Monotone Iterative Technique for a Class of Slanted Cantilever Beam Equations

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: u(4)(t) = f(t, u(t), u󸀠(t)), t ∈ [0, 1], u(0) = u󸀠(0) = u󸀠󸀠(1) = u󸀠󸀠󸀠(1) = 0, where f : [0, 1] × R → R 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.


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 fourthorder equation of nonlinearity.Due to the different support conditions of elastic beams, a variety of boundary value problems are derived; see [1].
In [6,7,10,16,17], for the fourth-order ordinary differential equation with the boundary value condition (0) =   (0) =   (1) = 0,   (1) = ((1)), 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 2 Mathematical Problems in Engineering 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][19][20][21][22][23][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][19][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.

Preliminaries
In this section, we introduce some basic concepts and preliminary facts which are used in this paper.
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 V() =   (); then we have Evidently, ‖ 0 ‖ = 1.Therefore, the fourth-order LBVP ( 6) is equivalent to the following third-order boundary value problem: We have known that, for any ℎ ∈ (), the third-order linear boundary value problem has a unique solution V ∈  3 (), 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.
This completes the proof of Lemma 1.
Proof.According to the above analysis, if there exists the unique solution V ∈  3 () of LBVP (8), then  =  0 V ∈  4 () 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,  +  +  0 creates bounded inverse operator.
Since  0 and  are the positive operators in (), and ℎ ≤ ‖ℎ‖, then from the definition of operator  0 , we have  0 ℎ ≤ ‖ℎ‖, and by Lemma 2(c), it is obvious that Mathematical Problems in Engineering for any  ∈ .By Lemma 2, we can obtain for any  ∈ .Since  0 and  are the positive operators, we can obtain that Therefore, the solution of LBVP ( 6) satisfies  =  0 ℎ ≥ 0, and   = ℎ ≥ 0. This completes the proof of Lemma 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).
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  :  × R × R → R 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 then BVP ( 1) has one maximal solution  and minimal solution  between  and .
Clearly,  is a bounded nonempty convex closed set in  1 ().
For any  ∈ , we define an operator  :  → () as follows: () () =  (,  () ,   ()) +   () +  () . (29) 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  =  ∘  :  →  1 ().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 2. We show that if In fact, similar to the first step, let  1 =  1 ,  2 =  2 , and then by the assumption (F1), we can obtain By the boundary conditions, we can get that then applying Lemma (10) to  2 −  1 , we have which means that  2 ≥  1 , ( 2 )  ≥ ( 1 )  .
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 [  ,   ].
Step 4. We testify that  and  are the minimal and maximal solutions of BVP (1) between  and , respectively.
(39) 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  :  × R × R → R 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 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  :  × R × R → R 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  1 and  2 satisfying then BVP (1) has a unique solution  * in , and, for every  0 ∈ , 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  = .