Monotone Iterative Technique for a Class of Slanted Cantilever Beam Equations

Abstract

In this paper, we deal with the existence and uniqueness of the solutions of two-point boundary value problem of fourth-order ordinary differential equation: ,  ,   where is a continuous function. The problem describes the static deformation of an elastic beam whose left end-point is fixed and right is freed, which is called slanted cantilever beam. Under some weaker assumptions, we establish a new maximum principle by the perturbation of positive operator and construct the monotone iterative sequence of the lower and upper solutions, and, based on this, we obtain the existence and uniqueness results for the slanted cantilever beam.

1. Introduction

In mechanics, the two-point boundary value problems of fourth-order ordinary differential equations are mainly used to describe the static deformation of elastic beam under external force, and especially a model to study travelling waves in suspension bridges can be furnished by the fourth-order equation of nonlinearity. Due to the different support conditions of elastic beams, a variety of boundary value problems are derived; see [1].

In this paper, we deal with the existence and uniqueness results of solutions to the two-point boundary value problem of fourth-order ordinary differential equation where is a continuous function. The problem is called slanted cantilever beam which describes the static deformation of an elastic beam whose left end-point is fixed and right is freed. For the equation, the physical meaning of the first-order derivative of unknown function is the slope, which reflects the curving degree of the elastic beam; see [1–5].

There are many results on the cantilever beam equation; see [4–17]. Specially, in [4, 14, 15], Agarwal et al. used the fixed point theorems of cone mapping to research the special case of BVP (1) that the nonlinear term does not contain the derivative term , namely, In these works, since there is no derivative term, the research about the solutions is simple and feasible relatively.

In [6, 7, 10, 16, 17], for the fourth-order ordinary differential equation with the boundary value condition ,  , which means that the left end of the beam is fixed and the right is attached to a bearing device, the existence and multiplicity of solutions have been discussed by using the variational methods and critical point theory.

For the case of BVP (1), in [11], Yao constructed a successively iterative sequence by using the monotone iterative technique and applying the successively approximate method to prove an existence theorem. Recently, in [5], by using the fixed point index theory in cones, Li researched the existence of positive solutions of cantilever beam equation in which the nonlinear term contains all order derivatives of unknown function.

However, there are still many limitations in the study of this problem in recent years. First of all, most conclusions of the existences were obtained only by roughly estimating the properties of the corresponding Green function; secondly, most of the conditions for nonlinear term are very harsh, so the existence results of the solutions are not optimal.

For the solvability of elastic beam equations with other types of boundary conditions, many results have been obtained; see [18–24] and references therein. Specially, in [23], Li dealt the fourth-order boundary value problem and obtained the existence and uniqueness of solutions by utilizing the perturbation of positive operator and the monotone iterative technique of upper and lower solutions. It is well known that the monotone iterative method of lower and upper solutions has been widely used in solving the boundary value problem of ordinary differential equations. However, as far as we know, no researchers studied BVP (1) by monotone iterative method of lower and upper solutions.

Motivated by the papers mentioned above, we will use the monotone iterative technique of lower and upper solutions to discuss the existence and uniqueness of BVP (1). It is well known that the theoretical basis of the monotone iterative technique is the maximum principle. It often requires two aspect of works for this method. One is to construct the iterative sequence and judge its monotonicity, and the other is to verify the convergence of the constructed sequence. Generally, For the case of BVP (2), the nonlinear item , if the linear differential operator at the left satisfies the maximum principle, then the monotone iterative technique is feasible; see [18–20]. However, in BVP (1), the nonlinear term contains the derivative; the general maximum principle cannot guarantee the monotonicity of the iterative sequence. Therefore, in order to ensure the feasibility of the monotone iterative technique, we should strengthen the maximum principle.

The purpose of this paper is to construct a new maximum principle for fourth-order differential operator where are constants satisfying and establish the monotone iterative technique in the case of the lower and upper solutions existing in BVP (1). To the best of our knowledge, using this method to solve the problem of the solvability of cantilever beam equation is rare. It means that our conclusions are new and meaningful.

The paper is organized as follows. Section 2 provides the preliminary results which are used in theorems stated and proved in the article, and Section 3 presents the main results and its proof of the article.

2. Preliminaries

In this section, we introduce some basic concepts and preliminary facts which are used in this paper.

Let , , be a continuous function space endowed the maximum norm , are -order continuous differentiable function spaces which are defined in , and denote a cone in the form of all nonnegative functions in . Evidently,

Let constants satisfy the expression (5). In order to study the existence of solutions of the BVP (1), we establish a new maximum principle for the differential operator (4). To this end, we consider the corresponding fourth-order linear boundary value problem (LBVP) Assume that ; then we have Evidently, Therefore, the fourth-order LBVP (6) is equivalent to the following third-order boundary value problem:

We have known that, for any , the third-order linear boundary value problem has a unique solution , which can be expressed as where is the Green function of LBVP (9) given by the following expressions: Clearly, is continuous, and the following lemma is established.

Lemma 1. has the following properties:(a), for any (b), for any (c), for any

Proof. From the expression of (11), it follows that (a) holds.
(b) For , we have For , (c) From the expression (11), for any holds obviously.
This completes the proof of Lemma 1.

From Lemma 1, we can obtain the following result which is needed in the proof of our main results.

Lemma 2. The solution operator of LBVP (9) is the completely continuous linear operator, and its norm satisfies Furthermore, if , then , for every

Proof. From (10), we can easily obtain that the solution operator of LBVP (9) is a completely continuous linear operator.
For any , by (10) and (11), we obtain It is easy to see that , which implies that holds.
Furthermore, for , from (10) and the second inequality of Lemma 1(b), we get It follows that From Lemma 1(b), we have This completes the proof of Lemma 2.

In order to establish the new maximum principle, we also need to prove the following lemma.

Lemma 3. Let there exist constants and satisfying the assumption (5); then LBVP (6) has a unique solution for any , and the solution operator is completely continuous. Specifically, when , then the solution satisfies

Proof. According to the above analysis, if there exists the unique solution of LBVP (8), then is the unique solution of LBVP (6). By the Lemma 2, LBVP (8) is equivalent to the operator equation where is the unit operator in . By Lemma 2, it follows that Therefore, creates bounded inverse operator. According to the Neumann expansion, we can obtain that its norm satisfies Therefore, the operator equation (18) has a uniqueness solution Thus, LBVP (6) has the uniqueness solution , where Since the operator is completely continuous and is a bounded linear operator, then the operator is completely continuous. Thus, according to the boundedness of , we can get that is a completely continuous operator.
Now, we prove that, for any , the solution of LBVP (6) satisfies
Since and are the positive operators in , and , then from the definition of operator , we have , and by Lemma 2(c), it is obvious that for any . By Lemma 2, we can obtain for any . Since and are the positive operators, we can obtain that Therefore, the solution of LBVP (6) satisfies , and This completes the proof of Lemma 3.

According to the conclusion of Lemma 3, the following maximum principle can be obtained.

Lemma 4. Let there exist constants and satisfying the assumption (5), if satisfies then for any .

3. Main Results

Now, we are in the position to state and prove our main results. We will apply monotone iterative method of the lower and upper solutions to obtain the existence and uniqueness of solutions for cantilever beam equation (1). To this end, we define the lower and upper solutions of BVP (1).

Definition 5. If satisfies then is called a lower solution of BVP (1). If the inequality of (27) is inverse, then is called an upper solution of BVP (1).

Theorem 6. Let be continuous, and there are lower and upper solutions and for BVP (1), satisfying If satisfies the following condition:(F1) there exist positive constants and satisfying (5), such that  for arbitrary , then BVP (1) has one maximal solution and minimal solution between and

Proof. Let , Clearly, is a bounded nonempty convex closed set in .
For any , we define an operator as follows: According to the continuity of , it is easy to see that is the continuous bounded operator in . Let be the solution operator of LBVP (6); then the solution of BVP (1) in is equivalent to the fixed point of the composition operator . We can easily obtain that operator as completely continuous by the complete continuity of and the boundedness of . In the following, we will take four steps to prove the conclusion.
Step  1. We prove that .
To this end, we let for every And define , and then is the solution of LBVP (6). Thus, satisfy Then by the definition of the lower and upper solutions and the assumption (F1), it is clear that By the boundary conditions, we can get that Applying Lemma (10) to and , we have which means ,   in . Therefore, we can conclude that
Step  2. We show that if satisfy , then holds.
In fact, similar to the first step, let , and then by the assumption (F1), we can obtain By the boundary conditions, we can get that then applying Lemma (10) to , we have which means that
Step  3. We demonstrate that there exist solutions between and .
We use and as the initial element for constructing iterative sequence According to the definition of the operator , Steps  1 and 2, we can easily see that which means that and are monotone increasing and decreasing in the order interval , respectively; and are also monotonous in the order interval .
By the compactness of , we know that are the relatively compact sets in , and, therefore, they have the uniformly convergent subsequence in . Then by (38), are all uniformly convergent in ; therefore, and are uniformly convergent in , which means there exist and , such that Since is a convex closed set in , it is obvious that In the expression (37), we let , and then, from the continuity of , we can easily see , for any Thus, and are the solutions of BVP (1) between and .
Step  4. We testify that and are the minimal and maximal solutions of BVP (1) between and , respectively.
Let be an arbitrary solution of LBVP (6); then satisfies By Step  2, using acting times for the last expression, it can be easily obtained that Taking , we can see It can be easily obtained that and are the minimum and maximum solutions of BVP (1) between and , respectively.
This completes the proof of Theorem 6.

From the above proof process, the next corollary can be easily obtained.

Corollary 7. Let be continuous, and there exist lower and upper solutions and for BVP (1), satisfying If satisfies the assumption (F1), we use and as the initial elements to construct iterative sequences and by linear iterative equation then we can obtain that uniformly hold for arbitrary , where and are the minimal and maximal solutions of BVP (1) in the set respectively.

Theorem 6 gives the existence of the solution of BVP (1). Now, we can further discuss the uniqueness result of the solutions by strengthening the assumption (F1).

Theorem 8. Let be continuous, and there exist lower and upper solutions and for BVP (1), satisfying If satisfies the assumption (F1) and the following condition: (F2) there exist positive constants and satisfying  such that  for every , , then BVP (1) has a unique solution in , and, for every , the monotone iterative sequence constructed by (42) uniformly converges to the unique solution .

Proof. By the proof of Theorem 6, when the assumption (F1) holds, then the BVP (1) has maximal solution and minimal solution in , and for every solution , we have Next, we need to prove that
According to the proof of Lemma (9), the operator defined by (22) is a positive linear operator, and its norm satisfies Since , for any , we have
Assuming that are the monotone iterative sequences constructed in Theorem 6, by the assumptions (F1) and (F2) and the positivity of operator , we can see which implies that Therefore, we can get By (47) and the assumption F2, we have Thus, we have Therefore, by the conclusions of Corollary 7, we can obtain Consequently, is the unique solution of BVP (1).
Now, we need to testify that the monotone iterative sequence constructed by (42) uniformly converges to the unique solution .
Assuming , then the monotone iterative sequence used as the initial element constructed by (42) satisfying . According to Step  2 of the proof process of Theorem 6, it is easy to see that Taking , it follows that in . Therefore, the conclusion is established.

Finally, we give a numerical example to illustrate our theoretical results.

Example 1. Consider the following nonlinear problem: Clearly, is a lower solution of problem (55). Letting , we can obtain that it is means that is a upper solution of problem (55).
On the other hand, for arbitrary , when , and , we can easily obtain which implies that satisfies the condition (F1) for Then, by Theorem 6, problem (55) has at least one maximal solution and minimal solution between and .
Furthermore, it is obvious that which implies that satisfies the condition (F2) for Then, by Theorem 8, the problem (55) has a unique solution which satisfies .

Conflicts of Interest

The authors declare that they have no conflicts of interest.