### Abstract

This paper is concerned with the existence, nonexistence, and uniqueness of convex
monotone positive solutions of elastic beam equations with a parameter *λ*. The boundary conditions mean that the beam is fixed at one end and attached to a
bearing device or freed at the other end. By using fixed point theorem of cone expansion,
we show that there exists
*λ*. In particular, we give an estimate for critical value of parameter *λ*.

**MSC: **
34B18, 34B15.

##### Keywords:

elastic beam equation; positive solution; fixed point; cone### 1 Introduction and preliminaries

In this paper, we consider the following nonlinear fourth-order two-point boundary value problem (BVP) for elastic beam equation:

where
*x* is a solution of BVP (1) and

Because of characterization of the deformation of the equilibrium state, fourth-order
boundary value problems for elastic beam equations are extensively applied to mechanics
and engineering; see [1-3]. Some nonlinear elastic beam equations have been studied extensively. For a small
sample of such work, we refer the reader to the work of Bai and Wang [4], Bai [5], Bonanno and Bellaa [6], Li [7], Liu and Li [8], Liu [9], Ma and Xu [10], and Ma and Thompson [11] on an elastic beam whose two ends are simply supported, the works of Yang [12] and Zhang [13] on an elastic beam of which one end is embedded and another end is fastened with
a sliding clamp, and the work of Graef *et al.*[14] on multipoint boundary value problems.

BVP (1) with
*q* at the right end. When the elastic beam equation does not contain parameter *λ*, the existence of multiple positive solutions and unique positive solution was presented
in [17] by variational methods and in [18] by a fixed point theorem, respectively; monotone positive solutions were obtained
by using the monotone iteration method in [19]. However, there are few papers concerned with positive solutions for BVP (1) with
parameter, especially with the solution’s dependence on parameter *λ* in the existing literature. The aim of this paper is to show that the existence and
number of convex monotone positive solutions of BVP (1) are affected by the parameter
*λ*.

The paper is organized as follows. In Section 2, we present that a nontrivial and
nonnegative solution of BVP (1) is convex monotone positive solution. In Section 3,
we obtain some results on the existence, multiplicity and nonexistence of positive
solutions for BVP (1). These results show that the number of positive solutions for
BVP (1) depends on the parameter *λ*. In Section 4, we establish some uniqueness criteria for positive solutions for BVP
(1) and show that such a positive solution
*λ*. In particular, we give an estimate for the critical value of the parameter *λ*.

In the rest of this section, we introduce some notations and known results. For the reader’s convenience, we suggest that one refer to [20-22], and [23] for details.

Let *E* be a real Banach space and *θ* denote the zero element of *E*. A nonempty closed convex set
*E* if it satisfies (i)
*E* is partially ordered by the cone *P*, *i.e.*,
*P* is said to be normal if there exists a positive number *N*, called the normal constant of *P*, such that

For all
*i.e.*,

Let
*T* if

**Lemma 1.1** (Fixed point theorem of cone expansion) [21,22]

*Assume that*
*and*
*are bounded open subsets of**E**with*
*Let*
*be a completely continuous operator such that*
*if*
*and*
*if*
*Then**T**has a fixed point in*

**Lemma 1.2**[23]

*Let**P**be a normal cone in**E*,
*be increasing and for all*
*and*
*there exists*
*such that*
*Then**T**has a unique fixed point*
*in*
*Moreover*, *constructing successively the sequence*
*for any*
*we have*

### 2 Solutions

In what follows, set
*P* is a normal cone and its normality constant is 1.

From [18] and [19], it is evident that BVP (1) has an integral formulation given by

where

It is easy to see that

Define operators

Then

It is clear from (2) that solving BVP (1) is equivalent to finding fixed points of
the operator
*x* is a fixed point of *B* iff *x* is a solution of the following BVP:

and *x* is a fixed point of *λA* iff *x* is a solution of the following cantilever beam problem:

**Lemma 2.1***If*
*satisfies*

*then*

(i)
*is nondecreasing in*
*moreover*,

(ii)
*that is*,
*is a convex function on*

*Proof* From (6), we have

Now, let

then, it is easy to show that
*E*, and if

**Lemma 2.2**

*Proof*

By Lemma 2.1,

that is,

**Lemma 2.3**

(i)
*is a completely continuous operator*;

(ii) *if*
*is nondecreasing*, *then*
*is a completely continuous operator*.

*Proof* Similarly to the proof of Theorem 1 in [19], applying the Arzela-Ascoli Theorem, the proof can be completed. □

From the proof of Lemma 2.2 we can show the following result.

**Theorem 2.4***If*
*is a solution for BVP* (1), *then**x**is a convex monotone positive solution for BVP* (1).

So, in the following sections, we only need to study solutions for BVP (1) in

### 3 Existence and nonexistence results

It is obvious from Lemma 2.2 that if
*K*. The following conditions will be assumed:

(H1)

(H2)

(H3)

(H4)

(H5)

(H6)

Set

and

**Lemma 3.1***Suppose that* (H1)-(H3) *hold*. *If*
*then*

*Proof*

Let

**Theorem 3.2***Suppose that* (H1)-(H3) *hold*.

(i) *If* (H4) *holds*, *then*
*has minimal and maximal fixed points in*
*for*
*Moreover*, *there exists*
*such that*
*has at least one and has no fixed points in*
*for*
*and*
*respectively*.

(ii) *If*
*then when*
*there exists*
*such that*
*has at least one and no fixed points in*
*for*
*and*
*respectively*; *when*
*has at least one fixed point in*
*for*

*Proof* (i) From (H1), (H3), and (H4) we have

Set

Lemma 2.3 implies that

(ii) For any

Similarly to the proof of Lemma 3.1, we can show

**Lemma 3.3***Suppose that* (H1)-(H3) *hold and that one of* (H5) *and* (H6) *holds*. *If* Λ *is nonempty*, *then*

(i) Λ *is bounded from above*, *that is*,

(ii)

*Proof* (i) Suppose to the contrary that there exists an increasing sequence

Case 1.

which is a contradiction.

Case 2.

When (H5) holds, take
*K*, we have

which is a contradiction.

When (H6) holds, choose

Moreover,

which is a contradiction.

Consequently, we find that Λ is bounded from above.

(ii) By the definition of
*K*, that is, there exists a constant

we see that
*K*. Consequently, by an application of the Arzela-Ascoli Theorem we conclude that
*K*. So, there exists a subsequence

By taking the limit we have

**Theorem 3.4***Suppose that* (H1)-(H4) *hold and that one of* (H5) *and* (H6) *holds*. *Then*, *there exists a*
*such that BVP* (1) *has at least two*, *one*, *and no positive solutions for*
*and*
*respectively*.

*Proof* Theorem 3.2 implies

Now, given

Let
*i.e.*,

When (H5) holds, take

When (H6) holds, from the proof of Lemma 3.3 we can set

Consequently, in virtue of Lemma 1.1 we find that

Equation (9) implies that
*K* with

**Remark 3.1** In the above results, we can replace (H5) with the following condition: there exists

In the following, we give some sufficient conditions that BVP (1) has no positive solutions.

**Theorem 3.5***Suppose that there exists a nonnegative integrable function*
*such that*
*and*
*Then BVP* (1) *has no positive solutions for*

*Proof* Assume to the contrary that

Similarly to the proof of Theorem 3.5, we can easily obtain the following results.

**Theorem 3.6***Suppose that there exist an integrable function*
*and a number*
*such that*
*and*
*Then BVP* (1) *has no positive solutions for*

**Theorem 3.7***Suppose that*
*Then BVP* (1) *has no positive solutions for*

**Remark 3.2** When
*q* in Theorems 3.2, 3.4-3.6 and obtain the following corresponding results for BVP (5).

Suppose that (H1) and (H3) hold. Then BVP (5) has minimal and maximal solutions in

Suppose that (H1), (H3), and (H5) hold. Then

Under the conditions in Theorem 3.5, BVP (5) has no positive solutions for

Suppose that

**Remark 3.3** (i) We give an example to illustrate Theorem 3.2. Let

By straightforward calculations we see that

We give another example to illustrate Theorem 3.4. Let

A straightforward calculation can show that

(ii) In Theorems 3.5-3.7, we do not require *f* and *q* to be monotone in *x*. For example, let

Take

### 4 Uniqueness and dependence on parameter

In this section, we will apply cone theory to further study the uniqueness of solution
for BVP (1) in
*λ*. The following hypotheses are needed:

(H7)

(H8)

(H9) for all

**Remark 4.1** The inequalities in (H7), (H8), and (H9) are equivalent to the following inequalities,
respectively:

Let

**Remark 4.2** (H2) and (H7) imply

**Remark 4.3** Let

we conclude

So, in this section, we only need to consider the unique solution for BVP (1) in

**Lemma 4.1***Assume that* (H2) *and* (H7) *hold*. *Then**B**has a unique fixed point*
*in*
*moreover*, *constructing successively the sequence*
*for any initial value*
*we have*

*Proof* For any

**Lemma 4.2***Assume that* (H1), (H2), (H7), *and* (H8) *hold*. *Then*

(i)
*is an increasing operator*;

(ii) *for any*
*and*
*there exists*
*such that*

(iii) *for*
*and*
*there exists*
*such that*

*Proof* The conclusion (i) follows from (H1), (H2), (H7), and (4).

The proof of (ii). For given

Let

then

The proof of (iii). For any

Moreover, from (H7) and (H8) we have

where

**Lemma 4.3***Assume that* (H1), (H2), (H7), *and* (H8) *hold*. *Then*
*has a unique fixed point*
*in*
*iff there exists*
*such that*
*Moreover*, *constructing successively the sequence*
*for any initial value*
*we have*

*Proof* ‘⇒’ Let
*i.e.*,

‘⇐’ By virtue of Lemma 4.1, *B* has a unique fixed point

Now, we are going to prove

Let

By the definition of

Set

Lemma 2.3 implies that

To prove that

then

From (16)-(18) we infer that

By (17), we have

which means that

Now, we prove that

It is evident that

Thus, from (19) we have

Finally, the iterative scheme and (12) can be proved in a similar way to the proof of Theorem 3.4 of [21], here it is omitted. The proof is complete. □

**Theorem 4.4***Assume that* (H1), (H2), (H7), *and* (H8) *hold*. *Then there exists a*
*such that BVP* (1) *has a unique solution*
*in*
*for*
*and does not have any solution in*
*for*
*Moreover*, *set*
*for any*
*then* (12) *holds*.

*Proof* By Lemma 4.1, *B* has the unique fixed point

Set

Similarly to the proof of Lemma 3.1, we can show that

Now, take

Let

Denote

Set

This means that

In what follows, we assume that
*B* in

**Theorem 4.5***Assume that* (H1), (H2), (H7), *and* (H8) *hold*. *Then*
*depends upon the parameter**λ**as follows*:

(i)
*is nondecreasing with respect to**λ**for*

(ii)
*is continuous with respect to**λ**for*

(iii)
*and*

*Proof* (i) Let

(ii) Let

By virtue of the above conclusion (i) we have

which implies that
*P*. Further, similarly to the proof of the conclusion (ii) in Lemma 3.3 we see that

By taking the limit we have

A similar argument can show that for any

(iii) It is obvious from the above conclusion (ii) that

In order to finish the proof of

Case 1.

Since

Case 2.

By the above conclusion (i) we have

Now, we give an estimate for critical value

Moreover,

**Theorem 4.6***Assume that* (H1), (H2), (H7), *and* (H8) *hold*. *Then*

*Proof* For any

Note that
*k* such that

Let

Moreover, taking

Consequently, from (21) we obtain

**Remark 4.4** Different from Theorems 3.2 and 3.4, the estimate of

**Corollary 4.7***Assume that* (H1), (H2), (H7), *and* (H9) *hold*. *Then*

(i) *BVP* (1) *has a unique positive solution*
*in*
*for*
*Moreover*, *for any*
*set*
*then*

(ii)
*is nondecreasing with respect to**λ**for*

(iii)
*is continuous with respect to**λ**for*

(iv)
*and*

*Proof* From (H1), (H2), (H7), and (4), we see that

where

From (H9), we have

When
*B* satisfies (H2) and (H7). So we can obtain the following two results.

**Corollary 4.8***Assume that* (H1) *and* (H8) *hold*. *If*
*then*

(i) *there exists*
*such that BVP* (1) *with*
*has a unique positive solution*
*in*
*for*
*and does not have any solution in*
*for*
*Moreover*, *for any*
*set*
*then*

(ii)
*is nondecreasing with respect to**λ**for*

(iii)
*is continuous with respect to**λ**for*

(iv)
*and*

**Corollary 4.9***Assume that* (H1) *and* (H8) *hold*. *If*
*then*

(i) *for any*
*BVP* (1) *with*
*has a unique positive solution*
*in*
*moreover*, *for any*
*set*
*then*

(ii)
*is nondecreasing in**λ**for*

(iii)