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.

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

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.

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

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.