Multi-point boundary value problem for first order impulsive integro-differential equations with multi-point jump conditions

In this article we introduce a new definition of impulsive conditions for boundary value problems of first order impulsive integro-differential equations with multi-point boundary conditions. By using the method of lower and upper solutions in reversed order coupled with the monotone iterative technique, we obtain the extremal solutions of the boundary value problem. An example is also discussed to illustrate our results.

Mathematics Subject Classification 2010: 34B15; 34B37.

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 [1-3]). In this article, we consider the following boundary value problem for first order impulsive integro-differential equations (BVP):

where f ∈ C(J × R³, R), 0 = t₀ < t₁ < t₂ < · · · < t_m < t_m+1 = T,

k ∈ C(D, R⁺), D = {(t, s) ∈ J × J: t ≥ s}, h ∈ C(J × J, R⁺). I_k ∈ C(R, R), , 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 [4-14]). 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 [15-20]). 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

It should be noticed that the terms and 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

for . The new jump conditions mean that a sudden change of the state at impulse point t_k depends on the multi-point of past states on (t_k-1, t_k]. We note that if c_k = 1, and , 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 \ {t₁, t₂, ..., t_m}. PC(J, R) = {x: J → R; x(t) is continuous everywhere except for some t_k at which and exist and , k = 1, 2, ..., m}, PC¹(J, R) = {x ∈ PC(J, R); x'(t) is continuous everywhere except for some t_k at which and exist and }. Let E = PC(J, R) and , then E and are Banach spaces with the nomes ||x||_E = sup_t∈J |x(t)| and , respectively. A function is called a solution of BVP (1.1) if it satisfies (1.1).

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

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

where , 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):

where M > 0, H, K ≥ 0, 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. is a solution of (2.1) if and only if x ∈ E is a solution of the impulsive integral equation

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

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

Putting t = T in (2.3), we have

From , we obtain

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

Differentiating (2.2) for t ≠ t_k, we obtain

It is easy to see that

Since G(0, s) = G(T, s), then . This completes the proof.   □

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

Then BVP (2.1) has a unique solution.

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

where G(t, s) is defined as in Lemma 2.1. Since , we have for any x, y ∈ E, that

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

Lemma 2.3. Assume that satisfies

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

where . Then, x(t) ≤ 0 for all t ∈ J.

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

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

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

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

and u(0) ≥ e^MT u(T). In consequence,

can be obtained.

If t_* = 0, then

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

If t_* > 0, we obtain from (2.11),

This joint to (2.12) yields

Therefore,

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 , we denote

and we write β₀ ≤ α₀ if β₀(t) ≤ α₀(t) for all t ∈ J.

Theorem 3.1. Let the following conditions hold.

(H₁) The functions α₀ and β₀ are lower and upper solutions of BVP (1.1), respectively, such that β₀(t) ≤ α₀(t) on J.

(H₂) The function f ∈ C(J × R³, R) satisfies

for .

(H₃) The function I_k ∈ C(R, R) satisfies

whenever , l = 1, 2, ..., c_k , c_k ∈ N = {1, 2, ...}, L_k ≥ 0, k = 1, 2, ..., m.

(H₄) Inequalities (2.6) and (2.9) hold.

Then there exist monotone sequences 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

on J, where x is any solution of BVP (1.1) such that β₀ (t) ≤ x(t) ≤ α₀(t) on J.

Proof. For any σ ∈ [β₀, α₀], we consider BVP (2.1) with

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 [β₀, α₀] to and A has the following properties.

(i) β₀ ≤ Aβ₀, Aα₀ ≤ α₀;

(ii) For any σ_l, σ₂ ∈ [β₀, α₀], σ_l ≤ σ₂ implies Aσ_l ≤ Aσ₂.

To prove (i), set φ = β₀ - β₁, where β₁ = Aβ₀. Then from (H_l) and (2.1) for t ∈ J^-, we have

and

By Lemma 2.3, we get that φ(t) ≤ 0 for all t ∈ J , i.e., β₀ ≤ Aβ₀. Similarly, we can prove that Aα₀ ≤ α₀.

To prove (ii), let u_l = Aσ_l, u₂ = Aσ₂, where σ_l ≤ σ₂ on J and σ_l, σ₂ ∈ [β₀, α₀]. Set φ = u_l - u₂. Then for t ∈ J^- and by (H₂), we obtain

and by (H3);

It is easy to see that

Then by using Lemma 2.3, we have φ(t) ≤ 0, which implies that Aσ_l ≤ Aσ₂.

Now, we define the sequences {α_n}, {β_n} such that α_n+l = Aα_n and β_n+l = Aβ_n. From (i) and (ii) the sequence {α_n}, {β_n} satisfy the inequality

for all n ∈ N. Obviously, each α_n, β_n (n = 1, 2, ...) satisfy

and

Therefore, there exist x_* and x*, such that lim_n→∞ β_n = x_* and lim_n→∞ α_n = x* uniformly on J. Clearly, x_*, x* are solutions of BVP (1.1).

Finally, we are going to prove that x_*, x* are minimal and maximal solutions of BVP (1.1). Assume that x(t) is any solution of BVP (1.1) such that x ∈ [β₀, α₀] and that there exists a positive integer n such that β_n(t) ≤ x(t) ≤ α_n (t) on J. Let φ = β_n+1 - x, then for t ∈ J^-,

and

Then by using Lemma 2.3, we have φ(t) ≤ 0, which implies that β_n+1 ≤ x on J. Similarly we obtain x ≤ α_n+1 on J. Since β₀ ≤ x ≤ α₀ on J , by induction we get β_n ≤ × ≤ α_n on J for every n. Therefore, x_* (t) ≤ x(t) ≤ x*(t) on J by taking n → ∞. The proof is complete.   □

In this section, in order to illustrate our results, we consider an example.

Example 4.1. Consider the BVP

where k(t, s) = h(t, s) = ts, m = 1, .

Obviously, α₀ = 0, are lower and upper solutions for (4.1), respectively, and β₀ ≤ α₀.

Let

Then,

where It is easy to see that

whenever , l = 1, ..., 5.

Taking , it follows that

and

Therefore, (4.1) satisfies all conditions of Theorem 3.1. So, BVP (4.1) has minimal and maximal solutions in the segment [β₀, α₀].

The authors declare that they have no competing interests.

All authors contributed equally in this article. They read and approved the final manuscript.

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

1. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

2. Bainov, DD, Simeonov, PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific & Technical, New York (1993). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

3. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995)

4. Ding, W, Mi, J, Han, M: Periodic boundary value problems for the first order impulsive functional differential equations. Appl Math Comput. 165, 433–446 (2005). Publisher Full Text

5. Zhang, F, Li, M, Yan, J: Nonhomogeneous boundary value problem for first-order impulsive differential equations with delay. Comput Math Appl. 51, 927–936 (2006). Publisher Full Text

6. Chen, L, Sun, J: Nonlinear boundary problem of first order impulsive integro-differential equations. J Comput Appl Math. 202, 392–401 (2007). Publisher Full Text

7. Liang, R, Shen, J: Periodic boundary value problem for the first order impulsive functional differential equations. J Comput Appl Math. 202, 498–510 (2007). Publisher Full Text

8. Ding, W, Xing, Y, Han, M: Anti-periodic boundary value problems for first order impulsive functional differential equations. Appl Math Comput. 186, 45–53 (2007). Publisher Full Text

9. Yang, X, Shen, J: Nonlinear boundary value problems for first order impulsive functional differential equations. Appl Math Comput. 189, 1943–1952 (2007). Publisher Full Text

10. Luo, Z, Jing, Z: Periodic boundary value problem for first-order impulsive functional differential equations. Comput Math Appl. 55, 2094–2107 (2008). Publisher Full Text

11. Wang, X, Zhang, J: Impulsive anti-periodic boundary value problem of first-order integro-differential equations. J Comput Appl Math. 234, 3261–3267 (2010). Publisher Full Text

12. Song, G, Zhao, Y, Sun, X: Integral boundary value problems for first order impulsive integro-differential equations of mixed type. J Comput Appl Math. 235, 2928–2935 (2011). Publisher Full Text

13. Nieto, JJ, Rodríguez-López, R: Existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions. Comput Appl Math. 40, 433–442 (2000). Publisher Full Text

14. Nieto, JJ, Rodríguez-López, R: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations. J Math Anal Appl. 318, 593–610 (2006). Publisher Full Text

15. He, Z, Yu, J: Periodic boundary value problem for first-order impulsive functional differential equations. J Comput Appl Math. 138, 205–217 (2002). Publisher Full Text

16. He, Z, Yu, J: Periodic boundary value problem for first-order impulsive ordinary differential equations. J Math Anal Appl. 272, 67–78 (2002). Publisher Full Text

17. Chen, L, Sun, J: Nonlinear boundary value problem of first order impulsive functional differential equations. J Math Anal Appl. 318, 726–741 (2006). Publisher Full Text

18. Chen, L, Sun, J: Nonlinear boundary value problem for first order impulsive integro-differential equations of mixed type. J Math Anal Appl. 325, 830–842 (2007). Publisher Full Text

19. Wang, G, Zhang, L, Song, G: Extremal solutions for the first order impulsive functional differential equations with upper and lower solutions in reversed order. J Comput Appl Math. 235, 325–333 (2010). Publisher Full Text

20. Zhang, L: Boundary value problem for first order impulsive functional integro-differential equations. J Comput Appl Math. 235, 2442–2450 (2011). Publisher Full Text

21. Zhang, Y, Zhang, F: Multi-point boundary value problem of first order impulsive functional differential equations. J Appl Math Comput. 31, 267–278 (2009). Publisher Full Text

22. Tariboon, J: Boundary value problems for first order functional differential equations with impulsive integral conditions. J Comput Appl Math. 234, 2411–2419 (2010). Publisher Full Text

23. Liu, Z, Han, J, Fang, L: Integral boundary value problems for first order integro-differential equations with impulsive integral conditions. Comput Math Appl. 61, 3035–3043 (2011). Publisher Full Text