Research

Positive solutions for elastic beam equations with nonlinear boundary conditions and a parameter

Wenxia Wang*, Yanping Zheng, Hui Yang and Junxia Wang

Author Affiliations

Department of Mathematics, Taiyuan Normal University, Taiyuan, 030012, P.R. China

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Boundary Value Problems 2014, 2014:80  doi:10.1186/1687-2770-2014-80

 Received: 7 October 2013 Accepted: 26 March 2014 Published: 9 April 2014

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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 such that the beam equation has at least two, one, and no positive solutions for , and , respectively; furthermore, by using cone theory we establish some uniqueness criteria for positive solutions for the beam and show that such solution depends continuously on the parameter 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:

(1)

where is a parameter. Throughout this paper, we assume that , , . is called a positive solution of BVP (1) if x is a solution of BVP (1) and , . A convex monotone positive solution means convex nondecreasing positive solution.

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 is called a cantilever beam equation, it describes the deflection of the elastic beam fixed at the left end and free at the right end. Existence and multiplicity of positive solutions of cantilever beam problems without parameter have been studied by some authors; see Yao [15,16] and references therein. BVP (1) with describes the deflection of the elastic beam fixed at the left end and attached to a bearing device given by the function −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 depends continuously on the parameter 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 is called a cone of E if it satisfies (i) , ; (ii) , . E is partially ordered by the cone P, i.e., iff . A cone P is said to be normal if there exists a positive number N, called the normal constant of P, such that implies . For , , denote .

For all , the notation means that there exist and such that . Clearly, ∼ is an equivalence relation. Given (i.e., and ), we denote by the set . It is easy to see that .

Let . An operator is said to be increasing if for , . An element is called a fixed point of T if .

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

Assume thatandare bounded open subsets ofEwith. Letbe a completely continuous operator such thatifandif. ThenThas a fixed point in.

Lemma 1.2[23]

LetPbe a normal cone inE, be increasing and for alland, there existssuch that. ThenThas a unique fixed pointin. Moreover, constructing successively the sequence () for any, we have.

2 Solutions

In what follows, set , the Banach space of all continuous functions on with the norm . . It is clear that 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

(2)

where

(3)

It is easy to see that and

(4)

Define operators by

Then , , and .

It is clear from (2) that solving BVP (1) is equivalent to finding fixed points of the operator . In particular, 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:

(5)

Lemma 2.1Ifsatisfies

(6)

then

(i) is nondecreasing in, moreover, , ;

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

Proof From (6), we have . Moreover, . So, . Thus, we complete the proof of the lemma. □

Now, let

then, it is easy to show that is also a cone in E, and if , then .

Lemma 2.2, , .

Proof implies , so and . Moreover, for ,

By Lemma 2.1, is convex and nondecreasing in . From (3) and (4) we have

that is, for . Thus, we obtain . From the above proof, we can show that and . This ends the proof. □

Lemma 2.3

(i) is a completely continuous operator;

(ii) ifis nondecreasing, thenis 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 2.4Ifis a solution for BVP (1), thenxis 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 is a solution for BVP (1) then . So in this section, we will apply Lemma 1.1 to study the existence, multiplicity and nonexistence of solutions for BVP (1) in . It is reasonable that the domain of is restricted on K. The following conditions will be assumed:

(H1) is nondecreasing in for fixed ;

(H2) is nondecreasing in ;

(H3) ;

(H4) ;

(H5) ;

(H6) .

Set

(7)

and .

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

Proof means that there exists such that . Therefore, for any , we have . Set , ,  . From (H1) and (H2) we obtain . By Lemma 2.3 and (H3), converges to a fixed point of in . Thus . This completes the proof. □

Let , , , and

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

(i) If (H4) holds, thenhas minimal and maximal fixed points infor. Moreover, there existssuch thathas at least one and has no fixed points inforand, respectively.

(ii) If, , then when, there existssuch thathas at least one and no fixed points inforand, respectively; when, has at least one fixed point infor.

Proof (i) From (H1), (H3), and (H4) we have . For any , we obtain

Set , ,  , then from (H1) and (H2) we have

(8)

Lemma 2.3 implies that and converge to fixed points and of , respectively. From (8) it is evident that are the minimal fixed point and maximal fixed point of in , respectively. From the definition of we can complete the rest of the proof.

(ii) For any , there exists such that and for , . Let and , then and

Similarly to the proof of Lemma 3.1, we can show . The conclusion (ii) follows from Lemma 3.1 and the definition of . This completes the proof of Theorem 3.2. □

Lemma 3.3Suppose 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 such that . Set is a fixed point of , that is, . There are two cases to be considered.

Case 1. is bounded, that is, there exists a constant such that for  . Hence, from (H1), (H3), and (4) we have

Case 2. is unbounded, that is, there exists a subsequence of , still denoted by , such that .

When (H5) holds, take , there exists such that for , . Choose such that . Thus, , . Moreover, from (H1) and the definition of K, we have

When (H6) holds, choose such that . There exists such that for . Choose such that , so

Moreover,

Consequently, we find that Λ is bounded from above.

(ii) By the definition of , there exists a nondecreasing sequence such that . Let be a fixed point of . Arguing similarly as above in case 2, we can show that is a bounded subset in K, that is, there exists a constant such that ,  ; on the other hand, note that

we see that is an equicontinuous subset in K. Consequently, by an application of the Arzela-Ascoli Theorem we conclude that is a relatively compact set in K. So, there exists a subsequence converging to . Note that

By taking the limit we have , that is, . The proof is complete. □

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

Proof Theorem 3.2 implies , so . From Lemmas 3.1 and 3.3, we have . Therefore, from the definition of we only to prove that has at least two fixed points in for .

Now, given . Theorem 3.2 means that has at least one fixed point which satisfies .

Let . Note that for , so for with , i.e., , we have

(9)

When (H5) holds, take , there exists such that for , . Set . Then . If , we have

When (H6) holds, from the proof of Lemma 3.3 we can set . Then . If , we have .

Consequently, in virtue of Lemma 1.1 we find that has another fixed point with

Equation (9) implies that has no fixed points in . In conclusion, for , has at least two fixed points and in K with . The proof is complete. □

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

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

Theorem 3.5Suppose that there exists a nonnegative integrable functionsuch that, , and. Then BVP (1) has no positive solutions for.

Proof Assume to the contrary that is a solution of BVP (1), then , which is a contradiction. The proof is complete. □

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

Theorem 3.6Suppose that there exist an integrable functionand a numbersuch that, , , and. Then BVP (1) has no positive solutions for.

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

Remark 3.2 When , BVP (1) becomes a cantilever beam problem (5). In this case, we can delete the conditions on 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 for . Further, if , then there exists such that BVP (5) has at least one and has no positive solutions for and , respectively; if then BVP (5) has at least one positive solution for .

Suppose that (H1), (H3), and (H5) hold. Then and BVP (5) has at least two, one and has no positive solutions for , and , respectively.

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

Suppose that and satisfy the conditions in Theorem 3.6, then BVP (5) has no positive solutions for .

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

By straightforward calculations we see that , , , , , and . So the conditions in Theorem 3.2 are satisfied. Therefore, by Theorem 3.2 we find that there exists such that BVP (1) has minimal and maximal solutions in for , has at least one positive solution for and has no positive solutions for , where and .

We give another example to illustrate Theorem 3.4. Let , and

A straightforward calculation can show that , , , , and Therefore, the conditions (H1)-(H5) hold. Thus, by Theorem 3.4 we see that there exists such that BVP (1) has at least two, one, and no positive solutions for , , and , respectively.

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

Take , , then the conditions in Theorem 3.6 are satisfied and . So by Theorem 3.6 we find that BVP (1) has no positive solutions for .

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 and the dependence of such a positive solution on the parameter λ. The following hypotheses are needed:

(H7) and for all and , there exists such that ;

(H8) and for , , ;

(H9) for all , and , there exists such that .

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

Let and define as in Section 1. It is obvious that and if then and , .

Remark 4.2 (H2) and (H7) imply for . Moreover, for .

Remark 4.3 Let be a solution for BVP (1) in . If (H2) and (H7) hold, then . Indeed, from Theorem 2.4 we have . So Remark 4.2 implies . Note that

we conclude .

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

Lemma 4.1Assume that (H2) and (H7) hold. ThenBhas a unique fixed pointin, moreover, constructing successively the sequence () for any initial value, we have.

Proof For any , we have , which means . For all , , from (H7) we have . Consequently, the conclusion follows from Lemma 1.2. This completes the proof. □

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

(i) is an increasing operator;

(ii) for anyand, there existssuch that;

(iii) forand, there existssuch that

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

The proof of (ii). For given , , from (H1) and (4) we have

Let

(10)

then and

(11)

The proof of (iii). For any , , from (10) and (11) we have

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

where . This completes the proof. □

Lemma 4.3Assume that (H1), (H2), (H7), and (H8) hold. Thenhas a unique fixed pointiniff there existssuch that. Moreover, constructing successively the sequence () for any initial value, we have

(12)

Proof ‘⇒’ Let be a fixed point of in , i.e., . Taking , we obtain .

‘⇐’ By virtue of Lemma 4.1, B has a unique fixed point in . Moreover,

(13)

Now, we are going to prove

(14)

Let , then . Otherwise, , from Lemma 4.2 we have

By the definition of , we get a contradiction . Thus, (14) holds.

Set , , ,  . From (13) and (14) we have

(15)

Lemma 2.3 implies that and converge to fixed points and of , respectively. From (15), we have

(16)

To prove that has only one fixed point in , let

(17)

then

(18)

From (16)-(18) we infer that , which means that . We assert that . Otherwise, for , then by Lemma 4.2 we deduce that

By (17), we have , moreover, , which is a contradiction. So . Thus, by (16) and (18) we have

which means that is the unique fixed point of in .

Now, we prove that is the unique fixed point of in . By the above proof, we only need to show that does not have any fixed point in . If is a fixed point of in . Let

(19)

It is evident that . If , then . By Lemma 4.2 we have

Thus, from (19) we have , which is a contradiction. So . Moreover, , which implies the contradiction: and .

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

Proof By Lemma 4.1, B has the unique fixed point in . So , moreover, . Let , we have

(20)

Set . Lemma 4.3 implies that

(21)

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

Now, take and let and , then and . By (H7), (H8), and (20), we have , that is, . Moreover, .

Let , then . We assert that . Indeed, if , from the definition of it is obvious that . Suppose that and . Then by (14) and (21) there exists such that . Similarly to the proof of (20), we have

Denote , then

(22)

Set for given , then . Since , we can choose such that . Therefore, from (22) we have

This means that , which is a contradiction to the definition of . So, . Consequently, an application of Lemma 4.3 completes the proof. □

In what follows, we assume that is the unique fixed point of B in , is the unique fixed point of in and .

Theorem 4.5Assume that (H1), (H2), (H7), and (H8) hold. Thendepends upon the parameterλas follows:

(i) is nondecreasing with respect toλfor;

(ii) is continuous with respect toλfor;

(iii) and.

Proof (i) Let with . Since , from the proof of Lemma 4.3, we find that the unique fixed of belongs to , which means that .

(ii) Let . In order to prove , let sequence satisfy

By virtue of the above conclusion (i) we have

(23)

which implies that is a bounded subset in P. Further, similarly to the proof of the conclusion (ii) in Lemma 3.3 we see that converges to . From (23) we have , which leads to . Note that

By taking the limit we have . Since has only one fixed point in , then . This means that as .

A similar argument can show that for any , as . Thus, the proof of (ii) is complete.

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

In order to finish the proof of , we consider two cases.

Case 1. .

Since , then , which means .

Case 2. .

By the above conclusion (i) we have . Suppose to the contrary that . Similarly to the case 2 in the proof of Lemma 3.3, we conclude that has a fixed point . From Remark 4.3 we have . So , which is a contradiction. This ends the proof. □

Now, we give an estimate for critical value in Theorem 4.4. If (H1) and (H8) hold, then

Moreover, .

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

(24)

Proof For any , there exists such that

(25)

Note that , we can choose a sufficiently large positive integer number k such that , that is,

(26)

Let , then, from (4), (25), and (26) we have

Moreover, taking , we have

Consequently, from (21) we obtain , that is, , which implies that (24) holds. This completes the proof. □

Remark 4.4 Different from Theorems 3.2 and 3.4, the estimate of in Theorem 4.6 does not take into account effect of . This is valuable, because the conditions (H2) and (H7) cannot ensure as . Certainly, if , then . In particular, if , then .

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

(i) BVP (1) has a unique positive solutioninfor. Moreover, for any, set (), then;

(ii) is nondecreasing with respect toλfor;

(iii) is continuous with respect toλfor;

(iv) and.

Proof From (H1), (H2), (H7), and (4), we see that is increasing for any given . Further, for any given we have

where . Thus, the conclusion (i) follows from Lemma 1.2.

From (H9), we have for , and . Therefore, in the same way as in the proof of Theorem 4.5, we can complete the rest of the proof. □

When is a constant function, and . It is evident that B satisfies (H2) and (H7). So we can obtain the following two results.

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

(i) there existssuch that BVP (1) withhas a unique positive solutioninforand does not have any solution infor. Moreover, for any, set (), then;

(ii) is nondecreasing with respect toλfor;

(iii) is continuous with respect toλfor;

(iv) and.

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

(i) for any, BVP (1) withhas a unique positive solutionin, moreover, for any, set (), then;

(ii) is nondecreasing inλfor;

(iii) is continuous with respect toλfor;

(iv) and.

Corollary 4.10Assume that (H1) and (H9) hold. Then the conclusions (i), (ii), (iii), and (iv) in Corollary 4.9 hold.

Finally, we give two concrete examples to illustrate those results in the section.

Example 1 In BVP (1), let

it is obvious that the conditions (H1) and (H2) are satisfied. For any ,

as , we have ;

as and , we have ;

as and , we have ,

that is, for and . Similarly, we can obtain for . Therefore, the conditions (H7) and (H8) are satisfied. Note that

By Theorems 4.4, 4.5 and Remarks 4.3, 4.4 we see that there exists such that BVP (1) has a unique positive solution for and does not have any positive solution for . Moreover, for any , set (), then , and such solution satisfies the properties (i), (ii), and (iii) in Theorem 4.5.

Example 2 In BVP (1), let , , , , it is easy to see that (H1) holds. For any , we have

So, (H8) holds. Note that

by Corollary 4.9 we find that BVP (1) has a unique positive solution for . Moreover, for any , set (), then , and such a solution satisfies the properties (ii), (iii), and (iv) in Corollary 4.9 with .

In this example, by using Wolfram Mathematica 9.0, we can plot the graphs of solutions for BVP (1) with , as the Figure 1 shows.

Figure 1. Solutions for BVP (1) withand

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors participated in drafting, revising and commenting on the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments. This research was supported by the NNSF of China (11361047), the University Natural Science Research Develop Foundation of Shanxi Province of China (20111021, 2013156), Research Project Supported by Shanxi Scholarship Council of China (2013-102) and the Science Foundation of Qinghai Province of China (2012-Z-910).

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