Abstract
The aim of the present paper is to consider a fully elastic beam equation with left-end fixed and right-end simply supported, i.e., , where is a continuous function. By applying Leray–Schauder fixed point theorem of the completely continuous operator, the existence and uniqueness of solutions are obtained under the conditions that the nonlinear function satisfies the linear growth and superlinear growth. For the case of superlinear growth, a Nagumo-type condition is introduced to limit that is quadratical growth on at most.
1. Introduction
This paper focuses on the existence and uniqueness of solutions for the boundary value problems (BVP) of the fourth-order ordinary differential equation:where is continuous and involves all derivatives below the fourth order of unknown functions. The problem simulates the deformation of an elastic beam with the left-end fixed and the right simply supported.
The boundary value problems for the nonlinear fourth-order differential equations are the mathematical models used to characterize the deflection of elastic beams under external forces. Elastic beams are one of the most basic structures in architecture and engineering, and some different boundary conditions are derived due to the diversity of stress states on its two ends.
In the past few years, owing to its actual mathematical model and wide application background, the research on nonlinear fourth-order two-point BVP has been very active. Its solvability has attracted the close attention of many scholars, and some profound results have been obtained through various nonlinear analysis methods and techniques (See [1–22] and its references).
There are many results on the solvability for the special cases of BVP (1) whose nonlinear term is independent of the third-order derivative term of (see [2, 8, 10, 15]). However, only few articles have studied the existence of solutions for BVP (1). It is worth noting that, in [12], Yao proved the existence of the solutions for BVP (1) by calculating the maximum value of Green function and its partial derivatives as well as constructing height functions of on bounded sets. In addition, the research on its solvability under some excellent growth conditions, especially the superlinear growth conditions, is even more rare.
In the mechanical analysis of beams, the physical meaning of the derivatives of are slope, bending moment, shear force, and load density, respectively (see [1, 2, 17–19]). The existence of slope, bending moment, shear force, and load density is undoubtedly very beneficial to the complete stress analysis of beams. Nevertheless, the dependence of on the third derivative increases the difficulty for our study, but this is also a fundamental difference from the previous problems. In recent years, the research on the solvability of the elastic beam equation that involves all lower-order derivatives of deformation function has become a hot topic (see [5, 6, 11–22]). For example, the elastic beam equation whose both ends are simply supported (see [14, 16, 20]):and one end is simply supported, and the other end is sliding clamped (see [5, 11, 22]):and one end is fixed, and the other end is free (see [13, 17–19, 21]):
The solvability of equations (2)–(4) has been studied by various nonlinear functional methods, including fixed point index theory in cone, fixed point theorem, lower and upper solutions’ method, topological degree theory, and variational method. In particular, in [16], Li and Liang researched the existence and uniqueness of solutions for BVP (2) under the condition that is linear growth; in [19], by supplementing the Nagumo-type condition to limit the growth of on , Li and Chen obtained the existence and uniqueness of solutions for BVP (4) under the condition that satisfies one-side superlinear growth.
Inspired by the literature listed above, in the present article, we discuss the existence and uniqueness of solutions for BVP (1) under the linear growth and one-side superlinear growth condition. The results of existence and uniqueness under the linear growth are presented in Section 3, and the existence results under the one-side superlinear growth are presented in Section 4. It should be noted that, in this paper, the estimation of the maximum value of Green function and its partial derivatives is no longer needed. In addition, with the help of efficient norm estimation and Leray–Schauder fixed point theorem of completely continuous operator, our discussion is carried out in the whole workspace without the restrictions of boundedness of and its derivatives. Therefore, our conclusions greatly improve and generalize the case of the bounded domain in the existing literature, which is new and significant. In order to prove the conclusion, we introduce some necessary properties of the solutions for the corresponding linear equation in the following section.
2. Preliminaries
Let . We introduce the following common spaces on :(i) be the continuous function space with the norm (ii) denotes the Hilbert space formed by inner product of all Lebesgue square integrable functions on , and its inner product norm is (iii) be the -order continuous differentiable function space with the norm (iv) be the Sobolev space constituted by norm , shows that , is absolutely continuous on , and
Firstly, we consider the following linear boundary value problem (LBVP) corresponding to BVP (1):
Lemma 1. For , LBVP (5) exists a unique solution , and the solution operator is a bounded linear operator. Furthermore, is completely continuous.
Proof. For each , one can easily test thatis a solution of LBVP (5), whereis the Green function corresponding to the homogeneous linear problem, and it is obvious that . Through specific calculations, we can getTherefore, by applying the contraction mapping principle to the solution operator and combining with (6), it is easy to prove that has only one fixed point on , that is, shown in (6) is the unique solution of LBVP (5). When , is a unique classical solution of LBVP (5). It is clear that is continuous; then, according to (6), is a bounded linear operator.
Furthermore, is obviously completely continuous since the Sobolev embedding is compact and the embedding is continuous.
The proof is finished.
Lemma 2. The unique solution of LBVP (5) has the following properties:(a)(b)(c)
Proof. (a)With a simple calculation, we get Then, we can verify that, for given , Since for every , and thus, .(b)By (10), we can check that, for given , Since for every , and hence, combining with (11) and (12), .(c)Considering the boundary condition of BVP (5) and using the Hlder inequality, we can obtain that, for every ,and then,In the same way, we haveTherefore,According to the conclusion of (b) and the continuity of , there exists such that . Thus,We can find out in just the same way thatThose show that (c) is valid.
At this point, the proof is finished.
Finally, we introduce the famous Leray–Schsuder fixed point theorem, which will be used to establish our main theorems.
Lemma 3. (see [7]). Let be a Banach space, and is a completely continuous operator. If the solution set of the homotopy family equationis bounded in , then there exists a fixed point of in .
3. Solvability under Linear Growth
Theorem 1. Suppose that satisfies the following:
there exist constants with such that, for every ,
Then, BVP (1) has at least one solution.
Proof. Define operator byand then, is continuous and maps bounded sets of into bounded sets of . According to the definition of defined in Lemma 1, the solution of BVP (1) is equivalent to the fixed point of the composition operator:By Lemma 1, is completely continuous. Now, we apply Lemma 3 to verify that has a fixed point.
Consider the homotopy family equationLet be the solution of equation (25) corresponding to ; then, denotes , and then, is the unique solution of LBVP (5), so it satisfiesTaking norm for both ends of BVP (26), together with and Lemma 2 (c), we can obtainHence,Consequently, combining with Lemma 2 (c),That is, the solutions’ set of equation (25) is bounded in . Then, in accordance with Lemma 3, there exists is the fixed point of , which is the solution of BVP (1).
The proof of Theorem 1 is completed.
Next, we establish the uniqueness result for BVP (1).
Theorem 2. Suppose that satisfies the following:
there exist constants with such that, for every ,
Then, BVP (1) has a unique solution.
Proof. Let . Then, from , it can be clearly deduced that is valid. Therefore, by Theorem 1, BVP (1) has at least one solution. Now, we just need to prove the uniqueness.
Let be the solutions of BVP (1), then . Denote , thus is the unique solution of LBVP (5) for . According to and Lemma 2 (c),Since , it shows that ; hence, by (29), , that is, . Consequently, there is only one solution for BVP (1).
The proof of the theorem is completed.
If the partial derivatives of exist, then we can obtain the following conclusion by means of the differential mean value theorem.
Corollary 1. Suppose that . If the partial derivatives of on exist and satisfy,
there exist constants with such that, for every ,
Then, BVP (1) has a unique solution.
Example 1. Consider the fully nonlinear BVPThe nonlinear termsSinceTherefore,Set , then , which implies that satisfies the condition ; then, by Theorem 1, BVP (33) has at least one solution.
Remark 1. In fact, for any , letthen corresponding BVP (1) has at least one solution.
4. Solvability under Superlinear Growth
In this section, in order to facilitate the establishment of the theorem, we need a Nagumo-type condition to limit the growth of on and an important Lemma for the Nagumo-type condition,
for any , there is a function withsuch that, for every ,
Lemma 4. Suppose that satisfies . If there exists such that the solution of BVP (1) satisfies , then there exists , such that .
Proof. Let , then by (38), there is a constant satisfyingSet be a solution of BVP (1) that satisfies ; now, we check . From Lemma 2 (b), it follows that there exists , such that . Suppose , that is, , then there exists such thatTherefore, . Thus, there are four cases:(1)(2)(3)(4)For case 1, let (see Figure 1)and according to the definition of supremum and the continuity of , we have andSincehence, by BVP (1) and (39),Then,Taking integral operation on for both sides of the inequality, and making variable substitution on the left. Then,Thereby,Then, based on (40),Similarly, we can discuss cases , and the conclusions are the same.
Theorem 3. Suppose that satisfies the following:
and there exist constants with such that, for every ,
Then, BVP (1) has at least one solution.
Proof. Define operator by (23). According to Lemma 1, it is evident that is a completely continuous linear operator. Hence, is completely continuous, and the solution of BVP (1) is equivalent to the fixed point of operator .
Consider homotopy family equation (25), and let be the solution of equation (25) corresponding to , denote , then is the unique solution of LBVP (5), so it satisfies (26). Multiplying both sides of equation (26) by and combining with , one can obtainIntegrating both sides of the above inequality on , together with Lemma 2 (c),By Lemma 2 (a), ; combining with (52) and ,Accordingly, by Lemma 2 (c),On the basis of the boundedness of Sobolev embedding ,where is the embedding constant of becauseHence, it can be deduced from Lemma 4 that there exists , such that . Combining with (55), we can infer thatIt shows that the solution set of equation (25) is bounded in . Then, by Lemma 3, there is a fixed point of in , which is a solution of BVP (1).
This completes the proof of Theorem 3.
Remark 2. Theorem 3 guarantees the existence of the solutions for BVP (1) under the condition that satisfies superlinear growth on , and . is a well-known Nagumo-type condition, which allows to quadratical growth on at most, and it is used to restrict the growth of on .
If exist, then we can obtain the following existence result without the superlinear growth condition by the differential mean value theorem.
Corollary 2. Suppose that satisfies . If the partial derivatives of on exist and satisfy,
there exist constants with such that, for every ,
Then, BVP (1) has at least one solution.
Proof. Let ; then, for every ,where . SinceTherefore, we can deduce thatCombining with , we can seewhich implies that is established by denoting . Consequently, by Theorem 3, BVP (1) has at least one solution.
The proof is completed.
Example 2. Consider the fully nonlinear BVPDenoteIt is obvious that is quadratical growth on , so is valid. In addition, for every ,Set , then , andwhich implies that satisfies the condition , and then, by Theorem 3, BVP (63) has at least one solution.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
M. Wei completed the proof of the main results and the writing of the first draft. Y. Li revised the first draft and put forward some suggestions for revision. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by NNSF of China (nos. 11661071 and 12061062) and NSF of Shanxi, China (no.201901D211399).