We are now going to state our existence results.

Theorem 3.1.

Assume that (H1) and (H2) hold, then in each of the following case:

- (i)
,
,

- (ii)
,
,

the BVP(1.6)-(1.7) has at least one positive solution.

Proof.

To prove Theorem 3.1, we just show that the mapping
defined by (2.18) has a nonzero fixed point in the cases, respectively.

Case(i): since

, by the definition of

, we may choose

and

so that

Let

, we now prove that

for every

and

. In fact, if there exist

and

such that

, then, by definition of

,

satisfies differential equation the following:

and boundary condition (1.7). Multiplying (3.2) by

and integrating on

, then using integration by parts in the left side, we have

By Lemma 2.4,

, and then

. We see that

, which is a contradiction. Hence,

satisfies the hypotheses of Lemma 2.6, in

. By Lemma 2.6 we have

On the other hand, since
, there exist
and
such that

Let

, then it is clear that

Choose

. Let

. Since

, from (3.5) we see that

from which we see that

, namely the hypotheses (i) of Lemma 2.7 holds. Next, we show that if

is large enough, then

for any

and

. In fact, if there exist

and

such that

, then

satisfies (3.2) and boundary condition (1.7). Multiplying (3.2) by

and integrating, from (3.6) we have

Consequently, we obtain that

from which and from (3.11) we get that

Let

, then for any

and

,

. Hence, hypothesis (ii) of Lemma 2.7 also holds. By Lemma 2.7, we have

Now, by the additivity of fixed point index, combine (3.4) and (3.14) to conclude that

Therefore,
has a fixed point in
, which is the positive solution of BVP(1.6)-(1.7).

Case (ii): since

, there exist

and

such that

Let

, then for every

, through the argument used in (3.9), we have

Hence,

. Next, we show that

for any

and

. In fact, if there exist

and

such that

, then

satisfies (3.2) and boundary (1.7). From (3.2) and (3.16), it follows that

Since

, we see that

, which is a contradiction. Hence, by Lemma 2.7, we have

On the other hand, since
, there exist
and
such that

Set

, we obviously have

If there exist

and

such that

, then (3.2) is valid. From (3.2) and (3.21), it follows that

By the proof of (3.13), we see that

. Let

, then for any

and

,

. Therefore, by Lemma 2.6, we have

From (3.19) and (3.23), it follows that

Therefore,
has a fixed point in
, which is the positive solution of BVP(1.6)-(1.7). The proof is completed.

From Theorem 3.1, we immediately obtain the following.

Corollary 3.2.

Assume that

and

hold, then in each of the following cases:

- (i)
,
,

- (ii)
,
,

the BVP(1.6)-(1.7) has at least one positive solution.