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:
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 , Bai , Bonanno and Bellaa , Li , Liu and Li , Liu , Ma and Xu , and Ma and Thompson  on an elastic beam whose two ends are simply supported, the works of Yang  and Zhang  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. 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  by variational methods and in  by a fixed point theorem, respectively; monotone positive solutions were obtained by using the monotone iteration method in . 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.
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 .
and x is a fixed point of λA iff x is a solution of the following cantilever beam problem:
Proof Similarly to the proof of Theorem 1 in , applying the Arzela-Ascoli Theorem, the proof can be completed. □
From the proof of Lemma 2.2 we can show the following result.
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:
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. □
Theorem 3.2Suppose that (H1)-(H3) hold.
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.
Lemma 3.3Suppose that (H1)-(H3) hold and that one of (H5) and (H6) holds. If Λ is nonempty, then
which is a contradiction.
which is a contradiction.
which is a contradiction.
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
In the following, we give some sufficient conditions that BVP (1) has no positive solutions.
Similarly to the proof of Theorem 3.5, we can easily obtain the following results.
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 .
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 .
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.
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:
Remark 4.1 The inequalities in (H7), (H8), and (H9) are equivalent to the following inequalities, respectively:
Lemma 4.2Assume that (H1), (H2), (H7), and (H8) hold. Then
Proof The conclusion (i) follows from (H1), (H2), (H7), and (4).
Moreover, from (H7) and (H8) we have
Now, we are going to prove
Finally, the iterative scheme and (12) can be proved in a similar way to the proof of Theorem 3.4 of , here it is omitted. The proof is complete. □
By virtue of the above conclusion (i) we have
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. □
Theorem 4.6Assume that (H1), (H2), (H7), and (H8) hold. Then
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
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
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.
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
The authors declare that they have no competing interests.
All authors participated in drafting, revising and commenting on the manuscript. All authors read and approved the final manuscript.
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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