Theorem 3.1.

Suppose that

holds, then a necessary and sufficient condition for BVP (1.1) to have a pseudo-

positive solution is that the following integral condition holds:

Proof.

The proof is divided into two parts, necessity and suffeciency.

Necessity.

Suppose that

is a pseudo-

positive solution of (1.1). Then both

and

exist. By Lemma 2.4, there exist two constants

such that

Without loss of generality, we may assume that

. This together with condition

implies that

On the other hand, since

is a pseudo-

positive solution of (1.1), we have

Otherwise, let

. By the proof of Lemma 2.3, we have that

,

, that is,

which contradicts that

is a pseudo-

positive solution. Therefore, there exists a positive

such that

. Obviously,

. By (1.6) we have

Consequently,

, which implies that

It follows from (3.3) and (3.6) that

which is the desired inequality.

Sufficiency.

First, we prove the existence of a pair of upper and lower solutions. Since

is integrable on

, we have

Otherwise, if

, then there exists a real number

such that

when

, which contradicts the condition that

is integrable on

. In view of condition

and (3.8), we obtain that

where
.

Suppose that (3.1) holds. Firstly, we define the linear operators

and

as follows:

where

is given by (2.27). Let

It is easy to know from (3.11) and (3.12) that

By Lemma 2.4, we know that there exists a positive number

such that

Take

sufficiently small, then by (3.10), we get that

, that is,

Thus, from (3.14) and (3.16), we have

Considering

, it follows from (3.15), (3.17), and condition

that

From (3.13) and (3.16), it follows that

Thus, we have shown that
and
are lower and upper solutions of BVP (1.1), respectively.

Additionally, when

,

, by (3.17) and condition

, we have

From (3.1), we have
So, it follows from Lemma 2.5 that BVP (1.1) admits a pseudo-
positive solution such that

Remark 3.2.

Lin et al. [

23,

24] considered the existence and uniqueness of solutions for some fourth-order and

conjugate boundary value problems when

, where

under the following condition:

for

and

, there exists

such that

Lei et al. [25] and Liu and Yu [26] investigated the existence and uniqueness of positive solutions to singular boundary value problems under the following condition:

for all
,
where
and
is nondecreasing on
and nonincreasing on
.

Obviously, (3.21)-(3.22) imply condition
and condition
implies condition
. So, condition
is weaker than conditions
and
. Thus, functions considered in this paper are wider than those in [23–26].

In the following, when

admits the form

, that is, nonlinear term

is not mixed monotone on

, but monotone with respect

, BVP (1.1) becomes

If
satisfies one of the following:

is continuous, nondecreasing on

, for each fixed

, there exists a function

,

and

is integrable on

such that

Theorem 3.3.

Suppose that

holds, then a necessary and sufficient condition for BVP (3.23) to have a pseudo-

positive solution is that the following integral condition holds

Proof.

The proof is similar to that of Theorem 3.1; we omit the details.

Theorem 3.4.

Suppose that

holds, then a necessary and sufficient condition for problem (3.23) to have a

positive solution is that the following integral condition holds

Proof.

The proof is divided into two parts, necessity and suffeciency.

Necessity.

Assume that

is a

positive solution of BVP (3.23). By Lemma 2.4, there exist two constants

and

,

, such that

Let

be a constant such that

. By condition

, we have

By virtue of (3.28), we obtain that

By boundary value condition, we know that there exists a

such that

For

by integration of (3.29), we get

Integrating (3.31), we have

Exchanging the order of integration, we obtain that

Similarly, by integration of (3.29), we get

Equations (3.33) and (3.34) imply that

Since

is a

positive solution of BVP (1.1), there exists a positive

such that

. Obviously,

. On the other hand, choose

, then

. By condition

, we have

Consequently,

, which implies that

It follows from (3.35) and (3.37) that

which is the desired inequality.

Sufficiency.

Suppose that (3.26) holds. Let

It is easy to know, from (3.11) and (3.26), that

Thus, (3.12), (3.39), and (3.40) imply that

By Lemma 2.4, we know that there exists a positive number

such that

Take

sufficiently small, then by (3.10), we get that

, that is,

Thus, from (3.41) and (3.43), we have

Notice that

, it follows from (3.42)–(3.44) and condition

that

From (3.39) and (3.43), it follows that

Thus, we have shown that
and
are lower and upper solutions of BVP (1.1), respectively.

From the first conclusion of Lemma 2.5, we conclude that problem (1.1) has at least one
positive solution
.