The basic space used in this paper is a real Banach space

with norm

defined by

, where

. Let

It is obvious that
is a cone in
.

Lemma 2.1 (see [7]).

Let
, then
,
.

Lemma 2.2.

Let
, then there exists a constant
such that
.

Proof.

The mean value theorem guarantees that there exists

, such that

Moreover, the mean value theorem of differential guarantees that there exists

, such that

Denote
; then the proof is complete.

Lemma 2.3.

Assume that (H1), (H2) hold. If
is a solution of BVP(1.1), there exists a unique
, such that
and
,
.

Proof.

From the fact that
, we know that
is strictly decreasing. It follows that
is also strictly decreasing. Thus,
is strictly concave on [0, 1]. Without loss of generality, we assume that
. By the concavity of
, we know that
,
. So we get
. By
, it is obvious that
. Hence,
,
.

On the other hand, from the concavity of
, we know that there exists a unique
where the maximum is attained. By the boundary conditions and
, we know that
or 1, that is,
such that
and then
.

Lemma 2.4.

Assume that (H1), (H2) hold. Suppose

is a solution of BVP(1.1); then

Proof.

First, by integrating (1.1) on

, we have

According to the boundary condition, we have

By a similar argument in [5],
; then the proof is completed.

Now we define an operator

by

Lemma 2.5.

is completely continuous.

Proof.

Let

; then from the definition of

, we have

So
is monotone decreasing continuous and
. Hence,
is nonnegative and concave on [0, 1]. By computation, we can get
. This shows that
. The continuity of
is obvious since
is continuous. Next, we prove that
is compact on
.

Let

be a bounded subset of

and

is a constant such that

for

. From the definition of

, for any

, we get

Hence,
is uniformly bounded and equicontinuous. So we have that
is compact on
. From (2.13), we know for
,
, such that when
, we have
. So
is compact on
; it follows that
is compact on
. Therefore,
is compact on
.

Thus,
is completely continuous.

It is easy to prove that each fixed point of
is a solution for BVP(1.1).

Lemma 2.6 (see [1]).

Assume that (H1) holds. Then for

,

To obtain positive solution for BVP(1.1), the following definitions and fixed point theorems in a cone are very useful.

Definition 2.7.

The map

is said to be a nonnegative continuous concave functional on a cone of a real Banach space

provided that

is continuous and

for all

and

. Similarly, we say the map

is a nonnegative continuous convex functional on a cone of a real Banach space

provided that

is continuous and

for all
and
.

Let

and

be a nonnegative continuous convex functionals on

,

a nonnegative continuous concave functional on

, and

a nonnegative continuous functional on

. Then for positive real number

, and

, we define the following convex sets:

Theorem 2.8 (see [11]).

Let
be a real Banach space and
a cone. Assume that
and
are two bounded open sets in
with
,
. Let
be completely continuous. Suppose that one of following two conditions is satisfied:

(1)
,
, and
,
;

(2)
,
, and
,
.

Then
has at least one fixed point in
.

Theorem 2.9 (see [12]).

Let

be a cone in a real Banach space

. Let

and

be a nonnegative continuous convex functionals on

,

a nonnegative continuous concave functional on

, and

a nonnegative continuous functional on

satisfying

for

, such that for positive number

and

,

for all
. Suppose
is completely continuous and there exist positive numbers
, and
with
such that

and
for
;

()
for
with
;

()
and
for
with
.

Then
has at least three fixed points
, such that

for
,

,

with
,

.