Abstract
In this article we introduce a new definition of impulsive conditions for boundary value problems of first order impulsive integrodifferential equations with multipoint 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.
Keywords:
impulsive integrodifferential equations; multipoint boundary value problem; lower and upper solutions; monotone iterative technique.1 Introduction
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 [13]). In this article, we consider the following boundary value problem for first order impulsive integrodifferential equations (BVP):
where f ∈ C(J × R^{3}, R), 0 = t_{0 }< t_{1 }< t_{2 }< · · · < 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 [414]). 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 [1520]). However, the discussion of multipoint 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 shortterm 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_{k1 }on (t_{k1}, t_{k1 }+ σ_{k1}], respectively.
The aim of the present article is to discuss the new impulsive multipoint condition
for . The new jump conditions mean that a sudden change of the state at impulse point t_{k }depends on the multipoint of past states on (t_{k1}, 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.
2 Preliminaries
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 and exist and , k = 1, 2, ..., m}, PC^{1}(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 integrodifferential 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
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^{MT}u(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. □
3 Main results
In this section, we are in a position to prove our main results concerning the existence criteria for solutions of BVP (1.1).
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
(H_{3}) 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.
(H_{4}) 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 β_{0 }(t) ≤ x(t) ≤ α_{0}(t) on J.
Proof. For any σ ∈ [β_{0}, α_{0}], 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 [β_{0}, α_{0}] to 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
and
By Lemma 2.3, we get that φ(t) ≤ 0 for all t ∈ J , i.e., β_{0 }≤ Aβ_{0}. Similarly, we can prove that Aα_{0 }≤ α_{0}.
To prove (ii), let u_{l }= Aσ_{l}, u_{2 }= Aσ_{2}, where σ_{l }≤ σ_{2 }on J and σ_{l}, σ_{2 }∈ [β_{0}, α_{0}]. Set φ = u_{l } u_{2}. Then for t ∈ J^{ }and by (H_{2}), 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 }≤ Aσ_{2}.
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 ∈ [β_{0}, α_{0}] 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 β_{0 }≤ x ≤ α_{0 }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. □
4 An example
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, α_{0 }= 0, are lower and upper solutions for (4.1), respectively, and β_{0 }≤ α_{0}.
Let
Then,
and
Therefore, (4.1) satisfies all conditions of Theorem 3.1. So, BVP (4.1) has minimal and maximal solutions in the segment [β_{0}, α_{0}].
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed equally in this article. They read and approved the final manuscript.
Acknowledgements
This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
References

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

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

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

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

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

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

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

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

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

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

Wang, X, Zhang, J: Impulsive antiperiodic boundary value problem of firstorder integrodifferential equations. J Comput Appl Math. 234, 3261–3267 (2010). Publisher Full Text

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

Nieto, JJ, RodríguezLópez, 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

Nieto, JJ, RodríguezLópez, R: Periodic boundary value problem for nonLipschitzian impulsive functional differential equations. J Math Anal Appl. 318, 593–610 (2006). Publisher Full Text

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

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

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

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

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

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

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

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

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