Now, we can state and prove our main result.

Theorem 2.1.

Suppose that there exist two distinct positive constants

,

and a function

with

and

such that

Then (BVP

_{
λ
}) has a positive solution

with

between

and

if

Proof.

It is clear that (BVP

_{
λ
}) has a solution

if, and only if,

is the solution of the operator equation

It follows from the definition of

in our observation

and Lemma B that

Hence,

, which implies

. Furthermore, it is easy to check that

is completely continuous. If there exists a

such that

, then we obtain the desired result. Thus, we may assume that

where
and
. We now separate the rest proof into the following three steps.

Step 1.

It follows from the definitions of

and

that, for

,

In fact, if there exist

and

such that

then, by (2.11),

which gives a contradiction. This proves that (2.13) holds. Thus, by Lemma C,

Step 2.

Suppose to the contrary that there exist

and

such that

It is clear that (2.17) is equivalent to

Since

and

it follows that there exists a

such that

Then

. From

on

, we see that

on

on

and

on

. It follows from

and

on

that

This contradiction implies

Step 3.

It follows from Steps (1) and (2) and the property of the fixed point index (see, for example, [10, Theorem
]) that the proof is complete.

Remark 2.2.

It follows from the conclusion of Theorem 2.1 that the positive constant

and nonnegative function

satisfy

There are many functions
and positive constants
satisfying (2.27). For example, Suppose that
and
. Let
on
, then
on
and

Remark 2.3.

A simple calculation shows that

Then, we have the following results.

(i)Suppose that

. Taking

, there exists

(

can be chosen small arbitrarily) such that

It follows from Remark 2.2 that the hypothesis (2.2) of Theorem 2.1 is satisfied if
.

(ii)Suppose that

. Taking

, there exists

(

can be chosen large arbitrarily) such that

which satisfies the hypothesis (2.1) of Theorem 2.1.

(iii)Suppose that

. Taking

, there exists

(

can be chosen small arbitrarily) such that

which satisfies the hypothesis (2.1) of Theorem 2.1.

(iv)Suppose that

. Taking

, there exists a

(

can be chosen large arbitrarily) such that

Hence, we have the following two cases.

Case i.

Assume that

is bounded, say

for some constant

. Taking

(since

can be chosen large arbitrarily,

can be chosen large arbitrarily, too),

Case ii.

Assume that

is unbounded, then there exist a

(

can be chosen large arbitrarily) and

such that

It follows from

and (2.37) that

By Cases (i), (ii) and Remark 2.2, we see that the hypothesis (2.2) of Theorem 2.1 is satisfied if
.

We immediately conclude the following corollaries.

Corollary 2.4.

(BVP_{
λ
}) has at least one positive solution for
if one of the following conditions holds:

Proof.

It follows from Remark 2.3 and Theorem 2.1 that the desired result holds, immediately.

Corollary 2.5.

Let

on
for some
and
.

Then, for

, (BVP

_{
λ
}) has at least two positive solutions

and

such that

Proof.

It follows from Remark 2.3 that there exist two real numbers

satisfying

Hence, by Theorem 2.1 and Remark 2.2, we see that for each

, there exist two positive solutions

and

of (BVP

_{
λ
}) such that

Thus, we complete the proof.

Corollary 2.6.

Let

on
, for some
.

Then, for

, (BVP

_{
λ
}) has at least two positive solutions

and

such that

Proof.

It follows from Remark 2.3 that there exist two real numbers

satisfying

Hence, by Theorem 2.1 and Remark 2.2, we see that, for each

, (BVP

_{
λ
}) has two positive solutions

and

such that

Thus, we completed the proof.