We note that

is a solution of (1.1) if and only if

Let
be endowed with the norm
and
is concave and nonnegative valued on
, and
.

Clearly,
is a Banach space with the norm
and
is a cone in
. For each
, extend
to
with
for
.

Define

as

We seek a fixed point,

, of

in the cone

. Define

Then,
denotes a positive solution of BVP (1.1).

It follows from (2.2) that

Lemma 2.1.

Let
be defined by (2.2). If
, then

(i)
.

(ii)
is completely continuous.

(iii)
,
.

(iv)
is decreasing on
.

The proof is similar to the proofs of Lemma 2.3 and Theorem 3.1 in [7], and is omitted.

Fix

such that

, and set

Throughout this paper, we assume
and
.

Now, we define the nonnegative, increasing, continuous functionals

,

, and

on

by

For the notational convenience, we denote

,

and

,

by

Theorem 2.2.

Suppose that there are positive numbers

such that

Assume
satisfies the following conditions:

for
, uniformly in
,

for
, uniformly in
.

Then, BVP (1.1) has at least two positive solutions of the form

where
,
and
,
.

Proof.

By the definition of operator
and its properties, it suffices to show that the conditions of Lemma 1.1 hold with respect to
.

First, we verify that
implies
.

Since

, one gets

for

. Recalling that (2.7), we know

for

. Then, we get

Secondly, we prove that
implies
.

Since

implies

, it holds that

for

, and for all

implies

It is obvious that

. On the other hand,

and (2.7) imply

By Lemma 1.1,

has at least two different fixed points

and

satisfying

which are twin positive solutions of BVP (1.1). The proof is complete.

In analogy to Theorem 2.2, we have the following result.

Theorem 2.3.

Suppose that there are positive numbers

such that

Assume
satisfies the following conditions:

for
, uniformly in
,

Then, BVP (1.1) has at least two positive solutions of the form

Now, we give theorems, which may be considered as the corollaries of Theorems 2.2 and 2.3.

From above, we deduce that
.

Theorem 2.4.

If the following conditions are satisfied:

,
, uniformly in
,

there exists a

such that for all

, one has

Then, BVP (1.1) has at least two positive solutions of the form

Proof.

First, choose

, one gets

Secondly, since

, there is

sufficiently small such that

Without loss of generality, suppose

. Choose

so that

. For

, we have

and

. Thus,

Thirdly, since

, there is

sufficiently large such that

Without loss of generality, suppose

. Choose

. Then,

We get now
, and then the conditions in Theorem 2.2 are all satisfied. By Theorem 2.2, BVP (1.1) has at least two positive solutions. The proof is complete.

Theorem 2.5.

If the following conditions are satisfied:

, uniformly in
;
,

there exists a

such that for all

, one has

Then, BVP (1.1) has at least two positive solutions of the form

The proof is similar to that of Theorem 2.4 and we omitted it.

The following Corollaries are obvious.

Corollary 2.6.

If the following conditions are satisfied:

,
, uniformly in
,

there exists a

such that for all

, one has

Then, BVP (1.1) has at least two positive solutions of the form

Corollary 2.7.

If the following conditions are satisfied:

, uniformly in
,
;

there exists a

such that for all

, one has

Then, BVP (1.1) has at least two positive solutions of the form