Periodic Boundary Value Problems for a Class of Impulsive Functional Differential Equations of Hybrid Type

Abstract

This paper is concerned with the existence of extreme solutions of periodic boundary value problems for a class of first-order impulsive functional differential equations of hybrid type. We obtain the sufficient conditions for existence of extreme solutions by using upper and lower solutions method coupled with monotone iterative technique.

1. Introduction

The theory of impulsive differential equations is now being recognized to be not only richer than the corresponding theory of differential equations without impulses but also represents a more natural framework for mathematical modeling of many real-world phenomena [1–3]. Significant progress has been made in the theory of systems of impulsive differential equations in recent years (see [4–18] and the references cited therein). It is well known that the monotone iterative technique offers an approach for obtaining approximate solutions of nonlinear differential equations; for details, see [19] and the references therein. There also exist several works devoted to the applications of this technique to periodic boundary value problems of impulsive differential equations; see [20–26]. In [27, 28], the authors introduce a new concept of upper and lower solutions for periodic boundary value problems of a class of first-order functional differential equations. In paper [23], the authors applied this new concept to study the periodic boundary value problems for first-order impulsive functional differential equations. Motivated by [23, 27, 28], we will study periodic boundary value problem for the first-order impulsive functional differential equation of hybrid type where , , , , denotes the jump of at , , and represent the right and left limits of at , respectively. Denote . The integral part in (1) is defined by where , , , , , , and .

Let , is continuous for , , , and exist, and , . is continuously differentiable for , . and are Banach spaces with the norms

By a solution of (1), we mean a for which problem (1) is satisfied.

Note that (1) has a very general form; as special instances resulting from (1), one can have impulsive differential equations with deviating arguments and impulsive differential equations with the Volterra or Fredholm operators. For example, if does not include and , then (1) reduces to periodic boundary problem for impulsive differential equations with deviating arguments, which is discussed in [22, 23]; when , (1) is the following periodic boundary problem for impulsive integrodifferential equations of mixed type: similar problems are also discussed in [24–26].

2. Preliminaries

To apply the method of upper and lower solutions, we need the concept of lower solution and upper solution for (1).

Definition 1. A function is a lower solution of (1) if there exist , , , and  , , such that similarly, a function is an upper solution of (1) if where In what follows, we define the set for and , .

For the sake of convenience, we list the following conditions.Assume that for , and there exist and , , such that for , , , and , .There exists with , , such that

Lemma 2 (see [5]). Assume that , , , , , , are constants such that then

Lemma 3. Assume that , , , , and , such that where If then on .

Proof
Case  1  . Let ; then and where Obviously, implies that . To show , suppose, on the contrary, that for some . Then there are two possible cases:(a) for all ;(b)there exists such that .
In case (a), (17) implies that for and (); hence, is nonincreasing in ; then and . On the other hand, , which is a contradiction.
In case (b), let , , . We only consider for some , as the proof is similar for the case . Next, we consider two subcases.
Subcase I  . From (17), we get From Lemma (14), we have hence, This yields It contradicts (16).
Subcase II  . If , similar to the subcase I, it also yields a contradiction. We assume without loss of generality that and for some . By Lemma 2, we have Noting that , we get Hence, From (20) and (25), we obtain By multiplying on both sides of (27), we have Noting that , , we get This implies which contradicts (16). This completes the proof for the case .
Case  2  . Let ; then , and In view of Case 1, we have , for . Therefore, for . The proof of Lemma 3 is complete.

Lemma 4. Assume that , , , , , and , . If then the impulsive differential problem has a unique solution.

Proof. Define a map by where It is easy to verify that is a solution of (33) if and only if is a fixed point of . For any , we have this implies Condition (32) implies that is a contraction mapping. Banach’s fixed point theorem implies that has a unique fixed point, and so (33) has a unique solution. The proof is complete.

3. Main Results

Theorem 5. Assume that and are the lower solution and upper solution for (1) with on , respectively. Further, suppose that , , (16), and (32) are satisfied. Then, (1) possesses in the minimal and maximal solution , respectively. Moreover, there exist the monotone sequences and uniformly on , respectively.

Proof. For all , we consider linear impulsive differential equation where By Lemma 4, one can see that (38) has a unique solution . Denote that is a unique solution of (38).
Let , and , . We will show that(a), ,(b) is nondecreasing in .
The Proof of Property (a). Let ; since , it follows that , and where By Lemma 3, we have , for ; that is, . Analogously, it is proved that .
Next, we prove the property (b). Let , with , , , and ; then, from and , we have that, for , and, for , it is clear that . It follows by Case 1 of Lemma 3 that for ; that is, is monotonely nondecreasing in .
It follows, from the properties and , that By standard arguments, we conclude that there exist and such that It is easy to show that and are solutions of (1) using , satisfy the relations Taking the limit as on both sides of above relations, we have Equation (47) show that and are solutions of (1). Finally, we prove that, if is any solution of (1), then on . To this end, we assume, without loss of generality, that for some , since . From property (b), we can get Since , , by induction, we can conclude that Passing to the limit as , we obtain This ends the proof.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Science Foundation (11171085) and Key Project of Hunan Province Education Department (no. 11A095), China.