Let

*E* =

*C*[0, 1] endowed norm ||

*u*|| = max

_{0≤t≤1}*|u|*, and define the cone

*P* ⊆

*E* by

Then, it is easy to prove that *E* is a Banach space and *P* is a cone in *E*.

Define the operator

*T*:

*E* →

*E* by

**Lemma 3.1**. *T*: *E* → *E* is completely continuous, and *Te now prove thatP* ⊆ *P*.

**Proof**. For any

*u* ∈

*P*, then from properties of

*G*(

*t, s*),

*T* (

*u*)(

*t*) ≥ 0,

*t* ∈ [0, 1], and it follows from the definition of

*T* that

Thus, it follows from above that

From the above, we conclude that *TP* ⊆ *P*. Also, one can verify that *T* is completely continuous by the Arzela-Ascoli theorem. □

Then, it is clear to see that 0 < *l* ≤ *L* < L.

**Theorem 3.2**. Assume (H1) to (H3) hold. In addition,

(H5) There exists a constant 2 ≤

*p*_{1} such that

(H6) There exists a constant

*p*_{2} with

such that

Then, problem (Equation 1) has one positive solution.

**Proof**. From (H4), there exists a 0 <

*η* < ∞ such that

Choosing

*R*_{1} ∈ (0,

*η*), set Ω

_{1} = {

*u* ∈

*E* : ||

*u*|| <

*R*_{1}}. We now prove that

Let

*u* ∈

*P* ∩ ∂Ω

_{1}. Since min

_{θ≤t≤1-θ}*u*(

*t*) ≥

*θ* ||

*u*|| and ||

*u*|| =

*R*_{1}, from Equation

3, (H1) and (H3), it follows that

Then, Equation 4 holds.

On the other hand, from (H5), there exists

such that

From (H6), there exists

such that

Choosing

, set Ω

_{2} = {

*u* ∈

*E* : ||

*u*|| <

*R*_{2}}. We now prove that

If

*u* ∈

*P* ∩ ∂Ω

_{2}, we have

From Equations

5,

6, we can prove

Then, Equation 7 holds.

Therefore, by Equations 4 and 7 and the second part of Lemma 2.3, *T* has a fixed point in
, which is a positive solution of Equation 1. □

**Example**. Let

*q* = 2,

*h*(

*t*) = 1, Φ(

*s*) = 2 +

*s*,

*φ*(

*t*) = 2

*t*,

and

, namely,

It is easy to see that (H1) to (H3) hold. We also can have

Take

*p*_{1} = 2, then it is clear to see that (H4) and (H5) hold. Since

then (H6) hold.

**Theorem 3.3**. Assume (H1) to (H3) hold. In addition,

(H7) There exists a constant 2 ≤

*p*_{1} such that

(H8) There exists a constant

*p*_{2} with

such that

Then, problem (Equation 1) has one positive solution.

**Proof**. From (H7), there exists

*η*_{1} > 0 such that

From (H8), there exists

*η*_{2} > 0 such that

Choosing

, set Ω

_{1} = {

*u* ∈

*E* : ||

*u*|| <

*R*_{1}}. We now prove that

If

*u* ∈

*P* ∩ ∂Ω

_{1}, we have

From Equations

8,

9, we can prove

Then, Equation 10 holds.

On the other hand, from (H7), there exists

such that

Choosing

, set Ω

_{2} = {

*u* ∈

*E* : ||

*u*|| <

*R*_{2}}. We now prove that

If

*u* ∈

*P* ∩ ∂Ω

_{2}, Since min

_{θ≤t≤1-θ}*u*(

*t*) ≥

*θ* ||

*u*|| and ||

*u*|| =

*R*_{2}, we have

By Equation

11, (H1) and (H3), it follows that

Then, Equation 12 holds.

Therefore, by Equations 10 and 12 and the first part of Lemma 2.3, *T* has a fixed point in
, which is a positive solution of Equation 1. □

**Example**. Let *q* = 2, *h*(*t*) = *t*, Φ(*s*) = 2 + *s*, *φ*(*t*) = 2*t*,
and *g*(*s*) = *s*^{2}.

**Theorem 3.4**. Assume that (H1) to (H3) hold. In addition, *φ*(1) ≥ 1, and the functions *f*, *g* satisfy the following growth conditions:

(H12) There exists a constant

*a* > 0 such that

Then, BVP (Equation 1) has at least three positive solutions.

**Proof**. For the sake of applying the Leggett-Williams fixed point theorem, define a functional

*σ*(

*u*) on cone

*P* by

Evidently, *σ*: *P* → *R*^{+} is a nonnegative continuous and concave. Moreover, *σ*(*u*) ≤ ||*u*|| for each *u* ∈ *P*.

Now, we verify that the assumption of Lemma 2.2 is satisfied.

Firstly, it can verify that there exists a positive number *c* with
such that
.

By (H10), it is easy to see that there exists

*τ* > 0 such that

If

, then

by (H1) to (H3) and (H10).

Next, from (H11), there exists

*d*' ∈ (0,

*a*) such that

Take

. Then, for each

, we have

Finally, we will show that {*u* ∈ *P* (*σ, a, b*): *σ*(*u*) > *a*} ≠ ∅ and *σ*(*Tu*) > *a* for all *u* ∈ *P*(*σ, a, b*).

For

*u* ∈

*P* (

*σ, a, b*), we have

for all

*t* ∈ [

*θ*, 1 -

*θ*]. Then, we have

by (H1) to (H3), (H12). In addition, for each

*u* ∈

*P* (

*θ, a, c*) with ||

*Tu*|| >

*b*, we have

Above all, we know that the conditions of Lemma 2.2 are satisfied. By Lemma 2.2, the operator

*T* has at least three fixed points

*u*_{
i
} (

*i* = 1, 2, 3) such that

The proof is complete. □

**Example**. Let

*q* = 2,

*h*(

*t*) =

*t*, Φ(

*s*) = 2 +

*s*,

*φ*(

*t*) = 2

*t*,

and,

, namely,

From a simple computation, we have

Then, it is easy to see that (H1) to (H3) and (H10) to (H11) hold. Especially, take *a* = 1, by
and (H1), then (H12) holds.