# Solvability for a Fully Elastic Beam Equation with Left-End Fixed and Right-End Simply Supported

Authors: Mei Wei; Yongxiang Li
Journal: Mathematical Problems in Engineering
 (2021)

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## 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.,u4t=ft,ut,u′t,u″t,u‴t,t∈0,1u0=u′0=u1=u″1=0, where f:0,1×ℝ4⟶ℝ 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 ft,x0,x1,x2,x3 is quadratical growth on x3 at most.


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## Body

## 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:(1)u4t=ft,ut,u′t,u″t,u‴t,t∈0,1,u0=u′0=u1=u″1=0,where f:0,1×ℝ4⟶ℝ 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 f is independent of the third-order derivative term of u (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 f 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 derivativesu′t,u″t,u‴t,andu4 of ut 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 f on the third derivative u‴ 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 f involves all lower-order derivatives of deformation function u 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]):(2)u4t=ft,ut,u′t,u″t,u‴t,t∈0,1,u0=u″0=u1=u″1=0,and one end is simply supported, and the other end is sliding clamped (see [5, 11, 22]):(3)u4t=ft,ut,u′t,u″t,u‴t,t∈0,1,u0=u″0=u′1=u‴1=0,and one end is fixed, and the other end is free (see [13, 17–19, 21]):(4)u4t=ft,ut,u′t,u″t,u‴t,t∈0,1,u0=u′0=u″1=u‴1=0.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 ft,x0,x1,x2,x3 is linear growth; in [19], by supplementing the Nagumo-type condition to limit the growth of f on x3, Li and Chen obtained the existence and uniqueness of solutions for BVP (4) under the condition that f 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 u 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

LetI=0,1,n∈ℕ. We introduce the following common spaces on I:(i)
CI be the continuous function space with the norm uC=maxt∈Iut(ii)
L2I denotes the Hilbert space formed by inner product u,v=∫01utvtdt of all Lebesgue square integrable functions on I, and its inner product norm is u2=∫01ut2dt1/2(iii)
CnI be the n-order continuous differentiable function space with the norm uCn=maxuC,u′C,…,unC(iv)
HnI be the Sobolev space constituted by norm un,2=∑i=0nui221/2, u∈HnI shows that u∈Cn−1I, un−1t is absolutely continuous on I, and un∈L2IFirstly, we consider the following linear boundary value problem (LBVP) corresponding to BVP (1):(5)u4t=ht,t∈I,u0=u′0=u1=u″1=0.Lemma 1.
Forh∈L2I, LBVP (5) exists a unique solution u≔Sh∈H4I, and the solution operator S:L2I⟶H4I is a bounded linear operator. Furthermore, S:L2I⟶H3I is completely continuous.Proof.
For eachh∈L2I, one can easily test that(6)ut=∫01Gt,shsds≔Sht∈H4I,t∈I,is a solution of LBVP (5), where(7)Gt,s=112t21−s31−t−1−s23−t,0≤t≤s≤1,s21−t31−s−1−t23−s,0≤s≤t≤1,is the Green function corresponding to the homogeneous linear problem, and it is obvious that Gt,s≥0,t,s∈I. Through specific calculations, we can get(8)0≤∫01Gt,sds=148t21−t3−2t<1.
Therefore, by applying the contraction mapping principle to the solution operatorS and combining with (6), it is easy to prove that S has only one fixed point on L2I, that is, u shown in (6) is the unique solution of LBVP (5). When h∈CI, ut≔Sh∈C4I is a unique classical solution of LBVP (5). It is clear that Gt,s is continuous; then, according to (6), S:L2I⟶H4I is a bounded linear operator.
Furthermore,S:L2I⟶H3I is obviously completely continuous since the Sobolev embedding H4I↪C3I is compact and the embedding C3I↪H3I is continuous.
The proof is finished.Lemma 2.
The unique solutionu of LBVP (5) has the following properties:(a)
u″0u‴0≤0(b)
u‴0u‴1≤0(c)
ui2≤1/2ui+12,i=0,1,2,and3Proof.
(a)
With a simple calculation, we get(9)∂2∂t2Gt,s=121−s1−3t−1−t1−s2,0≤t≤s≤1,s21−ts−3,0≤s≤t≤1,(10)∂3∂t3Gt,s=121−ss2−2s−2,0≤t<s≤1,s23−s,0≤s<t≤1.Then, we can verify that, for givenh∈CI,(11)u″0=12∫011−s1−1−s2hsds,u‴0=12∫011−ss−12−3hsds.Since for everys∈I,(12)1−s1−1−s2≥0,1−ss−12−3≤0,and thus,u″0u‴0≤0.(b)
By (10), we can check that, for given h∈CI,(13)u‴1=12∫01s23−shsds.Since for everys∈I,(14)s23−s≥0,and hence, combining with (11) and (12), u‴0u‴1≤0.(c)
Considering the boundary condition of BVP (5) and using the Ho¨lder inequality, we can obtain that, for every t∈I,(15)ut=|∫0tu′sds|≤∫0tu′sds≤∫0tu′s2ds1/2·∫0t1ds1/2≤t1/2u′2,and then,(16)u2=∫01ut2dt1/2≤∫01tu′22dt1/2≤12u′2.
In the same way, we have(17)u′t=∫0tu″sds≤t1/2u″2,u″t=−∫t1u‴sds≤1−t1/2u‴2.
Therefore,(18)u′2≤12u″2,u″2≤12u‴2.
According to the conclusion of (b) and the continuity ofu‴t, there exists t0∈0,1 such that u‴t0=0. Thus,(19)u‴t=∫t0tu4sds,t∈t0,1,−∫tt0u4sds,t∈0,t0.
We can find out in just the same way that(20)u‴2≤12u42.
Those 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 X be a Banach space, and T:X⟶X is a completely continuous operator. If the solution set of the homotopy family equation(21)x=λTx,0<λ<1,is bounded in X, then there exists a fixed point of T in X.


## 3. Solvability under Linear Growth

Theorem 1.
Suppose thatf∈CI×ℝ4,ℝ satisfies the following:
H1 there exist constants a˜>0anda0,a1,a2,anda3≥0 with a0/4+a1/22+a2/2+a3/2<1 such that, for every t,x0,x1,x2,x3∈I×ℝ4,(22)ft,x0,x1,x2,x3≤a0x0+a1x1+a2x2+a3x3+a˜.Then, BVP (1) has at least one solution.Proof.
Define operatorF:H3I⟶L2I by(23)Fut=ft,ut,u′t,u″t,u‴t,t∈I,and then, F is continuous and maps bounded sets of H3I into bounded sets of L2I. According to the definition of S defined in Lemma 1, the solution of BVP (1) is equivalent to the fixed point of the composition operator:(24)T≔S∘F.
By Lemma1, T:H3I⟶H3I is completely continuous. Now, we apply Lemma 3 to verify that T has a fixed point.
Consider the homotopy family equation(25)u=λTu,0<λ<1.
Letu∈H3I be the solution of equation (25) corresponding to λ∈0,1; then, u=λTu=λSFu=SλFu denotes h=λFu, and then, u=Sh∈H4I is the unique solution of LBVP (5), so it satisfies(26)u4t=λft,ut,u′t,u″t,u‴t,t∈I,u0=u′0=u1=u″1=0.
Taking⋅2 norm for both ends of BVP (26), together with H1 and Lemma 2 (c), we can obtain(27)u42=λFu2≤a0u2+a1u′2+a2u″2+a3u‴2+a˜≤a04+a122+a22+a32u42+a˜.
Hence,(28)u42≤a˜1−a0/4+a1/22+a2/2+a3/2≔C0.
Consequently, combining with Lemma2 (c),(29)u3,2=u22+u′22+u″22+u‴22≤116+18+14+12u42<u42≤C0.
That is, the solutions’ set of equation (25) is bounded in H3I. Then, in accordance with Lemma 3, there exists u0∈H3I is the fixed point of T, which is the solution of BVP (1).
The proof of Theorem1 is completed.
Next, we establish the uniqueness result for BVP (1).Theorem 2.
Suppose thatf∈CI×ℝ4,ℝ satisfies the following:
H2 there exist constants a0,a1,a2anda3≥0 with a0/4+a1/22+a2/2+a3/2<1 such that, for every t∈Iandxiandyi∈ℝ,i=0,1,2,and3,(30)ft,x0,x1,x2,x3−ft,y0,y1,y2,y3≤∑i=03aixi−yi.Then, BVP (1) has a unique solution.Proof.
Leta˜=maxt∈Ift,0,0,0,0+1. Then, from H2, it can be clearly deduced that H1 is valid. Therefore, by Theorem 1, BVP (1) has at least one solution. Now, we just need to prove the uniqueness.
Letu1andu2 be the solutions of BVP (1), then u1=SFu1,u2=SFu2. Denote u=u2−u1=SFu2−Fu1, thus u is the unique solution of LBVP (5) for h=Fu2−Fu1. According to H2 and Lemma 2 (c),(31)u42=Fu2−Fu12≤a0u2−u12+a1u2′−u1′2+a2u2″−u1″2+a3u2″′−u1″′2=a0u2+a1u′2+a2u″2+a3u‴2≤a04+a122+a22+a32u42.
Sincea0/4+a1/22+a2/2+a3/2<1, it shows that u42≤0; hence, by (29), u3,2≤0, that is, u1=u2. Consequently, there is only one solution for BVP (1).
The proof of the theorem is completed.
If the partial derivativesfx0,fx1,fx2,andfx3 of ft,x0,x1,x2,x3 exist, then we can obtain the following conclusion by means of the differential mean value theorem.Corollary 1.
Suppose thatf∈CI×ℝ4,ℝ. If the partial derivatives fx0,fx1,fx2,andfx3 of ft,x0,x1,x2,x3 on x0,x1,x2,andx3 exist and satisfy,
H3 there exist constants a0,a1,a2,anda3≥0 with a0/4+a1/22+a2/2+a3/2<1 such that, for every t,x0,x1,x2,x3∈I×ℝ4,(32)fxit,x0,x1,x2,x3≤ai,i=0,1,2,and3.Then, BVP (1) has a unique solution.Example 1.
Consider the fully nonlinear BVP(33)u4t=2u1/3t+t2u′t1/3+sintu″t1/3+e−tu‴t1/2+et,t∈I,u0=u′0=u1=u″1=0.
The nonlinear terms(34)ft,x0,x1,x2,x3=2x01/3+t2·x11/3+sint⋅x21/3+e−tx31/2+et.
Since(35)x1/2≤14x+1,x1/3≤14x+54.
Therefore,(36)ft,x0,x1,x2,x3≤12x0+14x1+14x2+14x3+6+e.
Seta0=1/2,a1=a2=a3=1/4,anda˜=6+e, then a0/4+a1/22+a2/2+a3/2=1+32/82<1, which implies that f satisfies the condition H1; then, by Theorem 1, BVP (33) has at least one solution.Remark 1.
In fact, for anyt∈I,α˜t,αit∈CI,andβi∈0,1,i=0,1,2,and3, let(37)ft,x0,x1,x2,x3=∑i=03αitxiβi+α˜t,then 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 off on x3 and an important Lemma for the Nagumo-type condition,H∗ for any M>0, there is a function HMρ∈Cℝ+,ℝ+\0 with(38)∫0+∞ρHMρdρ=+∞,such that, for every t,x0,x1,x2,x3∈I×−M,M3×ℝ,(39)ft,x0,x1,x2,x3≤HMx3.Lemma 4.
Suppose thatf∈CI×ℝ4,ℝ satisfies H∗. If there exists M>0 such that the solution u of BVP (1) satisfies uC2≤M, then there exists M1=M1M>0, such that u‴C≤M1.Proof.
LetM>0, then by (38), there is a constant M1>0 satisfying(40)∫0M1ρHMρdρ>2M.
Setu be a solution of BVP (1) that satisfies uC2≤M; now, we check u‴C≤M1. From Lemma 2 (b), it follows that there exists t0∈I, such that u‴t0=0. Suppose u‴C≠0, that is, u‴C>0, then there exists t1∈I such that(41)u‴C=maxt∈Iu‴t=u‴t1>0.
Therefore,t0≠t1. Thus, there are four cases:(1)
u‴t1>0andt0<t1(2)
u‴t1>0andt0>t1(3)
u‴t1<0andt0<t1(4)
u‴t1<0andt0>t1
For case 1, let (see Figure1)(42)t2=supt∈t0,t1|u‴t=0,and according to the definition of supremum and the continuity of u‴t, we have t0≤t2<t1≤1 and(43)u‴t2=0,u‴t>0andt∈t2,t1.
Since(44)ut,u′t,u″t≤uC2≤M,hence, by BVP (1) and (39),(45)u4t=ft,ut,u′t,u″t,u‴t≤HMu‴t,t∈t2,t1.
Then,(46)u4tu‴tHMu‴t≤u‴t,t∈t2,t1.
Taking integral operation ont2,t1 for both sides of the inequality, and making variable substitution ρ=u‴t on the left. Then,(47)∫0u‴t1ρdρHMu‴t≤u″t1−u″t2≤2u″C≤2uC2≤2M.
Thereby,(48)∫0u‴CρdρHMρ+C0≤2M.
Then, based on (40),(49)u‴C≤M1.
Similarly, we can discuss cases2°−4°, and the conclusions are the same.Figure 1
u‴t1>0andt0<t1.Theorem 3.
Suppose thatf∈CI×ℝ4,ℝ satisfies the following:
H∗ and H4 there exist constants b˜>0andb0,b1,b2,andb3≥0 with b0/8+b1/4+b2/2+b3<1 such that, for every t,x0,x1,x2,x3∈I×ℝ4,(50)−ft,x0,x1,x2,x3x2≤b0x02+b1x12+b2x22+b3x32+b˜.Then, BVP (1) has at least one solution.Proof.
Define operatorF:C3I⟶CI by (23). According to Lemma 1, it is evident that S:CI⟶C3I is a completely continuous linear operator. Hence, T=S°F:C3I⟶C3I is completely continuous, and the solution of BVP (1) is equivalent to the fixed point of operator T.
Consider homotopy family equation (25), and let u∈C3I be the solution of equation (25) corresponding to λ∈0,1, denote h=λFu, then u=Sh∈C4I is the unique solution of LBVP (5), so it satisfies (26). Multiplying both sides of equation (26) by −u″t and combining with H4, one can obtain(51)−u4tu″t=−λft,ut,u′t,u″t,u‴tu″t≤λb0u2t+b1u′t2+b2u″t2+b3u‴t2+b˜≤b0u2t+b1u′t2+b2u″t2+b3u‴t2+b˜.
Integrating both sides of the above inequality onI, together with Lemma 2 (c),(52)u‴22+u″0u‴0≤∑i=03bi∫01uit2dt+b˜=∑i=03biui22+b˜≤b08+b14+b22+b3u‴22+b˜.
By Lemma2 (a), b˜−u″0u‴0>0; combining with (52) and H4,(53)u‴22≤b˜−u″0u‴01−b0/8+b1/4+b2/2+b3≔M0.
Accordingly, by Lemma2 (c),(54)u3,2=∑i=03ui221/2≤18+14+12+11/2u‴2≤M01/22.
On the basis of the boundedness of Sobolev embeddingH3I↪C2I,(55)uC2≤cu3,2≤2cM01/2≔M,where c is the embedding constant of H3I↪C2I because(56)λft,x0,x1,x2,x3≤ft,x0,x1,x2,x3≤HMu‴t.
Hence, it can be deduced from Lemma4 that there exists M1=M1M>0, such that u‴C≤M1. Combining with (55), we can infer that(57)uC3=maxuC2,u‴C≤maxM,M1≔M2.
It shows that the solution set of equation (25) is bounded in C3I. Then, by Lemma 3, there is a fixed point of T in C3I, which is a solution of BVP (1).
This completes the proof of Theorem3.Remark 2.
Theorem3 guarantees the existence of the solutions for BVP (1) under the condition that ft,x0,x1,x2,x3 satisfies superlinear growth on x0,x1,x2, and x3. H∗ is a well-known Nagumo-type condition, which allows HMρ to quadratical growth on ρ at most, and it is used to restrict the growth of f on x3.
Iffx0,fx1,fx2,andfx3 exist, then we can obtain the following existence result without the superlinear growth condition by the differential mean value theorem.Corollary 2.
Suppose thatf∈CI×ℝ4,ℝ satisfies H∗. If the partial derivatives fx0,fx1,fx2,andfx3 of ft,x0,x1,x2,x3 on x0,x1,x2,andx3 exist and satisfy,
H5 there exist constants b0,b1,b2,andb3≥0 with b0/8+b1/4+b2/2+b3<1 such that, for every t,x0,x1,x2,x3∈I×ℝ4,(58)fx024b0+fx124b1−fx2+fx324b3≤0.Then, BVP (1) has at least one solution.Proof.
LetC0=maxt∈Ift,0,0,0,0+1; then, for every t,x0,x1,x2,x3∈I×ℝ4,(59)−ft,x0,x1,x2,x3x2=−ft,x0,x1,x2,x3−ft,0,0,0,0x2−ft,0,0,0,0x2=−fx0t,γ0,γ1,γ2,γ3x0x2−fx1t,γ0,γ1,γ2,γ3x1x2−fx2t,γ0,γ1,γ2,γ3x22−fx3t,γ0,γ1,γ2,γ3x2x3−ft,0,0,0,0x2,where γ0=θx0,γ1=θx1,γ2=θx2,andγ3=θx3,θ∈0,1. Since(60)−fxit,γ0,γ1,γ2,γ3xix2≤fxi2t,γ0,γ1,γ2,γ3x224bi+bixi2i=0,1,and3;−ft,0,0,0,0x2≤C0x2≤b2x22+C024b2.
Therefore, we can deduce that(61)−ft,x0,x1,x2,x3x2≤fx024b0+fx124b1−fx2+fx324b3x22+b0x02+b1x12+b2x22+b3x32+C024b2.
Combining withH5, we can see(62)−ft,x0,x1,x2,x3x2≤b0x02+b1x12+b2x22+b3x32+C024b2,which implies that H4 is established by denoting C02/4b2≔b˜>0. Consequently, by Theorem 3, BVP (1) has at least one solution.
The proof is completed.Example 2.
Consider the fully nonlinear BVP(63)u4t=ut+t2u′t+u″3t+12u″tu″′2t+2e−t,t∈I,u0=u′0=u1=u″1=0.
Denote(64)ft,x0,x1,x2,x3=x0+t2x1+x23+12x2x32+2e−t.
It is obvious thatf is quadratical growth on x3, so H∗ is valid. In addition, for every t,x0,x1,x2,x3∈I×ℝ4,(65)−ft,x0,x1,x2,x3x2=−x0x2−t2x1x2−x24−12x22x32−2e−tx2≤x0x2+x1x2+2x2=2x0·x22+2x1·x22+2·x22≤x02+x12+x22+1.
Setb0=b1=b2=1,b3=0,andb˜=1, then b0/8+b1/4+b2/2+b3=7/8<1, and(66)−ft,x0,x1,x2,x3x2≤b0x02+b1x12+b2x22+b3x32+b˜,which implies that f satisfies the condition H4, and then, by Theorem 3, BVP (63) has at least one solution.



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