Monotone Positive Solutions for an Elastic Beam Equation with Nonlinear Boundary Conditions

Abstract

This paper is concerned with the existence of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions. By applying monotone iteration method, we not only obtain the existence of monotone positive solutions but also establish iterative schemes for approximating the solutions. It is worth mentioning that these iterative schemes start off with zero function or quadratic function, which is very useful and feasible for computational purpose. An example is also included to illustrate the main results obtained.

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:𝑑²𝑑𝑡²𝐸𝐼_𝑧𝑑²𝑢𝑑𝑡²=𝑞(𝑡), (1.1) where 𝐸 is Yang's modulus constant, 𝐼_𝑧 is moment of inertia with respect to 𝑧 axes, and 𝑞(𝑡) is loading at 𝑡. If the loading of beam considered is in relation to deflection and rate of change of deflection, we need to study a more general equation:𝑢⁽⁴⁾(𝑡)=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡). (1.2) According to different forms of supporting, various boundary value problems (BVPs for short) should be considered.

Owing to its importance in engineering, physics, and material mechanics, fourth-order BVPs 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 null boundary conditions. When the boundary conditions are nonzero or nonlinear, fourth-order equations can model beams resting on elastic bearings located in their extremities. Up to now, a little work has been done for fourth-order BVPs with nonlinear boundary conditions. It is worth mentioning that, in 2009, Alves et al. [16] studied some fourth-order BVPs with nonlinear boundary conditions, which models an elastic beam whose left end is fixed and right end is attached to a bearing device or sliding clamped. Their main tool was monotone iterative method. For more on monotone iterative techniques, one can refer to [17–20] and the references therein.

Motivated greatly by the above-mentioned excellent works, in this paper we consider the existence and iteration of monotone positive solutions for the following fourth-order BVP with nonlinear boundary conditions:𝑢⁽⁴⁾(𝑡)=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡)(),𝑢(0),𝑡∈0,1=𝑢^′′(0)𝑢=0,_′(1)=0,𝑢^′′′(1)=𝑔(𝑢(1)), (1.3) which models an elastic beam whose left end is simply supported and right end is sliding clamped, given by the function 𝑔. By applying monotone iterative method, we not only obtain the existence of monotone positive solutions for the BVP (1.3) but also establish iterative schemes for approximating the solutions. These iterative schemes start off with zero function or quadratic function, which is very useful and feasible for computational purpose. Our main tool is the following theorem [21].

Theorem 1.1. Let 𝐾 be a normal cone of a Banach space 𝐸 and 𝑣₀≤𝑤₀. Suppose that (𝑎₁)𝑇∶[𝑣₀,𝑤₀]→𝐸 is completely continuous,(𝑎₂)𝑇 is monotone increasing on [𝑣₀,𝑤₀],(𝑎₃)𝑣₀ is a lower solution of 𝑇, that is, 𝑣₀≤𝑇𝑣₀,(𝑎₄)𝑤₀ is an upper solution of 𝑇, that is, 𝑇𝑤₀≤𝑤₀. Then the iterative sequences 𝑣_𝑛=𝑇𝑣_𝑛−1,𝑤_𝑛=𝑇𝑤_𝑛−1(𝑛=1,2,3,…) (1.4) satisfy 𝑣₀≤𝑣₁≤⋯≤𝑣_𝑛≤⋯≤𝑤_𝑛≤⋯≤𝑤₁≤𝑤₀ (1.5) and converge to, respectively, 𝑣 and 𝑤∈[𝑣₀,𝑤₀], which are fixed points of 𝑇.

Throughout this paper, we always assume that the following conditions are satisfied:(𝐴₁)𝑓∈𝐶([0,1]×[0,+∞)×[0,+∞),[0,+∞));(𝐴₂)𝑔∈𝐶([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 𝛾 be a constant and 𝑦∈𝐶[0,1]. Then the BVP 𝑢⁽⁴⁾(𝑡)(𝑡)(),𝑢(0)=𝑦,𝑡∈0,1=𝑢^′′(0)𝑢=0,_′(1)=0,𝑢^′′′(1)=𝛾 (2.1) has a unique solution 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠−𝛾𝜙(𝑡),𝑡∈[0,1], (2.2) where 𝐺()=1𝑡,𝑠6𝑠6𝑡−3𝑡²−𝑠²𝑡,0≤𝑠≤𝑡≤1,6𝑠−3𝑠²−𝑡²𝜙(𝑡)=1,0≤𝑡≤𝑠≤1,21𝑡−6𝑡³,𝑡∈[0,1]. (2.3)

Proof. Let 𝑢 be a solution of the BVP (2.1). Then we may suppose that 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠+𝐴𝑡³+𝐵𝑡²+𝐶𝑡+𝐷,𝑡∈[0,1]. (2.4) By the boundary conditions in (2.1), we have 𝛾𝐴=6𝛾,𝐵=0,𝐶=−2,𝐷=0. (2.5) Therefore, the BVP (2.1) has a unique solution 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠−𝛾𝜙(𝑡),𝑡∈[0,1]. (2.6)

Lemma 2.2. For any (𝑡,𝑠)∈[0,1]×[0,1], one has ()0≤𝜕𝐺𝑡,𝑠≤()1𝜕𝑡1−𝑡𝑠,3𝑡²()≤1𝑠≤𝐺𝑡,𝑠22𝑡−𝑡²𝑠. (2.7)

Proof. For any fixed 𝑠∈[0,1], it is easy to know that 𝜕𝐺(𝑡,𝑠)=()()1𝜕𝑡1−𝑡𝑠,0≤𝑠≤𝑡≤1,1−𝑡𝑠−2(𝑠−𝑡)²,0≤𝑡≤𝑠≤1, (2.8) which shows that ()0≤𝜕𝐺𝑡,𝑠≤()()∈[]×[],𝜕𝑡1−𝑡𝑠for𝑡,𝑠0,10,1 (2.9) and so 𝐺(𝑡,𝑠)=^𝑡₀𝜕𝐺(𝜏,𝑠)𝜕𝜏𝑑𝜏≤^𝑡₀𝑠()11−𝜏𝑑𝜏=22𝑡−𝑡²𝑠for(𝑡,𝑠)∈[0,1]×[0,1]. (2.10) 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. (2.11)

Lemma 2.3. For any 𝑡∈[0,1], one has (𝑡)≤10≤𝜙22𝑡−𝑡²,0≤𝜙^′(𝑡)≤1−𝑡. (2.12)

Proof. It is obvious.

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 the following conditions are satisfied:(𝐻₁)𝑓(𝑡,𝑢₁,𝑣₁)≤𝑓(𝑡,𝑢₂,𝑣₂)≤𝑎,0≤𝑡≤1,0≤𝑢₁≤𝑢₂≤𝑎,0≤𝑣₁≤𝑣₂≤𝑎; (𝐻₂) (−𝑎/2)≤𝑔(𝑧₂)≤𝑔(𝑧₁),0≤𝑧₁≤𝑧₂≤𝑎.Then the BVP (1.3) has monotone positive solutions.

Proof. Let 𝐸=𝐶¹[0,1] be equipped with the norm ‖𝑢‖=maxmax_[]𝑡∈0,1|𝑢(𝑡)|,max_[]𝑡∈0,1𝑢^′(𝑡,(𝑡))𝐾=𝑢∈𝐸∶𝑢≥0,𝑢^′(𝑡)≥0for𝑡∈[0,1]. (3.1) Then 𝐾 is a normal cone in Banach space 𝐸. Note that this induces an order relation ≤ in 𝐸 by defining 𝑢≤𝑣 if and only if 𝑣−𝑢∈𝐾. If we define an operator 𝑇∶𝐾→𝐸 by (𝑇𝑢)(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠)𝑑𝑠−𝑔(𝑢(1))𝜙(t),𝑡∈[0,1], (3.2) then (𝑇𝑢)^′(𝑡)=¹₀𝜕𝐺(𝑡,𝑠)𝑓(𝑠)𝜕𝑡𝑠,𝑢,𝑢^′(𝑠)𝑑𝑠−𝑔(𝑢(1))𝜙^′(𝑡),𝑡∈[0,1], (3.3) which together with (𝐴₁), (𝐴₂), and Lemmas 2.2 and 2.3 implies that 𝑇∶𝐾→𝐾. Obviously, fixed points of 𝑇 are monotone nonnegative solutions of the BVP (1.3).
Let 𝑣₀(𝑡)≡0 and 𝑤₀(𝑡)=𝑎(2𝑡−𝑡²)/2, 𝑡∈[0,1]. First, it is easy to verify that 𝑇∶[𝑣₀,𝑤₀]→𝐾 is completely continuous by an application of Arzela-Ascoli theorem. Now, we divide our proof into the following steps.
Step 1. We assert that 𝑇 is monotone increasing on [𝑣₀,𝑤₀].
Suppose that 𝑢,𝑣∈[𝑣₀,𝑤₀] and 𝑢≤𝑣. Then 0≤𝑢(𝑡)≤𝑣(𝑡)≤𝑎 and 0≤𝑢^′(𝑡)≤𝑣^′(𝑡)≤𝑎for𝑡∈[0,1]. By (𝐻₁) and (𝐻₂), we have (𝑇𝑢)(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),𝑢^′(𝑠)(𝑢(1𝜙(𝑡)≤𝑑𝑠−𝑔))₁₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣(𝑠),𝑣^′(𝑠(𝑣(1𝜙(𝑡)=(𝑡)[],())𝑑𝑠−𝑔))𝑇𝑣)(,𝑡∈0,1𝑇𝑢^′(𝑡)=¹₀𝜕𝐺(𝑡,𝑠)𝑓(𝑠)𝜕𝑡𝑠,𝑢,𝑢^′(𝑠)𝑑𝑠−𝑔(𝑢(1))𝜙^′(𝑡)≤¹₀𝜕𝐺(𝑡,𝑠)𝑓(𝑠)𝜕𝑡𝑠,𝑣,𝑣^′(𝑠)𝑑𝑠−𝑔(𝑣(1))𝜙^′(𝑡)=(𝑇𝑣)^′(𝑡),𝑡∈[0,1], (3.4) which shows that 𝑇𝑢≤𝑇𝑣.Step 2. We prove that 𝑣₀ is a lower solution of 𝑇.
For any 𝑡∈[0,1], we know that 𝑇𝑣₀(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓(𝑠,0,0)𝑑𝑠−𝑔(0)𝜙(𝑡)≥0=𝑣₀(𝑡),𝑇𝑣₀^′(𝑡)=¹₀𝜕𝐺(𝑡,𝑠)𝑓()(0)𝜙𝜕𝑡𝑠,0,0𝑑𝑠−𝑔^′(𝑡)≥0=𝑣^′₀(𝑡), (3.5) which implies that 𝑣₀≤𝑇𝑣₀.Step 3. We show that 𝑤₀ is an upper solution of 𝑇.
It follows from Lemmas 2.2 and 2.3, (𝐻₁), and (𝐻₂) that 𝑇𝑤₀(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑤₀(𝑠),𝑤^′₀(𝑠)𝑑𝑠−𝑔𝑤₀(1)𝜙(𝑡)≤2𝑡−𝑡²2¹₀𝑠𝑓𝑠,𝑤₀(𝑠),𝑤^′₀(𝑠𝑎)𝑑𝑠+2𝑡−𝑡²4≤𝑎2𝑡−𝑡²4+𝑎2𝑡−𝑡²4=𝑤₀(𝑡),𝑡∈[0,1],𝑇𝑤₀^′(𝑡)=¹₀𝜕𝐺(𝑡,𝑠)𝑓𝜕𝑡𝑠,𝑤₀(𝑠),𝑤^′₀(𝑠)𝑑𝑠−𝑔𝑤₀(1)𝜙^′(𝑡)≤(1−𝑡)¹₀𝑠𝑓𝑠,𝑤₀(𝑠),𝑤^′₀(𝑠)𝑎()𝑑𝑠+1−𝑡2≤𝑎(1−𝑡)2+𝑎(1−𝑡)2=𝑤^′₀(𝑡),𝑡∈[0,1], (3.6) which indicates that 𝑇𝑤₀≤𝑤₀.Step 4. We claim that the BVP (1.3) has monotone positive solutions.
In fact, if we construct sequences {𝑣_𝑛}^∞_𝑛=1 and {𝑤_𝑛}^∞_𝑛=1 as follows: 𝑣_𝑛=𝑇𝑣_𝑛−1,𝑤_𝑛=𝑇𝑤_𝑛−1,𝑛=1,2,3,…, (3.7) then it follows from Theorem 1.1 that 𝑣₀≤𝑣₁≤⋯≤𝑣_𝑛≤⋯≤𝑤_𝑛≤⋯≤𝑤₁≤𝑤₀, (3.8) and {𝑣_𝑛}^∞_𝑛=0 and {𝑤_𝑛}^∞_𝑛=0 converge to, respectively, 𝑣 and 𝑤∈[𝑣₀,𝑤₀], which are monotone solutions of the BVP (1.3). Moreover, for any 𝑡∈(0,1], by Lemmas 2.2 and 2.3, we know that 𝑇𝑣₀(𝑡)=¹₀𝐺()𝑓()(0)𝜙(𝑡)≥𝑡𝑡,𝑠𝑠,0,0𝑑𝑠−𝑔₂3¹₀𝑠𝑓(𝑠,0,0)𝑑𝑠>0, (3.9) and so 0<𝑇𝑣₀(𝑡)≤(𝑇𝑣)(𝑡)=𝑣(𝑡)≤𝑤(𝑡),𝑡∈(0,1], (3.10) which shows that 𝑣 and 𝑤 are positive solutions of the BVP (1.3).

4. An Example

Example 4.1. Consider the BVP 𝑢⁽⁴⁾(𝑡)=14𝑢(𝑡)+1(𝑢161+𝑡)^′(𝑡)²(),𝑢(0)+1,𝑡∈0,1=𝑢^′′(0)𝑢=0,_′(1)=0,𝑢^′′′(1)1=−4(𝑢(1))². (4.1)
If we let 𝑓(𝑡,𝑢,𝑣)=(1/4)𝑢+1/16(1+𝑡)𝑣²+1for(𝑡,𝑢,𝑣)∈[0,1]×[0,+∞)×[0,+∞)and𝑔(𝑢)=−(1/4)𝑢²for𝑢∈[0,+∞), then all the hypotheses of Theorem 3.1 are fulfilled with 𝑎=2. It follows from Theorem 3.1 that the BVP (4.1) has monotone positive solutions 𝑣 and 𝑤. Moreover, the two iterative schemes are 𝑣₀(𝑡)[],𝑣≡0,𝑡∈0,1_𝑛+1(𝑡)=16^𝑡₀𝑠6𝑡−3𝑡²−𝑠²14𝑣_𝑛(𝑠)+1(𝑣161+𝑠)^′_𝑛(𝑠)²+1+1𝑑𝑠6¹_𝑡𝑡6𝑠−3𝑠²−𝑡²14𝑣_𝑛(𝑠)+1()𝑣161+𝑠^′_𝑛(𝑠)²+1+1𝑑𝑠4𝑣_𝑛(1)²121𝑡−6𝑡³[]𝑤,𝑡∈0,1,𝑛=0,1,2,…,₀(𝑡)=2𝑡−𝑡²[],𝑤,𝑡∈0,1_𝑛+1(𝑡)=16^𝑡₀𝑠6𝑡−3𝑡²−𝑠²14𝑤_𝑛(𝑠)+1()𝑤161+𝑠^′_𝑛(𝑠)²+1+1𝑑𝑠6¹_𝑡𝑡6𝑠−3𝑠²−𝑡²14𝑤_𝑛(𝑠)+1()𝑤161+𝑠^′_𝑛(𝑠)²+1+1𝑑𝑠4𝑤_𝑛(1)²121𝑡−6𝑡³,𝑡∈[0,1],𝑛=0,1,2,…. (4.2) The first, second, and third terms of the two schemes are as follows: 𝑣₀(𝑡)𝑣≡0,₁(𝑡)=131𝑡−6𝑡³+1𝑡24⁴,𝑣₂(𝑡)=205151580608𝑡−169109𝑡967680³+145𝑡3456⁴+13𝑡17280⁵−1𝑡17280⁶−1𝑡15120⁷+19𝑡967680⁸+1𝑡580608⁹−1𝑡580608¹⁰+1𝑡4561920¹¹,𝑤₀(𝑡)=2𝑡−𝑡²,𝑤₁(𝑡)=263480𝑡−73𝑡288³+5𝑡96⁴+1𝑡480⁵−1𝑡720⁶+1𝑡3360⁷,𝑤₂(𝑡)=2184413040215053644199158425017821892349788160𝑡−53437308090412822335713𝑡292125089109461748940800³+3755569𝑡88473600⁴+574129𝑡442368000⁵−19199𝑡132710400⁶−12433𝑡103219200⁷+47291𝑡1238630400⁸+8953𝑡1592524800⁹−17377𝑡4644864000¹⁰+19373𝑡51093504000¹¹+161𝑡2919628800¹²−269𝑡12651724800¹³+31𝑡14760345600¹⁴+1𝑡6709248000¹⁵−1𝑡23003136000¹⁶+1𝑡210567168000¹⁷. (4.3)

Acknowledgment

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