Research Article | Open Access
Periodic Boundary Value Problems for First-Order Impulsive Functional Integrodifferential Equations with Integral-Jump Conditions
By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for first-order impulsive functional integrodifferential equations with integral-jump conditions. An example is also given to illustrate our results.
The theory of impulsive differential equations is now being recognized as not only being richer than the corresponding theory of differential equations without impulses, but also representing a more natural framework for mathematical modelling of many real world phenomena and applications; see [1–5] and the references therein. Monotone iterative technique coupled with the method of upper and lower solutions has provided an effective mechanism to prove constructive existence results for initial and boundary value problems for nonlinear differential equations; see . However, many papers have studied applications of the monotone iterative technique to impulsive problems; see, for example, [7–15]. In those articles, the authors assumed that , that is, a short-term rapid change of the state (jump condition) at impulse point , depends on the left side of the limit of .
In [16–18] the authors discussed some classes of first-order impulsive problems with the impulsive integral conditions: where and , . Furthermore, Thaiprayoon et al.  have used such technique to investigate the existence criteria of extremal solutions of multipoint impulsive problems to include multipoint jump conditions for , . Recently, Thiramanus and Tariboon  have given some results on impulsive differential inequalities with integral-jump conditions of the form: where . We note that if , , , then the above inequalities mean that the bound of jump condition at is a functional of past states on the interval before the impulse point .
In spirit of the results from , this paper considers the periodic boundary value problem for first-order impulsive functional integrodifferential equation (PBVP) with integral-jump conditions: where , , , , , , , , , , and .
We first introduce a new concept of lower and upper solutions, then establish a new comparison principle, and discuss the existence and uniqueness of the solutions for first-order impulsive functional integrodifferential equations with integral-jump conditions. By using the method of upper and lower solutions and monotone iterative technique, we obtain the existence of extreme solution of PBVP (4). Finally, we give an example to illustrate the obtained results.
Let , , , and be continuous everywhere except for some , at which and exist and , , and let be continuous everywhere except for some at which and exist and , . Clearly, is a Banach space with the norm . Let . A function is called a solution of PBVP (4) if it satisfies (4).
Definition 1. We say that the functions , are lower and upper solutions of PBVP (4) if there exist , , , , , and such that
Denote . We prove the comparison principle by using the following lemma (see ).
Lemma 2. Let , ,and, , , be constants and let , . If then, for ,
Now we are in the position to establish a new comparison principle, which plays an important role in monotone iterative technique.
Lemma 3. Assume that satisfies where constants , , , , , and , . If where , , and are given by Definition 1 with and then , .
Proof. Case 1. One has . Suppose that there exists such that and distinguish two cases.
Case (a). for all , ; then so that is nondecreasing in , and then . Since , then is a constant function , which implies that getting a contradiction.
Case (b). for some . Let ; then there exists , for some such that or . Without loss of generality, we only consider , and for the case the proof is similar.
From (11), it is easy to see that We consider the inequalities From Lemma 2, we have
Let in (18); then
If , then (20) with , for all implies
This contradicts the condition (12).
Suppose that . If for , then Lemma 2 provides that Since and integrating (11) from into , we obtain Hence, We note that if , then .
Let in (25); then From (26), we have which gives contradicting condition (12). For the case , the proof is similar.
Case 2. One has . Set . It follows that , and for , and for , , In view of Case 1, we see that on , and therefore on . This completes the proof.
Corollary 4. Assume that satisfies where constants , , , , , and , . If where and , are given by Lemma 3, then , for .
Corollary 5. Assume that satisfies where constants , , , , , and , , and condition (32) holds. Then , for .
Let us consider the following linear problem of PBVP (4): where , , , , , , , , , , and .
Lemma 6. A function is a solution of (35) if and only if is a solution of the following impulsive integral equation: where and
Proof. If is a solution of (35), by directly integrating, we obtain
If is a solution of the above-mentioned integral equation, then
The proof is complete.
Lemma 7. Let , , , , , , , , , , and and assume that
Then problem (35) has a unique solution in .
Corollary 8. Let , , , , , , , , , , and and assume that Then problem (35) has a unique solution in .
3. Main Results
In this section, we establish existence criteria for solutions of the PBVP (4) by the method of lower and upper solutions and monotone iterative technique. For , , we write if for all . In such a case, we denote , .
Theorem 9. Assume the existence of lower and upper solutions for PBVP (4) and also suppose that the following conditions hold.
The function satisfies where , , , and , , where , , , , and .
The functions satisfy where , , where and , .
If inequalities (12) and (40) hold, then there exists a solution of PBVP (4) such that , for .
Proof. We consider the following modified problem relative to PBVP (4):
If is such that on , then is a solution of PBVP (4) if and only if is a solution of (46). We will show that (46) is solvable and that every solution of (46) satisfies on . Suppose that is a solution of (46). We will show that . Let . Then, we have since and
By Lemma 3, we get on ; that is, . Similar arguments show that .
It remains to prove that (46) possesses at least one solution. By Lemma 6, PBVP (46) is equivalent to the following impulsive integral equation: where . For any , define a continuous compact operator by Let , such that , , , and take the compact sets, , , , and . Since is continuous, then we can choose a constant , such that , . For , we see that any solution of satisfies