Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 5707623, 8 pages
https://doi.org/10.1155/2017/5707623

Research Article

Monotone Iterative Technique for a Class of Slanted Cantilever Beam Equations

Mei Wei and Qiang Li

Department of Mathematics, Shanxi Normal University, Linfen 041000, China

Correspondence should be addressed to Qiang Li; lznwnuliqiang@126.com

Received 19 August 2017; Revised 1 December 2017; Accepted 13 December 2017; Published 31 December 2017

Academic Editor: Dan Simon

Copyright © 2017 Mei Wei and Qiang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

References

C. P. Gupta, “Existence and uniqueness theorems for the bending of an elastic beam equation,” Applicable Analysis: An International Journal, vol. 26, no. 4, pp. 289–304, 1988.

A. R. Aftabizadeh, “Existence and uniqueness theorems for fourth-order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 116, no. 2, pp. 415–426, 1986.

C. P. Gupta, “Existence and uniqueness results for the bending of an elastic beam equation at resonance,” Journal of Mathematical Analysis and Applications, vol. 135, no. 1, pp. 208–225, 1988.

R. P. Agarwal and D. O'Regan, “Multiplicity results for singular conjugate, focal, and problems,” Journal of Differential Equations, vol. 170, no. 1, pp. 142–156, 2001.

Y. Li, “Existence of positive solutions for the cantilever beam equations with fully nonlinear terms,” Nonlinear Analysis: Real World Applications, vol. 27, pp. 221–237, 2016.

A. Cabada and S. Tersian, “Multiplicity of solutions of a two point boundary value problem for a fourth-order equation,” Applied Mathematics and Computation, vol. 219, no. 10, pp. 5261–5267, 2013.

A. Hadjian and M. Ramezani, “Existence of infinitely many solutions for fourth-order equations depending on two parameters,” Electronic Journal of Differential Equations, vol. 117, pp. 1–10, 2017.

D. R. Anderson and J. Hoffacker, “Existence of solutions for a cantilever beam problem,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 958–973, 2006.

Q. A. Dang and T. K. Ngo, “Existence results and iterative method for solving the cantilever beam equation with fully nonlinear term,” Nonlinear Analysis: Real World Applications, vol. 36, pp. 56–68, 2017.

G. Bonanno, A. Chinnì, and S. A. Tersian, “Existence results for a two point boundary value problem involving a fourth-order equation,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2015, pp. 1–9, 2015.

Q. Yao, “Monotonically iterative method of nonlinear cantilever beam equations,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 432–437, 2008.

Q. Yao, “Solvability of singular cantilever beam equation,” Annals of Differential Equations. Weifen Fangcheng Niankan, vol. 24, no. 1, pp. 93–99, 2008.

Q. Yao, “Local existence of multiple positive solutions to a singular cantilever beam equation,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 138–154, 2010.

R. P. Agarwal and D. O'Regan, “Twin solutions to singular boundary value problems,” Proceedings of the American Mathematical Society, vol. 128, no. 7, pp. 2085–2094, 2000.

R. P. Agarwal, D. O'Regan, and V. Lakshmikantham, “Singular focal and higher order boundary value problems,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 42, no. 2, Ser. A: Theory Methods, pp. 215–228, 2000.

L. Yang, H. Chen, and X. Yang, “The multiplicity of solutions for fourth-order equations generated from a boundary condition,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1599–1603, 2011.

Y. Song, “A nonlinear boundary value problem for fourth-order elastic beam equations,” Tijdschrift voor Urologie, vol. 2014, no. 1, pp. 1–11, 2014.

A. Cabada, “The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 185, no. 2, pp. 302–320, 1994.

A. Cabada, J. A. Cid, and L. s. Sanchez, “Positivity and lower and upper solutions for fourth order boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1599–1612, 2007.

C. De Coster and L. Sanchez, “Upper and lower solutions, ambrosetti-prodi problem and positive solutions for fourth order O.D.E,” Rivista di Matemàtica Pura ed Applicata, no. 14, pp. 57–82, 1994.

R. Ma, J. Zhang, and S. Fu, “The method, of lower and upper solutions for fourth-order two-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 215, pp. 415–422, 1997.

Y. S. Yang, “Fourth-order two-point boundary value problems,” Proceedings of the American Mathematical Society, vol. 104, no. 1, pp. 175–180, 1988.

Y. Li, “A monotone iterative technique for solving the bending elastic beam equations,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2200–2208, 2010.

Z. Bai, “The upper and lower solution method for some fourth-order boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 6, pp. 1704–1709, 2007.