#### Abstract

We consider positive solutions and optimal control problem for a second order impulsive differential equation with mixed monotone terms. Firstly, by using a fixed point theorem of mixed monotone operator, we study positive solutions of the boundary value problem for impulsive differential equations with mixed monotone terms, and sufficient conditions for existence and uniqueness of positive solutions will be established. Also, we study positive solutions of the initial value problem for our system. Moreover, we investigate the control problem of positive solutions to our equations, and then, we prove the existence of an optimal control and its stability. In addition, related examples will be given for illustrations.

#### 1. Introduction

Mixed monotone operators have been introduced by Guo and Lakshmikantham [1] in 1987. Recently, many authors have investigated those kinds of operators in Banach spaces and obtained a lot of interesting and important results (see [2–9]). In this work, by using a fixed point theorem of mixed monotone operator, we study the existence and uniqueness of positive solutions to the boundary value problem of impulsive differential equations with mixed monotone terms: Here, , and with on any subinterval of . A function is given on , denotes the jump of at and , where and represent the right and left limits of at , respectively. Also, is a given function in . Furthermore, is a given constant.

For convenience, we put , and .

Then, we study the existence and uniqueness of positive solutions to initial value problem as follows: where and and are given constants. Moreover, we consider the related optimal control problem (OP) of (2) as follows:

*Problem (OP).* Find an optimal control such that
Here, is a control space defined by
where is a fixed positive number and is the cost functional defined by
where is the control, function is a unique positive solution to the state problem (IP; ), and is the given desired target profiles in .

The existence and uniqueness of solutions for boundary value problem have been discussed by many authors, and the boundary value problem of impulsive differential equation is a new and important branch of the differential equation theory, which has an extensive physical, chemical, biological, and engineering background, realistic mathematical model, and so forth (see [10–14]). The theory on mixed monotone operators has attracted much attention and has been widely studied, such as Guo and Lakshmikantham [1] have applied the monotone iterative technique to discuss an initial value problem of differential equations without impulse: They obtained the existence of the coupled quasisolutions by mixed monotone sequence of coupled quasi upper and lower solutions. Zhai and Zhang [7] showed a new fixed point theorem for differential equations with mixed monotone term. Jinli and Yihai [15] considered the following problem: They used the coupled fixed point theorem for mixed monotone condensing operators to obtain the existence and uniqueness of solutions.

Also, there is a vast literature on optimal control problems (see [16–19]). For instance, with a fixed point theorem of generalized concave operator, the authors [19] have studied the optimal control problem of positive solutions to the following second order impulsive differential equation:

In this paper, we investigate impulsive differential equations with mixed monotone terms, which have variable coefficient nonlinear terms. Then, we prove the existence and uniqueness of positive solutions to (BP; ) and (IP; ). Moreover, we prove the existence of an optimal control to (OP) and its stability.

The plan of this paper is as follows. In Section 2, we recall the fundamentals of a fixed point theorem of mixed monotone operators. In Section 3, we deal with the existence and uniqueness of positive solutions to (BP; ). In Section 4, we show the existence and uniqueness of positive solutions to (IP; ). In Section 5, we prove the existence of an optimal control to (OP) and its stability. In the final Section 6, related examples on the main results are given.

##### 1.1. Notations

Throughout this paper, we use the following notations.

Let is continuous at and left continuous at exists, . Then, we can easily find that is a Banach space with the norm .

We put with the usual real Hilbert structure and denote by the norm in , for simplicity, and is a usual Sobolev space, namely, where denoted the th derivative of .

Also, and , denote positive (or nonnegative) constants only depending on their arguments.

#### 2. A Fixed Point Theorem of Mixed Monotone Operator

In this section, we recall the fundamentals of a fixed point theorem of mixed monotone operator.

Suppose that is a real Banach space which is partially ordered by a cone , that is, if and only if . If and , then we denote or . By we denote the zero element of . Recall that a nonempty closed convex set is a cone if it satisfies(i); (ii).

Putting is an interior point of , a cone is said to be solid if its interior is nonempty. Moreover, is called normal if there exists a constant such that, for all implies ; in this case is called the normality constant of . If , the set is called the order interval between and .

For all , the notation means that there exist and such that . Clearly, is an equivalence relation. Giving (i.e., and , we denote by the set . It is easy to see that is convex and for all . If and , it is clear that . For other detailed properties of cones, we refer to the monograph by Guo and Lakshmikantham [20].

*Definition 1 (cf. [1, 2]). * is said to be a mixed monotone operator if is increasing in and decreasing in , that is, implies that . Element is called a fixed point of if .

Here, one recalls the following fixed point theorem of mixed monotone operator which has been established by Zhai and Zhang [7].

Proposition 2 (cf. [7, Theorem 2.1]). *Let be a normal cone of a real Banach space . Also, let be a mixed monotone operator. Assume that*)* there exists with such that ;
*()

*for any*and ,*there exists**such that*.*Then operator has a unique fixed point in . Moreover, for any initial , constructing successively the sequences*

*one has and as .*

By applying Proposition 2, one shows the existence and uniqueness of the positive solution to (BP; ) and (IP; ) on .

#### 3. Boundary Value Problem (BP; )

In this section, we show the existence and uniqueness of the positive solution to (BP; ) by applying a fixed point theorem of mixed monotone operator (Proposition 2).

Throughout this section, we assume the following conditions ()–():() is nondecreasing in for each and , and is nonincreasing in for each and . Also, for all .()For each , is nondecreasing in for each and is nonincreasing in for each .()For all , there exists a constant such that for any , any , and any .

We give the definition of solutions to (BP; ).

*Definition 3. *Let , and let be a given constant. Then, a function is called a solution to (BP; ) on if it satisfies (1).

Now, we mention our first main theorem in this paper, which is concerned with the existence-uniqueness of the positive solution to (BP; ) on .

Theorem 4. *Assuming the conditions ()–(), and having has a fixed positive constant, then for each function with a.e. , there exists a unique positive solution to (BP; ) on .*

Here, we give the key lemma, which is concerned with the characterization of solutions to (BP; ).

Lemma 5. *Assume the same conditions as in Theorem 4. Then, is a solution to ( BP; ) on if and only if satisfies the following integral equation:
*

*where*

*Proof. *Firstly, integrating from to , we obtain
Again, integrating (14), we have
From (14) with , we infer that
Hence, from , (15), and (16), we can find that
where is the function defined as in (13). Thus, the proof is completed.

By Lemma 5, we can show the solvability of (BP; ). In fact, we define an operator by where is the function defined as in (13).

We can easily find that the following lemma holds.

Lemma 6. *Assume the same conditions as in Theorem 4. Then, is a solution to ( BP; ) on if and only if is the fixed point of the operator defined by (18).*

Taking account of Proposition 2 and Lemmas 5 and 6, one can prove Theorem 4 concerning the existence and uniqueness of the positive solution to (BP; ) on .

*Proof of Theorem 4. *By applying a fixed point theorem of mixed monotone operator (Proposition 2), we show the existence and uniqueness of the positive solution to (BP; ) on .

To do so, set
Clearly, is a normal cone in and the normality constant is .

Let be the operator defined by (18). Then, we infer from (), (), (13), and a.e. that
Thus, we see that .

Firstly, by (), (), and (18), we can easily prove that is a mixed monotone operator.

Next, we show . Put
Then, we see from () that . Therefore, for any and , we observe from ()–(), (13), and a.e. that
which implies that
Thus, the condition holds.

Now, we show (), defining a function by
hence, for all , then we can easily see that for all .

Now we show that . Set
then, .

Note that has maximum and minimum on , since with on any subinterval of . So, let

Here, put . Then, from (), (), (13), and a.e. , it follows that

Also, we have
Thus, we observe that
which implies that .

By arguments as above, we see that the operator defined by (18) satisfies conditions () and () in Proposition 2. Therefore, by applying Proposition 2, we conclude that an operator equation has a unique solution in ; hence there exists a unique positive solution to (BP; ) on .

#### 4. Initial Value Problem (IP; )

In this section, we show the existence-uniqueness of the positive solution to (IP; ) on by arguments similar to (BP; ).

Throughout this section, we assume the following conditions , :, such that for all and . Also, is nonincreasing in for each and and is nondecreasing in for each and . Moreover, for all .For all , there exists a constant , such that for any , any , and any .

Here, we give the definition of solutions to (IP; ).

*Definition 7. *Let and and as given constants. Then, a function is called a solution to (IP; ) on if it satisfies (2).

Now, we mention our second main theorem in this paper, which is concerned with the existence-uniqueness of the positive solution to (IP; ) on .

Theorem 8. *Assuming the conditions , , and and having has a fixed positive constant. Then, for each function with a.e. , there exists a unique positive solution to (IP; ) on .*

Based on the proof of Lemma 5 (cf. (15)), one can get the following key lemma concerning the characterization of solutions to (IP; ).

Lemma 9. *Assume the same conditions as in Theorem 8. Then, is a solution to ( IP; ) on if and only if satisfies the following integral equations:
*

By Lemma 9 and Proposition 2, one can show Theorem 8 concerning the existence-uniqueness of the positive solution to (IP; ) on .

*Proof of Theorem 8. *Now, we define an operator by
Then, we have to find the fixed point of the operator in order to show the existence-uniqueness of the solution to (IP; ) on .

Let be the same space defined by (19). Then, we infer from , (), (32), and a.e. that
Thus, we see that .

Also, we observe from , (), and (32) that is a mixed monotone operator.

Next, we show . Put
Then, we see from that . Therefore, for any and , we observe from (), , , (32), and a.e. that
which implies that
Thus, the condition holds.

Now, we show (), defining a function by
hence, for all . Then, we can easily see that for all .

Now we show that . Set
then, .

From , (), (32), and a.e. , it follows that

Also, we have
Thus, we observe that
which implies that .

By arguments as above, we see that the operator defined by (32) satisfies conditions () and () in Proposition 2. Therefore, by applying Proposition 2, we conclude that an operator equation has a unique solution in ; hence there exists a unique positive solution to (IP; ) on .

#### 5. Optimal Control Problem (OP)

In this section, we consider an optimal control problem (OP) to (IP; ). Throughout this section, we assume all the conditions of Theorem 8. Also, we assume the following additional conditions.()There is a constant such that Also, for each , there exists a positive constant such that () is a given desired target profile in .

At first, we give the key lemma in order to show the result of continuous dependence of positive solutions to (IP; ).

Lemma 10 (cf. [19, Lemma 5.1]). *Let , and let be an operator given by
**
Assume that weakly in as for some . Then
*

For the detailed proof of Lemma 10, we refer to [19, Lemma 5.1].

Taking account of Lemma 10, one can show the following proposition concerning the result of continuous dependence of positive solutions to (IP; ).

Proposition 11 (cf. [19, Proposition 5.2]). *Assume the same conditions as in Theorem 8, (), and (). Let and , where is the control space defined by (4). Assume weakly in as . Then, the unique positive solution to (IP; on converges to one to (IP; on in the sense that
*

*Proof. *By arguments similar to [19, Proposition 5.2], we can prove (46). In fact, note from Lemma 9 that is a solution to (IP) on if and only if

Now, let , then, we obtain from () that
for all , where is a function defined in (44).

Applying a Gronwall-type inequality (e.g., [21, Proposition ]) to (48), we obtain
for all . Therefore, it follows from (48) and (49) that
for all .

By (50) and the assumption (), we also have
for all .

Next, we consider the time interval . Then, we see from (50) and () that
for any and . By the same arguments as before (cf. (49) and (50)), we can take some constants so that
for all .

Also, we obtain from () and (53) that
for some positive constants .

By repeating this procedure, we can take positive constants and such that
for all .

Here, put . Then, we infer from (55) that

Since weakly in as , we observe from Lemma 10 that
Hence, we see from (56) and (57) that
Thus, the proof of Proposition 11 has been completed.

Now, we mention our main result concerning the existence of an optimal control to (OP).

Theorem 12. *Assume the same conditions as in Theorem 8, (), and (). Then, the problem ( OP) has at least one optimal control such that
*

*where is a control space defined by (4) and is the cost functional defined in (5).*

*Proof. *By the quite standard method, we can prove Theorem 12. In fact, let be a minimizing sequence so that
By the definition (5) of , we see that is bounded in . Hence, there is a subsequence and a function such that and

For any , let be a unique positive solution to (IP on . Then, from (61) and Proposition 11, we observe that
where is a unique positive solution to (IP on .

Hence, it follows from (61), (62), and the weak lower semicontinuity of -norm that
which implies that is an optimal control to (OP).

Now, we mention our final main result in this paper, which is concerned with the stability of the optimal control to (OP).

Theorem 13. *Assume the same conditions as in Theorem 12. Let and for some and small positive constant . Also, let and be unique positive solutions to ( IP; ) and (IP;) on , respectively. Then,
*

*Proof. *Note from Lemma 9 that is a solution of (IP on if and only if

Now, let . Then, we obtain from () that

Applying a Gronwall-type inequality (e.g., [21, Proposition ]) to (66), we obtain
Therefore, it follows from (66) and (67) that

By (68) and the assumption (), we also have

Next, we consider the time interval . Then, we see from (68) and () that
for any . By the same arguments as before (cf. (67) and (68)), we can take some constant so that

Also, we obtain from () and (71) that
for some positive constant .

By repeating this procedure, we can take positive constants and such that

Here, put . Then, we infer from (73) that
Thus, the proof of Theorem 13 has been completed.

By Theorem 13 and the definition of (cf. (5)), we easily see that the following corollary holds.

Corollary 14. *Assume the same conditions as in Theorem 12. Let and for some and small positive constant . Then,
*

#### 6. Examples

In this section, we give an example of the main results.

*Example 1. *Consider the following boundary value problem of second order impulsive differential equation:

*Conclusion.* The boundary value problem (76) admits a unique positive solution, which is continuously differentiable on .

*Proof. *Let , , and . Evidently, and are increasing in for and are decreasing in for .

Set , , then,
Therefore, we easily see that ()–() hold. Hence, applying Theorem 4 to (76), we get a unique positive solution to (76) on for each with a.e. , where is a given constant.

*Example 2. *Consider the following initial value problem of the second order impulsive differential equation:

*Conclusion.* The initial value problem (78) admits a unique positive solution, which is continuously differentiable on . Moreover, the problem (OP) to (78) has at least one optimal control, and the stability result of optimal control holds.

*Proof. *Let , , , and . Clearly, is decreasing in for and is increasing in for .

Also, let , evidently, is increasing in for and is decreasing in for .

Set , , then,
Therefore, we easily see that conditions (), (), , , and hold. Hence, applying Theorem 8 to (78), we get a unique positive solution to (78) on for each with a.e. , where is a given constant.

In addition, let . Then, we easily see that () holds. Hence, applying Theorem 12, we see that Problem (OP) to (78) has at least one optimal control for each desired target profile in . Also, applying Theorem 13, we get the result on the stability of optimal control to (OP).

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The authors would like to thank the referees for their valuable suggestions and comments. The research was partially supported by the National Natural Science Foundation of China (no. 61250011), the Postdoctoral Science Foundation of China (no.2012M510786) and the Science Foundation of Shanxi Province (no. 2012011004-4).