Existence and Iteration of Monotone Positive Solution of BVP for an Elastic Beam Equation

Abstract

This paper is concerned with the existence of monotone positive solution of boundary value problem for an elastic beam equation. By applying iterative techniques, we not only obtain the existence of monotone positive solution but also establish iterative scheme for approximating the solution. It is worth mentioning that the iterative scheme starts off with zero function, which is very useful and feasible for computational purpose. An example is also included to illustrate the main results.

1. Introduction

It is well-known that beam is one of the basic structures in architecture. The deformations of an elastic beam in equilibrium state can be described by the following equation of deflection curve: 𝑑²𝑑𝑥²𝐸𝐼_𝑧𝑑²𝑢𝑑𝑥²=𝑞(𝑥), (1.1) where 𝐸 is Yang's modulus constant, 𝐼_𝑧 is moment of inertia with respect to 𝑧 axes, and 𝑞(𝑥) is loading at 𝑥. If the loading of beam considered is in relation to deflection and rate of change of deflection, we need to study more general equation 𝑢⁽⁴⁾(𝑥)=𝑓𝑥,𝑢(𝑥),𝑢^′(𝑥). (1.2) According to different forms of supporting, various boundary value problems (BVPs) should be considered.

Owing to its importance in engineering, physics, and material mechanics, BVPs for elastic beam equations have attracted much attention from many authors see, for example, [1–15] and the references therein. However, almost all of the papers we mentioned focused their attention on the existence of solutions or positive solutions. In the existing literature, there are few papers concerned with the computational methods of solutions or positive solutions. It is worth mentioning that Zhang [16] obtained the existence of positive solutions and established iterative schemes for approximating the solutions for an elastic beam equation with a corner. The main tools used were monotone iterative techniques. For monotone iterative methods, one can refer [17–19] and the references therein.

Motivated greatly by the above-mentioned excellent works, in this paper we investigate the existence and iteration of monotone positive solution for the following elastic beam equation BVP 𝑢⁽⁴⁾(𝑡)=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)(),𝑢(0),𝑡∈0,1=𝑢^′(1)=𝑢^′′(0)=𝑢^′′′(1)=0. (1.3) In material mechanics, the equation in (1.3) describes the deflection or deformation of an elastic beam under a certain force. The boundary conditions in (1.3) mean that the elastic beam is simply fixed at the end 𝑡=0 and fastened with a sliding clamp at the end 𝑡=1. By applying iterative techniques, we not only obtain the existence of monotone positive solution but also establish iterative scheme for approximating the solution. It is worth mentioning that the iterative scheme starts off with zero function, which is very useful and feasible for computational purpose. An example is also included to illustrate the main results.

Throughout this paper, we always assume that the following condition is satisfied:

(𝐴)𝑓∈𝐶([0,1]×[0,+∞)×[0,+∞),[0,+∞)).

2. Preliminary

In order to obtain the main results of this paper, we first present several fundamental lemmas in this section.

Lemma 2.1. Let 𝑦∈𝐶[0,1]. Then, the BVP 𝑢⁽⁴⁾(𝑡)(𝑡)(),𝑢(0)=𝑦,𝑡∈0,1=𝑢^′(1)=𝑢^′′(0)=𝑢^′′′(1)=0, (2.1) has a unique solution 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑦(s)𝑑𝑠,𝑡∈[0,1], (2.2) where 𝐺()=1𝑡,𝑠6𝑠6𝑡−3𝑡²−𝑠²𝑡,0≤𝑠≤𝑡≤1,6𝑠−3𝑠²−𝑡²,0≤𝑡≤𝑠≤1. (2.3)

Lemma 2.2. For any (𝑡,𝑠)∈[0,1]×[0,1], we have ()0≤𝜕𝐺𝑡,𝑠(),1𝜕𝑡≤𝑠1−𝑡3𝑡²()≤1𝑠≤𝐺𝑡,𝑠22𝑡−𝑡²𝑠. (2.4)

Proof. For any fixed 𝑠∈[0,1], it is easy to know that 𝜕𝐺(𝑡,𝑠)=𝑠()𝑠()−1𝜕𝑡1−𝑡,0≤𝑠≤𝑡≤1,1−𝑡2(𝑠−𝑡)²,0≤𝑡≤𝑠≤1, (2.5) which shows that ()0≤𝜕𝐺𝑡,𝑠()()∈[]×[],𝜕𝑡≤𝑠1−𝑡,for𝑡,𝑠0,10,1 (2.6) and so, 𝐺(𝑡,𝑠)=^𝑡₀𝜕𝐺(𝜏,𝑠)𝜕𝜏𝑑𝜏≤^𝑡₀𝑠()11−𝜏𝑑𝜏=22𝑡−𝑡²𝑠,for(𝑡,𝑠)∈[0,1]×[0,1]. (2.7) On the other hand, it follows from the expression of 𝐺(𝑡,𝑠) that 𝐺()=1𝑡,𝑠6𝑠6𝑡−3𝑡²−𝑠²≥16𝑠6𝑡−4𝑡²≥13𝑡²𝐺()=1𝑠,0≤𝑠≤𝑡≤1,𝑡,𝑠6𝑡6𝑠−3𝑠²−𝑡²≥16𝑡6𝑠−4𝑠²≥13𝑡²𝑠,0≤𝑡≤𝑠≤1. (2.8)

Let 𝐸=𝐶¹[0,1] be equipped with the norm ‖𝑢‖=max{‖𝑢‖_∞,‖𝑢^′‖_∞}. Then, 𝐸 is a Banach space. Denote (𝑡)≥2𝐾=𝑢∈𝐸∶𝑢3𝑡²‖𝑢‖_∞,𝑢^′(𝑡)≥0,for𝑡∈[0,1]. (2.9) Then, it is easy to verify that 𝐾 is a cone in 𝐸. Note that this induces an order relation ≲ in 𝐸 by defining 𝑢≲𝑣 if and only if 𝑣−𝑢∈𝐾. Now, we define an operator 𝑇 on 𝐾 by (𝑇𝑢)(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠)𝑑𝑠,𝑡∈[0,1]. (2.10) Obviously, fixed points of 𝑇 are monotone and nonnegative solutions of the BVP (1.3).

Lemma 2.3. 𝑇∶𝐾→𝐾 is completely continuous.

Proof. First, we prove 𝑇(𝐾)⊂𝐾. Suppose that 𝑢∈𝐾. In view of Lemma 2.2, on the one hand, (𝑡)=0≤𝑇𝑢)(₁₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠≤1)𝑑𝑠22𝑡−𝑡²¹₀𝑠𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠)≤1𝑑𝑠2¹₀𝑠𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠)𝑑𝑠,𝑡∈[0,1], (2.11) which shows that ‖𝑇𝑢‖_∞≤12¹₀𝑠𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠)𝑑𝑠. (2.12) On the other hand, (𝑇𝑢)(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠)≥1𝑑𝑠3𝑡²¹₀𝑠𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠)𝑑𝑠,𝑡∈[0,1], (2.13) which together with (2.12) implies that (𝑡)≥2𝑇𝑢)(3𝑡²‖𝑇𝑢‖_∞,𝑡∈[0,1]. (2.14) Again, by Lemma 2.2, we have (𝑇𝑢)^′(𝑡)=¹₀𝜕𝐺(𝑡,𝑠)𝑓(𝑠)𝜕𝑡𝑠,𝑢,𝑢^′(𝑠)𝑑𝑠≥0,𝑡∈[0,1]. (2.15) So, it follows from (2.14), (2.15) and the definition of 𝐾 that 𝑇(𝐾)⊂𝐾.
Next, we show that 𝑇 is a compact operator. Let 𝐷⊂𝐾 be a bounded set. Then, there exists 𝑀₁>0 such that ‖𝑢‖≤𝑀₁ for any 𝑢∈𝐷. For any {𝑦_𝑘}^∞_𝑘=1⊂𝑇(𝐷), there exist {𝑥_𝑘}^∞_𝑘=1⊂𝐷 such that 𝑦_𝑘=𝑇𝑥_𝑘. Denote 𝑀₂=sup𝑓𝑡,𝑥,𝑦∶𝑡,𝑥,𝑦∈[0,1]×0,𝑀₁×0,𝑀₁. (2.16) Then, for any positive integer 𝑘, it follows from Lemma 2.2 that 𝑦_𝑘(𝑡)=𝑇𝑥_𝑘(𝑡=₁₀)𝐺(𝑡,𝑠)𝑓𝑠,𝑥_𝑘(𝑠),𝑥^′_𝑘(𝑠≤1)𝑑𝑠22𝑡−𝑡²¹₀𝑠𝑓𝑠,𝑥_𝑘(𝑠),𝑥^′_𝑘(𝑠)≤𝑀𝑑𝑠₂4,𝑡∈[0,1], (2.17) which indicates that {𝑦_𝑘}^∞_𝑘=1 is uniformly bounded. Similarly, we have 𝑦^′_𝑘(𝑡)=𝑇𝑥_𝑘^′(𝑡=₁₀)𝜕𝐺(𝑡,𝑠)𝑓𝜕𝑡𝑠,𝑥_𝑘(𝑠),𝑥^′_𝑘(𝑠)≤()𝑑𝑠1−𝑡¹₀𝑠𝑓𝑠,𝑥_𝑘(𝑠),𝑥^′_𝑘(𝑠)≤𝑀𝑑𝑠₂2,𝑡∈[0,1]. (2.18) This shows that {𝑦^′_𝑘}^∞_𝑘=1 is uniformly bounded, which implies that {𝑦_𝑘}^∞_𝑘=1 is equicontinuous. By Arzela-Ascoli theorem, we know that {𝑦_𝑘}^∞_𝑘=1 has a convergent subsequence in 𝐶[0,1]. Without loss of generality, we may assume that {𝑦_𝑘}^∞_𝑘=1 converges in 𝐶[0,1].
On the other hand, for any 𝜖>0, by the uniform continuity of 𝜕𝐺(𝑡,s)/𝜕𝑡, we know that there exists a 𝛿>0 such that for any 𝑡₁,𝑡₂∈[0,1] with |𝑡₁−𝑡₂|<𝛿, 𝜕𝐺𝑡₁,𝑠−𝑡𝜕𝑡𝜕𝐺₂,𝑠<𝜖𝜕𝑡𝑀₂[]+1,𝑠∈0,1. (2.19) So, for any positive integer 𝑘, 𝑡₁,𝑡₂∈[0,1] with |𝑡₁−𝑡₂|<𝛿, we get 𝑦^′_𝑘𝑡₁−𝑦^′_𝑘𝑡₂=𝑇𝑥_𝑘^′𝑡₁−𝑇𝑥_𝑘^′𝑡₂=¹₀𝜕𝐺𝑡₁,𝑠−𝑡𝜕𝑡𝜕𝐺₂,𝑠𝑓𝜕𝑡𝑠,𝑥_𝑘(𝑠),𝑥^′_𝑘(𝑠)<𝑀𝑑𝑠₂𝜖𝑀₂+1<𝜖, (2.20) which shows that {𝑦^′_𝑘}^∞_𝑘=1 is equicontinuous. Again, it follows from Arzela-Ascoli theorem that {𝑦^′_𝑘}^∞_𝑘=1 has a convergent subsequence in 𝐶[0,1]. Therefore, {𝑦_𝑘}^∞_𝑘=1 has a convergent subsequence in 𝐾.
Finally, we prove that 𝑇 is continuous. Suppose that 𝑢_𝑚,𝑢∈𝐾 and ‖𝑢_𝑚−𝑢‖→0(𝑚→∞). Then, there exists 𝑀₃>0 such that for any positive integer 𝑚, ‖𝑢_𝑚‖≤𝑀₃. Denote 𝑀₄=sup𝑓𝑡,𝑥,𝑦∶𝑡,𝑥,𝑦∈[0,1]×0,𝑀₃×0,𝑀₃. (2.21) Then, for any positive integer 𝑚 and 𝑡∈[0,1], by Lemma 2.2, we have 𝐺(𝑡,𝑠)𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠)≤𝑀₄22𝑡−𝑡²𝑀𝑠≤⁴2[],()𝑠,𝑠∈0,1𝜕𝐺𝑡,𝑠𝑓𝜕𝑡𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠)≤𝑀₄(1−𝑡)𝑠≤𝑀₄𝑠,𝑠∈[0,1]. (2.22) So, it follows from Lebesgue dominated convergence theorem that lim_𝑚→∞𝑇𝑢_𝑚(𝑡)=lim_𝑚→∞¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠)=𝑑𝑠₁₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠=(𝑡)[],)𝑑𝑠𝑇𝑢)(,𝑡∈0,1lim_𝑚→∞𝑇𝑢_𝑚^′(𝑡)=lim_𝑚→∞¹₀𝜕𝐺(𝑡,𝑠)𝑓𝜕𝑡𝑠,𝑢_𝑚(𝑠),𝑢^′_𝑚(𝑠=)𝑑𝑠₁₀𝜕𝐺(𝑡,𝑠)𝑓(𝑠)𝜕𝑡𝑠,𝑢,𝑢^′(𝑠)=()𝑑𝑠𝑇𝑢^′(𝑡),𝑡∈[0,1], (2.23) which implies that 𝑇 is continuous. Therefore, 𝑇∶𝐾→𝐾 is completely continuous.

3. Main Results

Theorem 3.1. Assume that 𝑓(𝑡,0,0)≢0 for 𝑡∈(0,1), and there exists a constant 𝑎>0 such that 𝑓𝑡,𝑢₁,𝑣₁≤𝑓𝑡,𝑢₂,𝑣₂≤2𝑎,0≤𝑡≤1,0≤𝑢₁≤𝑢₂≤𝑎,0≤𝑣₁≤𝑣₂≤𝑎. (3.1) If we construct a iterative sequence 𝑣_𝑛+1=𝑇𝑣_𝑛, 𝑛=0,1,2,…, where 𝑣₀(𝑡)≡0 for 𝑡∈[0,1], then {𝑣_𝑛}^∞_𝑛=1 converges to 𝑣^∗ in 𝐶¹[0,1], which is a monotone positive solution of the BVP (1.3) and satisfy 0<𝑣^∗(𝑡)≤𝑎for𝑡∈(0,1],0≤𝑣^∗′(𝑡)≤𝑎for𝑡∈[0,1]. (3.2)

Proof. Let 𝐾_𝑎={𝑢∈𝐾∶‖𝑢‖≤𝑎}. We assert that 𝑇∶𝐾_𝑎→𝐾_𝑎. In fact, if 𝑢∈𝐾_𝑎, then 0≤𝑢(𝑠)≤max_0≤𝑠≤1𝑢(𝑠)≤‖𝑢‖≤𝑎,0≤𝑢^′(𝑠)≤max_0≤𝑠≤1𝑢^′(𝑠)≤‖𝑢‖≤𝑎,for𝑠∈[0,1], (3.3) which together with the condition (3.1) and Lemma 2.2 implies that 0≤(𝑇𝑢)(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠)[],()𝑑𝑠<𝑎,𝑡∈0,10≤𝑇𝑢^′(𝑡)=¹₀𝜕𝐺(𝑡,𝑠)𝑓(𝑠)𝜕𝑡𝑠,𝑢,𝑢^′(𝑠)𝑑𝑠≤𝑎,𝑡∈[0,1]. (3.4) Hence, we have shown that 𝑇∶𝐾_𝑎→𝐾_𝑎.
Now, we assert that {𝑣_𝑛}^∞_𝑛=1 converges to 𝑣^∗ in 𝐶¹[0,1], which is a monotone positive solution of the BVP (1.3) and satisfies 0<𝑣^∗(𝑡)≤𝑎for𝑡∈(0,1],0≤𝑣^∗′(𝑡)≤𝑎for𝑡∈[0,1]. (3.5)
Indeed, in view of 𝑣₀∈𝐾_𝑎 and 𝑇∶𝐾_𝑎→𝐾_𝑎, we have 𝑣_𝑛∈𝐾_𝑎, 𝑛=1,2,…. Since the set {𝑣_𝑛}^∞_𝑛=0 is bounded and 𝑇 is completely continuous, we know that the set {𝑣_𝑛}^∞_𝑛=1 is relatively compact.
In what follows, we prove that {𝑣_𝑛}^∞_𝑛=0 is monotone by induction. First, by Lemma 2.2, we have 13𝑡²¹₀𝑠𝑓(𝑠,0,0)𝑑𝑠≤𝑣₁(𝑡)−𝑣₀(𝑡)=𝑇𝑣₀(𝑡)=¹₀𝐺()𝑓()1𝑡,𝑠𝑠,0,0𝑑𝑠≤2¹₀𝑠𝑓(𝑠,0,0)𝑑𝑠,𝑡∈[0,1], (3.6) which implies that 𝑣₁(𝑡)−𝑣₀(𝑡)≥23𝑡²𝑣₁−𝑣_0∞,𝑡∈[0,1]. (3.7) At the same time, it is obvious that 𝑣^′₁(𝑡)−𝑣^′₀(𝑡)=𝑣^′₁(𝑡)=𝑇𝑣₀^′(𝑡)=¹₀𝜕𝐺(𝑡,𝑠)𝑓()[].𝜕𝑡𝑠,0,0𝑑𝑠≥0,𝑡∈0,1 (3.8) It follows from (3.7) and (3.8) that 𝑣₁−𝑣₀∈𝐾, which shows that 𝑣₀≲𝑣₁. Next, we assume that 𝑣_𝑘−1≲𝑣_𝑘. Then, in view of Lemma 2.2 and (3.1), we have 𝑣_𝑘+1(𝑡)−𝑣_𝑘(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣_𝑘(𝑠),𝑣^′_𝑘(𝑠)−𝑓𝑠,𝑣_𝑘−1(𝑠),𝑣^′_𝑘−1(𝑠)≤1𝑑𝑠2¹₀𝑠𝑓𝑠,𝑣_𝑘(𝑠),𝑣^′_𝑘(𝑠)−𝑓𝑠,𝑣_𝑘−1(𝑠),𝑣^′_𝑘−1(𝑠[],𝑣)𝑑𝑠,𝑡∈0,1_𝑘+1(𝑡)−𝑣_𝑘(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣_𝑘(𝑠),𝑣^′_𝑘(𝑠)−𝑓𝑠,𝑣_𝑘−1(𝑠),𝑣^′_𝑘−1(𝑠)≥1𝑑𝑠3𝑡²¹₀𝑠𝑓𝑠,𝑣_𝑘(𝑠),𝑣^′_𝑘(𝑠)−𝑓𝑠,𝑣_𝑘−1(𝑠),𝑣^′_𝑘−1(𝑠)𝑑𝑠,𝑡∈[0,1], (3.9) which implies that 𝑣_𝑘+1(𝑡)−𝑣_𝑘(𝑡)≥23𝑡²𝑣_𝑘+1−𝑣_𝑘∞,𝑡∈[0,1]. (3.10) At the same time, by Lemma 2.2 and (3.1), we also have 𝑣^′_𝑘+1(𝑡)−𝑣^′_𝑘(𝑡)=¹₀𝜕𝐺(𝑡,𝑠)𝑓𝜕𝑡𝑠,𝑣_𝑘(𝑠),𝑣^′_𝑘(𝑠)−𝑓𝑠,𝑣_𝑘−1(𝑠),𝑣^′_𝑘−1(𝑠)𝑑𝑠≥0,𝑡∈[0,1]. (3.11) It follows from (3.10) and (3.11) that 𝑣_𝑘+1−𝑣_𝑘∈𝐾, which indicates that 𝑣_𝑘≲𝑣_𝑘+1. Thus, we have shown that 𝑣_𝑛≲𝑣_𝑛+1, 𝑛=0,1,2….
Since {𝑣_𝑛}^∞_𝑛=1 is relatively compact and monotone, there exists a 𝑣^∗∈𝐾_𝑎 such that ‖𝑣_𝑛−𝑣^∗‖→0(𝑛→∞), which together with the continuity of 𝑇 and the fact that 𝑣_𝑛+1=𝑇𝑣_𝑛 implies that 𝑣^∗=𝑇𝑣^∗. Moreover, in view of 𝑓(𝑡,0,0)≢0 for 𝑡∈(0,1), we know that the zero function is not a solution of the BVP (1.3). Thus, ‖𝑣^∗‖_∞>0. So, it follows from 𝑣^∗∈𝐾_𝑎 that 0<𝑣^∗(𝑡)≤𝑎for𝑡∈(0,1],0≤𝑣^∗′(𝑡)≤𝑎for𝑡∈[0,1]. (3.12)

4. An Example

Example 4.1. Consider the BVP 𝑢⁽⁴⁾(𝑡)=12(𝑡)+1𝑡𝑢8𝑢^′(𝑡)²(),𝑢(0)+1,𝑡∈0,1=𝑢^′(1)=𝑢^′′(0)=𝑢^′′′(1)=0. (4.1)
If we let 𝑓(𝑡,𝑢,𝑣)=(1/2)𝑡𝑢+(1/8)𝑣²+1 for (𝑡,𝑢,𝑣)∈[0,1]×[0,+∞)×[0,+∞), then all the hypotheses of Theorem 3.1 are fulfilled with 𝑎=1. It follows from Theorem 3.1 that the BVP (4.1) has a monotone positive solution 𝑣^∗ satisfying 0<𝑣^∗(𝑡)≤1for𝑡∈(0,1],0≤𝑣^∗′(𝑡)≤1for𝑡∈[0,1]. (4.2) Moreover, the iterative scheme is 𝑣₀(𝑡)[],𝑣≡0,𝑡∈0,1_𝑛+1(𝑡)=16^𝑡₀𝑠6𝑡−3𝑡²−𝑠²12𝑠𝑣_𝑛(𝑠)+18𝑣^′_𝑛(𝑠)²+1+1𝑑𝑠6¹_𝑡𝑡6𝑠−3𝑠²−𝑡²12𝑠𝑣_𝑛(𝑠)+18𝑣^′_𝑛(𝑠)²+1𝑑𝑠,𝑡∈[0,1],𝑛=0,1,2,…. (4.3) The first, second, third, and fourth terms of the scheme are as follows: 𝑣₀(𝑡)𝑣≡0,₁(𝑡)=131𝑡−6𝑡³+1𝑡24⁴,𝑣₂(𝑡)=735216𝑡+𝑡144²−79𝑡432³+1𝑡64⁴+25𝑡1152⁵−23𝑡8640⁶−1𝑡192⁷+47𝑡11520⁸−11𝑡8640⁹+1𝑡6912¹⁰,𝑣₃(𝑡)=3785771119744𝑡+6935𝑡186624²−404629𝑡2239488³+60361𝑡8957952⁴+71689𝑡2985984⁵+110633𝑡14929920⁶−22181𝑡1492992⁷+373759𝑡119439360⁸+2467627𝑡477757440⁹−2449943𝑡716636160¹⁰−800341𝑡1592524800¹¹[phone]𝑡11466178560¹²−3556357𝑡7166361600¹³−655267𝑡4777574400¹⁴+70313𝑡477757440¹⁵−25471𝑡2388787200¹⁶−363911𝑡9555148800¹⁷+7781𝑡298598400¹⁸−25973𝑡2866544640¹⁹+8917𝑡4777574400²⁰−31𝑡143327232²¹+25𝑡2293235712²². (4.4)

Acknowledgment

This work is supported by the National Natural Science Foundation of China (10801068).