Nonlinear Boundary Value Problem of First-Order Impulsive Functional Differential Equations

Abstract

This paper investigates the nonlinear boundary value problem for a class of first-order impulsive functional differential equations. By establishing a comparison result and utilizing the method of upper and lower solutions, some criteria on the existence of extremal solutions as well as the unique solution are obtained. Examples are discussed to illustrate the validity of the obtained results.

1. Introduction

It is now realized that the theory of impulsive differential equations provides a general framework for mathematical modelling of many real world phenomena. In particular, it serves as an adequate mathematical tool for studying evolution processes that are subjected to abrupt changes in their states. Some typical physical systems that exhibit impulsive behaviour include the action of a pendulum clock, mechanical systems subject to impacts, the maintenance of a species through periodic stocking or harvesting, the thrust impulse maneuver of a spacecraft, and the function of the heart. For an introduction to the theory of impulsive differential equations, refer to [1].

It is also known that the method of upper and lower solutions coupled with the monotone iterative technique is a powerful tool for obtaining existence results of nonlinear differential equations [2]. There are numerous papers devoted to the applications of this method to nonlinear differential equations in the literature, see [3–9] and references therein. The existence of extremal solutions of impulsive differential equations is considered in papers [3–11]. However, only a few papers have implemented the technique in nonlinear boundary value problem of impulsive differential equations [5, 12]. In this paper, we will investigate nonlinear boundary value problem of a class of first-order impulsive functional differential equations. Such equations include the retarded impulsive differential equations as special cases [5, 12–14].

The rest of this paper is organized as follows. In Section 2, we establish a new comparison principle and discuss the existence and uniqueness of the solution for first order impulsive functional differential equations with linear boundary condition. We then obtain existence results for extremal solutions and unique solution in Section 3 by using the method of upper and lower solutions coupled with monotone iterative technique. To illustrate the obtained results, two examples are discussed in Section 4.

2. Preliminaries

Let , , with . We define that is continuous for any ; and exist and , is continuously differentiable for any ; , exist and . It is clear that and are Banach spaces with respective norms

(2.1)

Let us consider the following nonlinear boundary value problem (NBVP):

(2.2)

where is continuous in the second and the third variables, and for fixed , , , , and is continuous.

A function is called a solutions of NBVP (2.2) if it satisfies (2.2).

Remark 2.1.

• (i) If and the impulses depend only on , the equation of NBVP (2.2) reduces to the simpler case of impulsive differential equations:

  

  (2.3)

which have been studied in many papers. In some situation, the impulse depends also on some other parameters (e.g., the control of the amount of drug ingested by a patient at certain moments in the model for drug distribution [1, 3]).

• (ii) If , where , the equation of NBVP (2.2) can be regarded as retarded differential equation which has been considered in [5, 12–14].

We will need the following lemma.

Lemma 2.2 (see [1]).

Asumme that

the sequence satisfies with ,

is left continous at for ,

for ,,

(2.4)

where , and are real constants.

Then

(2.5)

In order to establish a comparison result and some lemmas, we will make the following assumptions on the function .

(H1) There exists a constant such that

(2.6)

(H2) The function satisfies Lipschitz condition, that is, there exists a such that

(2.7)

Inspired by the ideas in [5, 6], we shall establish the following comparison result.

Theorem 2.3.

Let such that

(2.8)

where , , , , and .

Suppose in addition that condition (H1) holds and

(2.9)

then .

Proof.

For simplicity, we let , . Set , then we have

(2.10)

Obviously, implies .

To show , we suppose, on the contrary, that for some . It is enough to consider the following cases.

(i)there exists a , such that , and for all ;

(ii)there exist , such that , .

Casedi.

By (2.10), we have for and , , hence is nonincreasing in , that is, . If , then , which is a contradiction. If , then which implies . But from (2.10), we get for . Hence, . It is again a contradiction.

Casedii.

Let , then . For some , there exists such that or . We only consider , as for the case , the proof is similar.

From (2.10) and condition (H1), we get

(2.11)

Consider the inequalities

(2.12)

By Lemma 2.2, we have

(2.13)

that is

(2.14)

First, we assume that . Let in (2.14), then

(2.15)

Noting that , we have

(2.16)

Hence

(2.17)

which is a contradiction.

Next, we assume that . By Lemma 2.2 and (2.10), we have

(2.18)

then

(2.19)

Setting in (2.14), we have

(2.20)

with (2.19), we obtain that

(2.21)

that is,

(2.22)

Therefore,

(2.23)

which is a contradiction. The proof of Theorem 2.3 is complete.

The following corollary is an easy consequence of Theorem 2.3.

Corollary 2.4.

Assume that there exist , , , for such that satisfies (2.8) with and

(2.24)

then , for .

Remark 2.5.

Setting , Corollary 2.4 reduces to the Theorem 2.3 of Li and Shen [6]. Therefore, Theorem 2.3 and Corollary 2.4 develops and generalizes the result in [6].

Remark 2.6.

We show some examples of function satisfying (H1).

(i), where , satisfies (H1) with ,

(2.25)

(ii), satisfies (H1) with ,

(2.26)

Consider the linear boundary value problem (LBVP)

(2.27)

where , , , , and .

By direct computation, we have the following result.

Lemma 2.7.

is a solution of LBVP (2.27) if and only if is a solution of the impulsive integral equation

(2.28)

where , , and

(2.29)

Lemma 2.8.

Let (H2) hold. Suppose further

(2.30)

where , , , then LBVP (2.27) has a unique solution.

By Lemma 2.7 and Banach fixed point theorem, the proof of Lemma 2.8 is apparent, so we omit the details.

3. Main Results

In this section, we use monotone iterative technique to obtain the existence results of extremal solutions and the unique solution of NBVP (2.2). We shall need the following definition.

Definition 3.1.

A function is said to be a lower solution of NBVP (2.2) if it satisfies

(3.1)

Analogously, is an upper solution of NBVP (2.2) if

(3.2)

For convenience, let us list the following conditions.

(H3) There exist constants , such that

(3.3)

wherever .

(H4) There exist constants for such that

(3.4)

wherever .

(H5) The function satisfies

(3.5)

(H6) There exist constants , with such that

(3.6)

wherever , and .

Let . Now we are in the position to establish the main results of this paper.

Theorem 3.2.

Let ()–() and inequalities (2.9) and (2.30) hold. Assume further that there exist lower and upper solutions and of NBVP (2.2), respectively, such that on . Then there exist monotone sequences with , such that , uniformly on . Moreover, , are minimal and maximal solutions of NBVP (2.2) in , respectively.

Proof.

For any , consider LVBP (2.27) with

(3.7)

By Lemma 2.8, we know that LBVP (2.27) has a unique solution . Define an operator by , then the operator has the following properties:

(a),

(b), if

To prove (a), let and .

(3.8)

By Theorem 2.3, we get for , that is, . Similarly, we can show that .

To prove (b), set , where and . Using (H3), (H4) and (H6), we get

(3.9)

By Theorem 2.3, we get for , that is, , then (b) is proved.

Let and for By the properties (a) and (b), we have

(3.10)

By the definition of operator , we have that and are uniformly bounded in . Thus and are uniformly bounded and equicontinuous in . By Arzela-Ascoli Theorem and (3.10), we know that there exist , in such that

(3.11)

Moreover, , are solutions of NBVP (2.2) in .

To prove that , are extremal solutions of NBVP (2.2), let be any solution of NBVP (2.2), that is,

(3.12)

By Theorem 2.3 and Induction, we get with and which implies that , that is, and are minimal and maximal solution of NBVP (2.2) in , respectively. The proof is complete.

Theorem 3.3.

Let the assumptions of Theorem 3.2 hold and assume the following.

(H7) There exist constants , such that

(3.13)

where .

(H8) There exist constants , such that

(3.14)

where .

(H9) There exist constants , with such that

(3.15)

whenever , and .

Then NBVP (2.2) has a unique solution in .

Proof.

By Theorem 3.2, we know that there exist , which are minimal and maximal solutions of NBVP (2.2) with .

Let . Using (H7), (H8), and (H9), we get

(3.16)

By Theorem 2.3, we have that , , that is, . Hence , this completes the proof.

4. Examples

To illustrate our main results, we shall discuss in this section some examples.

Example 4.1.

Consider the problem

(4.1)

where , , .

Let

(4.2)

Setting and , it is easy to verify that is a lower solution, and is an upper solution with .

For , and , we have

(4.3)

Setting , , , , and , , then conditions (H1)–(H6) are all satisfied:

(4.4)

then inequalities (2.9) and (2.30) are satisfied. By Theorem 3.2, problem (4.1) has extremal solutions .

Example 4.2.

Consider the problem

(4.5)

where , , .

Let

(4.6)

Setting and , then is a lower solution, and is an upper solution with .

For , and , we have , , and . Setting , , , , and , , then conditions (H1)–(H6) are all satisfied:

(4.7)

then inequalities (2.24) and (2.30) are satisfied. By Corollary 2.4 and Theorem 3.2, problem (4.5) has extremal solutions .

Moreover, let , , and , . It is easy to see that conditions (H7)–(H9) are satisfied. By Corollary 2.4 and Theorem 3.3, problem (4.5) has an unique solution in .