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).
Remark 2.1.
-
(i)
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]).
-
(ii)
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 [1]).
Asumme that
the sequence
satisfies
with
,
is left continous at
for
,
for
,
,
where
,
and
are real constants.
Then
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.
Theorem 2.3.
Let
such that
where
,
,
,
, and
.
Suppose in addition that condition (H1) holds and
then
.
Proof.
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
,
.
Casedi.
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.
Casedii.
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
that is
First, we assume that
. Let
in (2.14), then
Noting that
, we have
Hence
which is a contradiction.
Next, we assume that
. By Lemma 2.2 and (2.10), we have
then
Setting
in (2.14), we have
with (2.19), we obtain that
that is,
Therefore,
which is a contradiction. The proof of Theorem 2.3 is complete.
The following corollary is an easy consequence of Theorem 2.3.
Corollary 2.4.
Assume that there exist
,
,
, for
such that
satisfies (2.8) with
and
then
, for
.
Remark 2.5.
Setting
, Corollary 2.4 reduces to the Theorem 2.3 of Li and Shen [6]. Therefore, Theorem 2.3 and Corollary 2.4 develops and generalizes the result in [6].
Remark 2.6.
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.
Lemma 2.7.
is a solution of LBVP (2.27) if and only if
is a solution of the impulsive integral equation
where
,
,
and
Lemma 2.8.
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.