Research Article | Open Access
Integral BVPs for a Class of First-Order Impulsive Functional Differential Equations
The methods of lower and upper solutions and monotone iterative technique are employed to the study of integral boundary value problems for a class of first-order impulsive functional differential equations. Sufficient conditions are obtained for the existence of extreme solutions.
1. Introduction and Preliminaries
In this paper, we study the following integral boundary value problems (BVPs for short) of the impulsive functional differential equation
where . , where is continuous for exist, and . Furthermore, we will assume that is continuous and monotone nondecreasing, and for any bounded set is bounded. denotes the jump of at ; and represent the right and left limits of at , respectively. Denote
Let be continuously differentiable for . and are Banach spaces with the norms
Note that (1.1) has a very general form, as special instances resulting from (1.1), one can have impulsive differential equations with deviating arguments and impulsive differential equations with the Volterra or Fredholm operators. When , , (1.1) reduces to
In , Cao and Li. studied and understood existence and stability of solution of this equation by using fixed theorem and monotone iteration techniques.
When , , (1.1) reduces to
In , Li discussed and built the existence theorem of solutions of this equation by using fixed theorem, upper and lower solutions methods and monotone iterative techniques.
When , , , the equation (1.1) reduces to the periodic boundary value problem of the impulsive differential equation
There are plenty of results on studying the periodic boundary value problem of impulsive differential equations (see [3–8]). According to author’s know, there are no dependent references for studying the (1.1) yet. To fill in this void, we try to find the conditions on and , so that make sure that the (1.1) exists extremal solution.
It is well known that the monotone iterative technique offers an approach for obtaining approximate solutions of nonlinear differential equations, for details, see  and the references therein. There also exist several works devoted to the applications of this technique to boundary value problems of impulsive differential equations, see, for example, [1–3, 5–14]. In this paper, we consider (1.1) by using the method of upper and lower solutions combined with monotone iterative technique. This technique plays an important role in constructing monotone sequences which converge to the solutions of our problems. In presence of a lower solution and an upper solution with , we show under suitable conditions the sequences converge to the solutions of (1.1) by using the method of upper and lower solutions and monotone iterative technique.
Definition 1.1. The functions are called lower solution and upper solution of (1.1), respectively, if
In what follows we define the set
for and .
We list the following conditions.are lower and upper solutions of (1.1) such that There exists such that for There exist such that for
2. Main Results
To obtain our main results, we need the following lemmas.
Lemma 2.1 (see ). Suppose that the following conditions are satisfied. Sequence satisfies and and is left continuous at . For , where are constants, then
Lemma 2.2 (see ). If and where then
Lemma 2.3. If and , then the equation has one unique solution.
Proof. Firstly, we prove that (2.4) is equivalent to the integral equation
If is solution of (2.4), then, by directly integrating we obtain
If is solution of the above-mentioned integral equation, then
This yields . So (2.4) is equivalent to the integral equation
Now, we define operator as
For each ,
This indicates that is a contraction mapping. Then there is one unique such that , that is, (2.4) has an unique solution . The proof is complete.
Theorem 2.4. If the conditions are all satisfied, and, in addition, if there exist such that , then the impulsive equation (1.1) has minimal and maximal solutions in , and there are monotone sequences convergeing uniformly to in , respectively, where , and are lower and upper solutions of (1.1), respectively.
Proof. For each , we consider the equation
By Lemma 2.3, we know that (2.13) has a unique solution . Now, we define operator as .
We will prove that have the following properties.
(a).(b) is monotone nondecreasing on .
Proofs of properties are divided into three steps to proceed.
Step 1. Suppose that , then By Lemma 2.2, we obtain , so Step 2. Suppose that , then By Lemma 2.2, we obtain , so .
Similary we can show that , hence
Step 3. If then when , let . Then Furthermore, , thus By Lemma 2.2, we obtain , so .
Similarly, we can assume that When ,
Furthermore, , , thus when , hence by Lemma 2.2, we obtain , so .
In the same way we can prove that .
Thus by mathematical induction we can know that
So far, we finish the proof of the properties .
Now we prove that are lower and upper solutions of (1.1). Similarly, we can use mathematical induction to prove this.
When , are already lower and upper solutions of (1.1).
Thus is lower solution of (1.1).
Suppose that is lower solution of (1.1) when .
Then when ,
Thus by mathematical induction we can know that is lower solution of (1.1). In the same way we can prove that is upper solution of (1.1).
By , we can know that when , have limits respectively. Since they are independent of when , converge uniformly to and
According to , satisfying (2.13), that is,
when , we have Equation (2.24) indicates that are solutions of (1.1).
Lastly, we prove that are minimal and maximal solutions of the equation (1.1) in .
Suppose that is a solution of the equation and satisfies , obviously, we can assume that there is an such that .
If , then
And since , ,
Hence by Lemma 2.2, we can obtain , so . Similarly, we can obtain: This indicates that . Hence when , we can obtain that . This ends the proof.
Finally, we give an example to illustrate the efficiency of our results.
Example 2.5. Consider the problem of where Obviously, are the lower solution and upper solution for (2.27) with respectively. , . Let the conditions of Theorem 2.4 are all satisfied, so problem (2.27) has the maximal and minimal solutions in the segement .
This work was supported by the NNSF of China (10571050; 10871062), the Hunan Provincial Natural Science Foundation of China (09JJ6010), and the Zhejiang Provincial Natural Science Foundation of China (Y6090057).
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