 2, ..., c_k , c_k ∈ N = {1, 2, ...}, k = 1, 2, ..., m, μ ≥ 0.

The monotone iterative technique coupled with the method of lower and upper solutions is a powerful method used to approximate solutions of several nonlinear problems (see 4 5 6 7 8 9 10 11 12 13 14 ). Boundary value problems for first order impulsive functional differential equations with lower and upper solutions in reversed order have been widely discussed in recent years (see 15 16 17 18 19 20 ). However, the discussion of multi-point boundary value problems for first order impulsive functional differential equations is very limited (see 21 ). In all articles concerned with applications of the monotone iterative technique to impulsive problems, the authors have assumed that Δx(t_k ) = I_k (x(t_k )), that is a short-term rapid change of the state at impulse point t_k depends on the left side of the limit of x(t_k ).

Recently, Tariboon 22 and Liu et al. 23 studied some types of impulsive boundary value problems with the impulsive integral conditions

Δ x t k = I k ∫ t k - τ k t k x s d s - ∫ t k - 1 t k - 1 + σ k - 1 x s d s , k = 1 , 2 , … , m .

It should be noticed that the terms ∫ t k - τ k t k x s d s and ∫ t k - 1 t k - 1 + σ k - 1 x s d s of impulsive condition (1.2) illustrate the past memory state on [t_k - τ_k , t_k ] before impulse points t_k and the history effects after the past impulse points t _k-1 on (t _k-1, t _k-1 + σ _k-1], respectively.

The aim of the present article is to discuss the new impulsive multi-point condition

Δ x t k = I k ∑ l = 1 c k ρ l k x η l k = I k ρ 1 k x η 1 k + ⋯ + ρ l k x η l k + ⋯ + ρ c k k x η c k k ,

for t k - 1 < η 1 k < η 2 k < ⋯ < η c k k ≤ t k , k = 1 , 2 , … , m . The new jump conditions mean that a sudden change of the state at impulse point t_k depends on the multi-point η l k l = 1 , 2 , … , c k of past states on (t _k-1, t_k ]. We note that if c_k = 1, η c k k = t k and ρ c k k = 1 , then the impulsive condition (1.3) is reduced to the simple impulsive condition Δx(t_k ) = I_k (x(t_k )).

Firstly, we introduce the definitions of lower and upper solutions and formulate some lemmas which are used in our discussion. In the main results, we obtain the existence of extreme solutions for BVP (1.1) by using the method of lower and upper solutions in reversed order and the monotone iterative technique. Finally, we give an example to illustrate the obtained results.

Let J^- = J \ {t ₁, t ₂, ..., t_m }. PC(J, R) = {x: J → R; x(t) is continuous everywhere except for some t_k at which x t k - and x t k + exist and x t k - = x t k , k = 1, 2, ..., m}, PC ¹(J, R) = {x ∈ PC(J, R); x'(t) is continuous everywhere except for some t_k at which x ′ t k + and x ′ t k - exist and x ′ t k - = x ′ t k }. Let E = PC(J, R) and F = P C 1 J , R , then E and F   are Banach spaces with the nomes ||x||_E = sup _t∈J |x(t)| and x F = max x E , x ′ E , respectively. A function x ∈ F is called a solution of BVP (1.1) if it satisfies (1.1).

Definition 2.1. A function α 0 ∈ F is called a lower solution of BVP (1.1) if:

α ′ 0 t ≤ f t , α 0 t , F α 0 t , S α 0 t , t ∈ J - , Δ α 0 t k ≤ I k ∑ l = 1 c k ρ l k α 0 η l k , k = 1 , 2 , … , m , α 0 0 + μ ∑ k = 1 m ∑ l = 1 c k τ l k α 0 η l k ≤ α 0 T .

Analogously, a function β 0 ∈ F is called an upper solution of BVP (1.1) if:

β ′ 0 t ≥ f t , β 0 t , F β 0 t , S β 0 t , t ∈ J - , Δ β 0 t k ≥ I k ∑ l = 1 c k ρ l k β 0 η l k , k = 1 , 2 , … , m , β 0 0 + μ ∑ k = 1 m ∑ l = 1 c k τ l k β 0 η l k ≥ β 0 T ,

where t k - 1 < η l k ≤ t k , ρ l k , τ l k ≥ 0 , l = 1, 2, ..., c_k, c_k ∈ N = {1, 2, ...}, k = 1, 2, ..., m and μ ≥ 0.

Let us consider the following boundary value problem of a linear impulsive integro-differential equation (BVP):

x ′ t - M x t = H F x t + K S x t + v t , t ∈ J - , Δ x t k = L k ∑ l = 1 c k ρ l k x η l k + I k ∑ l = 1 c k ρ l k σ η l k - L k ∑ l = 1 c k ρ l k σ η l k , k = 1 , 2 , … , m , x 0 + μ ∑ k = 1 m ∑ l = 1 c k τ l k σ η l k = x T ,

where M > 0, H, K ≥ 0, L_k ≥ 0, t k - 1 < η l k ≤ t k , τ l k , ρ l k ≥ 0 , l = 1, 2, ..., c_k, c_k ∈ N = {1, 2, ...}, k = 1, 2,..., m are constants and v(t), σ(t) ∈ E.

Lemma 2.1. x ∈ F is a solution of (2.1) if and only if x ∈ E is a solution of the impulsive integral equation

x t = μ e M t e M T - 1 ∑ k = 1 m ∑ l = 1 c k τ l k σ η l k - ∫ 0 T G t , s P s d s - ∑ k = 1 m G t , t k L k ∑ l = 1 c k ρ l k x η l k + I k ∑ l = 1 c k ρ l k σ η l k - L k ∑ l = 1 c k ρ l k σ η l k , t ∈ J ,

where P(t) = H(Fx)(t) + K(Sx)(t) + v(t) and

G t , s = e M t - s e M T - 1 , 0 ≤ s < t ≤ T , e M T + t - s e M T - 1 , 0 ≤ t ≤ s ≤ T .

Proof. Assume that x(t) is a solution of BVP (2.1). By using the variation of parameters formula, we get

x t = x 0 e M t + ∫ 0 t e M t - s P s d s + ∑ 0 < t k < t e M t - t k L k ∑ l = 1 c k ρ l k x η l k + I k ∑ l = 1 c k ρ l k σ η l k - L k ∑ l = 1 c k ρ l k σ η l k .

Putting t = T in (2.3), we have

x T = x 0 e M T + ∫ 0 T e M T - s P s d s + ∑ k = 1 m e M T - t k L k ∑ l = 1 c k ρ l k x η l k + I k ∑ l = 1 c k ρ l k σ η l k - L k ∑ l = 1 c k ρ l k σ η l k .

From x 0 + μ ∑ k = 1 m ∑ l = 1 c k τ l k σ η l k = x T , we obtain

x 0 = - 1 e M T - 1 - μ ∑ k = 1 m ∑ l = 1 c k τ l k σ η l k + ∫ 0 T e M T - s P s d s + ∑ k = 1 m e M T - t k L k ∑ l = 1 c k ρ l k x η l k + I k ∑ l = 1 c k ρ l k σ η l k - L k ∑ l = 1 c k ρ l k σ η l k .

Substituting (2.5) into (2.3), we see that x ∈ E satisfies (2.2). Hence, x(t) is also the solution of (2.2).

Conversely, we assume that x(t) is a solution of (2.2). By computing directly, we have

G t ′ t , s = M e M t - s e M T - 1 , 0 ≤ s < t ≤ T , M e M T + t - s e M T - 1 , 0 ≤ t ≤ s ≤ T , = M G t , s .

Differentiating (2.2) for t ≠ t_k , we obtain

x ′ t = M x t + H F x t + K S x t + v t .

It is easy to see that

Δ x t k = L k ∑ l = 1 c k ρ l k x η l k + I k ∑ l = 1 c k ρ l k σ η l k - L k ∑ l = 1 c k ρ l k σ η l k .

Since G(0, s) = G(T, s), then x 0 + μ ∑ k = 1 m ∑ l = 1 c k τ l k σ η l k = x T . This completes the proof.   □

Lemma 2.2. Assume that M > 0, H, K ≥ 0, L_k ≥ 0, t k - 1 < η l k ≤ t k , ρ l k ≥ 0 , l = 1, 2, ..., c_k , c_k ∈ N = {1, 2, ...}, k = 1, 2, ..., m, and the following inequality holds:

e M T e M T - 1 ∫ 0 T H ∫ 0 s k s , r d r + K ∫ 0 T h s , r d r d s + e M T e M T - 1 ∑ k = 1 m L k ∑ l = 1 c k ρ l k < 1 .

Then BVP (2.1) has a unique solution.

Proof. For any x ∈ E, we define an operator A by

A x t = μ e M t e M T - 1 ∑ k = 1 m ∑ l = 1 c k τ l k σ η l k - ∫ 0 T G t , s H F x s + K S x s + v s d s - ∑ k = 1 m G t , t k L k ∑ l = 1 c k ρ l k x η l k + I k ∑ l = 1 c k ρ l k σ η l k - L k ∑ l = 1 c k ρ l k σ η l k , t ∈ J ,

where G(t, s) is defined as in Lemma 2.1. Since max t ∈ [ 0 , T ] G t , s = e M T e M T - 1 , we have for any x, y ∈ E, that

A x - A y E = - ∫ 0 T G t , s H ∫ 0 s k s , r x r - y r d r + K ∫ 0 T h s , r x r - y r d r d s - ∑ k = 1 m G t , t k L k ∑ l = 1 c k ρ l k x η l k - y η l k ≤ e M T e M T - 1 ∫ 0 T H ∫ 0 s k s , r d r + K ∫ 0 T h s , r d r d s + e M T e M T - 1 ∑ k = 1 m L k ∑ l = 1 c k ρ l k x - y E .

From (2.6) and the Banach fixed point theorem, A has a unique fixed point x ¯ ∈ E . By Lemma 2.1, x ¯ is also the unique solution of (2.1).   □

Lemma 2.3. Assume that x ∈ F satisfies

x ′ t ≥ M x t + H F x t + K S x t , t ∈ J - , Δ x t k ≥ L k ∑ l = 1 c k ρ l k x η l k , k = 1 , 2 , … , m , x 0 ≥ x T ,

where M > 0, H, K ≥ 0, L_k ≥ 0, t k - 1 < η l k ≤ t k , ρ l k ≥ 0 , l = 1, 2, ..., c_k , c_k ∈ N = {1, 2, ...}, k = 1, 2, ..., m. In addition assume that

e M T ∫ 0 T q s d s + ∑ k = 1 m L k ∑ l = 1 c k ρ l k e - M t k - η l k ≤ 1 ,

where q t = H ∫ 0 t k t , s e - M t - s d s + K ∫ 0 T h t , s e - M t - s d s . Then, x(t) ≤ 0 for all t ∈ J.

Proof. Set u(t) = x(t)e^-Mt for t ∈ J , then we have

u ′ t ≥ H ∫ 0 t k t , s e - M t - s u s d s + K ∫ 0 T h t , s e - M t - s u s d s , t ∈ J - , Δ u t k ≥ L k ∑ l = 1 c k ρ l k e - M t k - η l k u η l k , k = 1 , 2 , … , m , u 0 ≥ e M T u T .

Obviously, the function u(t) and x(t) have the same sign. Suppose, to the contrary, that u(t) > 0 for some t ∈ J. Then, there are two cases:

(i) There exists a t*∈ J , such that u(t*) > 0 and u(t) ≥ 0 for all t ∈ J.

(ii) There exists t*, t _* ∈ J, such that u(t*) > 0 and u(t _*) < 0.

Case (i): Equation (2.10) implies that u'(t) ≥ 0 for t ∈ J^- and Δu(t_k ) ≥ 0 for k = 1, 2, ..., m. This means that u(t) is nondecreasing in J. Therefore, u(T) ≥ u(t*) > 0 and u(T) ≥ u(0) ≥ u(T)e^MT , which is a contradiction.

Case (ii): Let t _* ∈ (t_i, t _i+1], i ∈ {0, 1, ..., m}, such that u(t _*) = inf {u(t): t ∈ J} < 0 and t* ∈ (t_j, t _j+1], j ∈ {0, 1, ..., m}, such that u(t*) > 0. We first claim that u(0) ≤ 0. Otherwise, if u(0) > 0, then by (2.10), we have

u t * - u 0 ≥ H ∫ 0 t * ∫ 0 s k s , r e - M s - r u r d r d s + K ∫ 0 t * ∫ 0 T h s , r e - M s - r u r d r d s + ∑ k = 1 i Δ u t k ≥ u t * ∫ 0 t * q s d s + ∑ k = 1 i L k ∑ l = 1 c k ρ l k e - M t k - η l k ≥ u t * ,

a contradiction, and so u(0) ≤ 0.

If t* < t _*, then j ≤ i. Integrating the differential inequality in (2.10) from t* to t _*, we obtain

u t * - u t * ≥ H ∫ t * t * ∫ 0 s k s , r e - M s - r u r d r d s + K ∫ t * t * ∫ 0 T h s , r e - M s - r u r d r d s + ∑ k = j + 1 i Δ u t k ≥ u t * ∫ t * t * q s d s + ∑ k = j + 1 i Δ u t k ≥ u t * ∫ t * t * q s d s + ∑ k = j + 1 i L k ∑ l = 1 c k ρ l k e - M t k - η l k u η l k ≥ u t * ∫ 0 T q s d s + ∑ k = 1 m L k ∑ l = 1 c k ρ l k e - M t k - η l k ≥ u t * ,

which is a contradiction to u(t*) > 0.

Now, assume that t _* < t*. Since 0 ≥ u(0) ≥ e^MTu(T), then u(T) ≤ 0. From (2.10), we have

u T - u t * ≥ H ∫ t * T ∫ 0 s k s , r e - M s - r u r d r d s + K ∫ t * T ∫ 0 T h s , r e - M s - r u r d r d s + ∑ k = j + 1 m Δ u t k ≥ u t * ∫ t * T q s d s + ∑ k = j + 1 m L k ∑ l = 1 c k ρ l k e - M t k - η l k ,

and u(0) ≥ e^MT u(T). In consequence,

u 0 ≥ e M T u T ≥ e M T u t * + u t * e M T ∫ t * T q s d s + ∑ k = j + 1 m L k ∑ l = 1 c k ρ l k e - M t k - η l k

can be obtained.

If t _* = 0, then

u t * ≥ e M T u t * + u t * e M T ∫ t * T q s d s + ∑ k = j + 1 m L k ∑ l = 1 c k ρ l k e - M t k - η l k ≥ e M T u t * + u t * .

This contradicts the fact that u(t*) > 0.

If t _* > 0, we obtain from (2.11),

u t * - u t * ∫ 0 t * q s d s + ∑ k = 1 i L k ∑ l = 1 c k ρ l k e - M t k - η l k ≥ u 0 .

This joint to (2.12) yields

u t * - u t * ∫ 0 t * q s d s + ∑ k = 1 i L k ∑ l = 1 c k ρ l k e - M t k - η l k ≥ e M T u t * + u t * e M T ∫ t * T q s d s + ∑ k = j + 1 m L k ∑ l = 1 c k ρ l k e - M t k - η l k .

Therefore,

u t * - e M T u t * ≥ u t * e M T ∫ t * T q s d s + ∑ k = j + 1 m L k ∑ l = 1 c k ρ l k e - M t k - η l k + u t * ∫ 0 t * q s d s + ∑ k = 1 i L k ∑ l = 1 c k ρ l k e - M t k - η l k ≥ u t * e M T ∫ t * T q s d s + ∑ k = j + 1 m L k ∑ l = 1 c k ρ l k e - M t k - η l k + u t * e M T ∫ 0 t * q s d s + ∑ k = 1 i L k ∑ l = 1 c k ρ l k e - M t k - η l k ≥ u t * e M T ∫ 0 T q s d s + ∑ k = 1 m L k ∑ l = 1 c k ρ l k e - M t k - η l k ≥ u t * .

This is a contradiction and so u(t) ≤ 0 for all t ∈ J. The proof is complete.   □

In this section, we are in a position to prove our main results concerning the existence criteria for solutions of BVP (1.1).

For β 0 , α 0 ∈ F , we denote

β 0 , α 0 = x ∈ F : β 0 t ≤ x t ≤ α 0 t , t ∈ J ,

and we write β ₀ ≤ α ₀ if β ₀(t) ≤ α ₀(t) for all t ∈ J.

Theorem 3.1. Let the following conditions hold.

(H ₁) The functions α ₀ and β ₀ are lower and upper solutions of BVP (1.1), respectively, such that β ₀(t) ≤ α ₀(t) on J.

(H ₂) The function f ∈ C(J × R ³, R) satisfies

f t , x , y , z - f t , x ¯ , y ¯ , z ¯ ≤ M x - x ¯ + H y - y ¯ + K z - z ¯ ,

for β 0 t ≤ x ¯ t ≤ x t ≤ α 0 t , F β 0 t ≤ y ¯ t ≤ y t ≤ F α 0 t , S β 0 t ≤ z ¯ t ≤ z t ≤ S α 0 t t ∈ J .

(H ₃) The function I_k ∈ C(R, R) satisfies

I k ∑ l = 1 c k ρ l k x η l k - I k ∑ l = 1 c k ρ l k y η l k ≤ L k ∑ l = 1 c k ρ l k x η l k - y η l k ,

whenever β 0 η l k ≤ y η l k ≤ x η l k ≤ α 0 η l k , l = 1, 2, ..., c_k , c_k ∈ N = {1, 2, ...}, L_k ≥ 0, k = 1, 2, ..., m.

(H ₄) Inequalities (2.6) and (2.9) hold.

Then there exist monotone sequences α n , β n ⊂ F such that lim _n→∞ α _n (t) = x*(t), lim _n→∞ β _n (t) = x _* (t) uniformly on J and x*, x _* are maximal and minimal solutions of BVP (1.1), respectively, such that

β 0 ≤ β 1 ≤ β 2 ≤ ⋯ ≤ β n ≤ x * ≤ x ≤ x * ≤ α n ≤ ⋯ ≤ α 2 ≤ α 1 ≤ α 0 ,

on J, where x is any solution of BVP (1.1) such that β ₀ (t) ≤ x(t) ≤ α ₀(t) on J.

Proof. For any σ ∈ [β ₀, α ₀], we consider BVP (2.1) with

v t = f t , σ t , F σ t , S σ t - M σ t - H F σ t - K S σ t .

By Lemma 2.2, BVP (2.1) has a unique solution x(t) for t ∈ J. We define an operator A by x = Aσ, then the operator A is an operator from [β ₀, α ₀] to F   and A has the following properties.

(i) β ₀ ≤ Aβ ₀, Aα ₀ ≤ α ₀;

(ii) For any σ _l, σ ₂ ∈ [β ₀, α ₀], σ _l ≤ σ ₂ implies Aσ _l ≤ Aσ ₂.

To prove (i), set φ = β ₀ - β ₁, where β ₁ = Aβ ₀. Then from (H _l) and (2.1) for t ∈ J ^-, we have

φ ′ t = β ′ 0 t - β ′ 1 t , ≥ f t , β 0 t , F β 0 t , S β 0 t - M β 1 t + H F β 1 t + K S β 1 t + f t , β 0 t , F β 0 t , S β 0 t - M β 0 t - H F β 0 t - K S β 0 t = M φ t + H F φ t + K S φ t ,

Δ φ t k = Δ β 0 t k - Δ β 1 t k ≥ I k ∑ l = 1 c k ρ l k β 0 η l k - L k ∑ l = 1 c k ρ l k β 1 η l k + I k ∑ l = 1 c k ρ l k β 0 η l k - L k ∑ l = 1 c k ρ l k β 0 η l k = L k ∑ l = 1 c k ρ l k φ η l k , k = 1 , 2 , … , m ,