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).
If and the impulses depend only on , the equation of NBVP (2.2) reduces to the simpler case of impulsive differential equations:
which have been studied in many papers. In some situation, the impulse depends also on some other parameters (e.g., the control of the amount of drug ingested by a patient at certain moments in the model for drug distribution [1, 3]).
If , where , the equation of NBVP (2.2) can be regarded as retarded differential equation which has been considered in [5, 12–14].
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 .
(H1) There exists a constant such that
(H2) The function satisfies Lipschitz condition, that is, there exists a such that
Inspired by the ideas in [5, 6], we shall establish the following comparison result.
Let such that
where , , , , and .
Suppose in addition that condition (H1) holds and
For simplicity, we let , . Set , then we have
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.
From (2.10) and condition (H1), we get
Consider the inequalities
By Lemma 2.2, we have
First, we assume that . Let in (2.14), then
Noting that , we have
which is a contradiction.
Next, we assume that . By Lemma 2.2 and (2.10), we have
Setting in (2.14), we have
with (2.19), we obtain that
which is a contradiction. The proof of Theorem 2.3 is complete.
The following corollary is an easy consequence of Theorem 2.3.
Assume that there exist , , , for such that satisfies (2.8) with and
then , for .
Setting , Corollary 2.4 reduces to the Theorem 2.3 of Li and Shen . Therefore, Theorem 2.3 and Corollary 2.4 develops and generalizes the result in .
We show some examples of function satisfying (H1).
(i), where , satisfies (H1) with ,
(ii), satisfies (H1) with ,
Consider the linear boundary value problem (LBVP)
where , , , , and .
By direct computation, we have the following result.
is a solution of LBVP (2.27) if and only if is a solution of the impulsive integral equation
where , , and
Let (H2) hold. Suppose further
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.