Nonlinear Boundary Value Problem of First-Order Impulsive Functional Differential Equations
© K. Zhang and X. Liu. 2010
Received: 8 December 2009
Accepted: 30 January 2010
Published: 21 February 2010
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.
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 .
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 . 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.
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
Let us consider the following nonlinear boundary value problem (NBVP):
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).
- (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)
We will need the following lemma.
Lemma 2.2 (see ).
the sequence satisfies with ,
is left continous at for ,
where , and are real constants.
In order to establish a comparison result and some lemmas, we will make the following assumptions on the function .
where , , , , and .
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 , .
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.
Let , then . For some , there exists such that or . We only consider , as for the case , the proof is similar.
which is a contradiction.
which is a contradiction. The proof of Theorem 2.3 is complete.
The following corollary is an easy consequence of Theorem 2.3.
then , for .
We show some examples of function satisfying (H1).
Consider the linear boundary value problem (LBVP)
where , , , , and .
By direct computation, we have the following result.
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.
For convenience, let us list the following conditions.
wherever , and .
Let . Now we are in the position to establish the main results of this paper.
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.
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:
(b) , if
By Theorem 2.3, we get for , that is, . Similarly, we can show that .
By Theorem 2.3, we get for , that is, , then (b) is proved.
Moreover, , are solutions of NBVP (2.2) in .
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.
Let the assumptions of Theorem 3.2 hold and assume the following.
whenever , and .
Then NBVP (2.2) has a unique solution in .
By Theorem 3.2, we know that there exist , which are minimal and maximal solutions of NBVP (2.2) with .
By Theorem 2.3, we have that , , that is, . Hence , this completes the proof.
To illustrate our main results, we shall discuss in this section some examples.
where , , .
Setting and , it is easy to verify that is a lower solution, and is an upper solution with .
then inequalities (2.9) and (2.30) are satisfied. By Theorem 3.2, problem (4.1) has extremal solutions .
where , , .
Setting and , then is a lower solution, and is an upper solution with .
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 .
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