On the fourth-order nonlinear beam equation of a small deflection with nonlocal conditions

1.   Introduction

Fourth-order boundary value problems have important applications in physics and mechanical engineering because they describe deflection (deformation) or bending of elastic beams. The state of the deflection of an elastic beam is modeled by the Euler-Bernoulli equation

This equation is widely used in mechanics and it is called the beam equation. Here; $u$ represents the deflection of the beam, $u^{\prime\prime}$ is the bending moment stiffness, $u^{(4)}$ is the load density stiffness, $f$ is the force per unit length, which represents the distributed load, and $\lambda$ is a parameter that represents the reciprocal of the flexural rigidity which measures the resistance to bend (see ^[1], page 199 for further details regarding the mechanics of the beam equation).

In ^[2], the author considered the linear fourth-order differential equation

subject to

where $q$ and $f$ are continuous functions on $[0, 1]$, and established a sufficient condition $\underset{0\leq x\leq1}{\sup}\left|q(x)\right| < \pi^{4}$ to guarantee a unique solution for this problem. In ^[3], the author investigated (1.1) under the condition that $f$ is continuous and bounded on $[0, 1]\times\mathbb{R}\times\mathbb{R},$ and subject to (1.3) as well as other types of boundary conditions, and established results on the existence and uniqueness theorems under suitable conditions. In ^[4] the author considered the same problem of ^[3] under more general conditions on $f,$ and established the existence of the solution of the equation subject to (1.3) under the condition that $f$ is continuous on $[0, 1]\times\mathbb{R}\times\mathbb{R},$ and replacing the boundedness condition with the growth condition

for some positive constants $a, b, c$ such that $a+b\pi^{2} < \pi^{4}.$ Since then, the problem of investigating existence of solutions of the equation received considerable attention from researchers in the last two decades (see ^{[5,6,7,8,9,10,11,12,13]} and references therein). The authors in ^{[9,11,13,14,15]} considered the equation under integral boundary conditions.

In this paper, we establish an existence and uniqueness theorem for the following boundary value problem

subject to the integral boundary conditions

where $A\in \mathbb{C}[0, 1],$ $p, q\in L^{1}[0, 1]$

and $f$ is continuous on $[0, 1]\times\mathbb{R}\times\mathbb{R}$ and satisfies a growth condition with variable parameters:

where $a, b, c$ are positive continuous functions on $[0, 1].$

The problem (1.5)-(1.6) generalizes the preceding problems in the following sense:

1. If $A = 0$ then (1.5) reduces to (1.1).

2. If $p = q = 0$ then (1.6) reduces to (1.3).

3. If $a, b, c$ are constants then (1.7) reduces to (1.4).

We are particularly interested in the case of small deflection. Deflections are small as long as they are below the elastic limit, and in this case the deflection curve might be almost flat, so the bending won't be noticed by eye. When the deflection is small, Hook's law is applicable, and this will give Euler-Bernoulli Equation. Here, shear distortion and effects of rotatory inertia are negligible due to the absence of the axial forces. If the axial force exists and becomes a function of transverse displacement, large deflection occurs and we need to seek help from nonlinear beam theory. It should be noted that most of the beams used in industry and constructions (Towers, bridges, aircrafts, ...) possess enery small deflections, since large deflections can cause cracks in the beams, and this may eventually lead to catastrophic damages. For more details about the beam theory we refer the reader to (^[16], pp 758–760). Small deflections usually occur when either the loaded force $f$ is small (so $a, b, c$ are small), or the material of the beam has high flexural rigidity which implies that $\lambda$ is small.

We propose the following hypothesis

where $d = \max\{a, b\}.$ Note that if $A_{1} = 0$ then $\lambda < \dfrac{16\alpha\beta}{5d}.$ This seems natural and reflects the fact that small amount of force and large amount of force loaded on a beam of high flexural rigidity will always produce small deflections.

2.   Existence and uniqueness theorems

2.1.   Existence theorem

The Pr.(1.5)-(1.6) can be converted into the following system:

Thus, we shall prove the following statement

Proposition 2.1. If (1.7) and (1.8) hold, then there exists a constant $M > 0$ such that for any $x\in [0, 1]$ and any solution $u$ to Pr.(1.5)-(1.6), we have

where $\parallel u \parallel_{\rho, 0} = \underset{0\leq x\leq1}{\max}\mid \rho(x) u(x)\mid,$ $\parallel u \parallel_{\rho, 1} = \parallel u'' \parallel_{\rho, 0}$ and $\rho(x) = x(1-x), \ x\in [0, 1].$

Proof. Multiplying both sides of the first equation of (2.1) by $\rho(x)u$ and integrating the resulting equation from $0$ to $1,$ then employing integration by parts, we obtain

Taking into account $u(0) = u(1) = \int_{0}^{1}p(x)u(x)dx,$ we have

The integrals $\int_{0}^{1}p(x)u(x)dx$ and $\int_{0}^{1}\rho(x)u(x)v(x)dx$ can be estimated by means of the Cauchy-Schwarz inequality

and

Since $\underset{0\leq x\leq1}{\sup} \rho(x) = \frac{1}{4},$ we have

Thus

Hence

Since

Thus

It follows that

where $C_{1} = \dfrac{1}{16\alpha^{2}}$ and $\alpha = 1-\int_{0}^{1}p^{2}(x)dx.$

Proceeding as before, multiplying both sides of the second equation of (2.1) by $\rho(x)v$ and integrating the resulting equation from $0$ to $1,$ then employing integration by parts, taking into account the nonlocal boundary conditions $v(0) = v(1) = \int_{0}^{1}q(x)v(x)dx,$ we obtain

Note that

and

Applying (1.7) to $f (x, \ u, \ v)$ by assuming that $a(x)\leq a, \ b(x)\leq b, \ c(x)\leq c, \ \forall x\in [0, 1]$ with $a, b, c > 0,$ to obtain

The integral $\int_{0}^{1}\mid c(x) v(x)\mid dx$ can be estimated by means of the $\epsilon-$inequality

Thus

Since

Substituting (2.19) into (2.14) and (2.18), we obtain

and

Now using (2.15), (2.20) and (2.21), we obtain

Let

Writing $\sqrt{C_{1}} = \dfrac{1}{4\alpha}$ and using the fact that $\alpha < 1$ and $\beta = 1-\intop_{0}^{1}q^{2}(x)dx,$ we see that

This implies that $1-\left(\intop_{0}^{1}q^{2}(x)dx+(\frac{A_{1}}{4}+\frac{\lambda a}{4})\sqrt{C_{1}}+\frac{\lambda b}{4}\right) > 0.$

Now, we can choose $\epsilon$ small such that $\gamma > 0.$ It follows that

Hence

where $M_{1} = \dfrac{\frac{ \lambda c^{2}}{8\epsilon}}{\gamma}.$ Combining (2.25) with (2.12), we have

where $M_{2} = C_{1}M_{1}.$

On the other hand, we have

Thus

Using $\underset{0\leq x\leq1}{\sup}\mid\rho'(x)\mid = 1,$ applying Hölder's inequality and using $\rho^{2}(x)\leq\rho(x), \ x\in [0, 1],$ we obtain

Similarly,

These two inequalities imply the required result, and complete the proof of the proposition.

The fundamental theorem used in proving the existence of the solution is Schauder's theorem. In order to make use of this theorem, it is sufficient to present the following lemmas.

Lemma 2.2. ^[17] Let $g:[0, 1]\rightarrow \mathbb{R}$ be a continuous function. If $1- \int_{0}^{1}p^{2}(x) dx = \alpha > 0,$ then the unique solution $u$ of the following boundary value problem

subject to the nonlocal boundary conditions $u(0) = u(1) = \int_{0}^{1}p(x)u(x)dx$ is given by

where $G_{1}(x; y)$ is the Green function of this BVP and given by (Eqs.(7)-(10) see ^[17]).

Thus from (2.1), we obtain an equivalent integral system

Define the Banach space

with norm $\parallel u \parallel_{\rho, 2} = \parallel u \parallel_{\rho, 0}+\parallel u\parallel_{\rho, 1},$ where $\parallel u \parallel_{\rho, 1} = \parallel u'' \parallel_{\rho, 0}.$ Also, define the operator $T: \mathbb{X} \longrightarrow \mathbb{X}$ by $T(u, v) = \left(T_{1}(u, v), T_{2}(u, v)\right),$ where $\mathbb{X} = \mathbb{Y}_{\rho}\times\mathbb{Y}_{\rho}$ with norm $\parallel (u, v) \parallel_{\rho, 2} = \parallel u \parallel_{\rho, 0}+\parallel v \parallel_{\rho, 0}$ and

and

Consider the closed and convex set

Lemma 2.3. For any $(u, v)\in \mathbb{S},$ $T(u, v)$ is contained in $\mathbb{S}.$

Proof. From the definition of $T(u, v)$ we have

Thus

Assume that $\underset{0\leq x\leq1}{\sup}\mid G_{1}(x, s)\mid\leq L$ and from (2.25), in particular, we have $\int_{0}^{1}v^{2}(x)dx \leq M_{1},$ thus

Similarly

Thus

It follows that $\parallel T(u, v) \parallel_{\rho, 2} \leq M,$ where $M = M^{*}_{1}+M^{*}_{2}.$ On account of the continuity of $f(x, u, v),$ $u$ and $v,$ it follows that $T(u, v)$ is continuous. This shows that $T(u, v)$ is also contained in $\mathbb{S}.$

To prove that $T(u, v)$ is compact, we use the Arzela-Ascoli lemma, that is $T(\mathbb{S})$ must be closed, bounded and equicontinuous.

In order to prove that $T(\mathbb{S})$ is equicontinuous, it is sufficient to prove that the inequality

is satisfied for any $x$ and $y$ in the interval $[0, 1].$ It follows by the definition of $T(u, v)$ that

Similarly, for $T_{2}(u(x), v(x)),$ we have

which proves the equicontinuous of $T(u, v).$

Consequently, $T(u, v)$ has a fixed point by the Schauder's fixed point theorem.

Thus, we have

Theorem 2.4. Under the hypothesis of Proposition 2.1, there exists a continuous solution $(u, v)$ which satisfies system (2.1).

2.2.   A uniqueness theorem

A uniqueness theorem can also be obtained if we assume that $f (x, \ u, \ u'')$ satisfies a Lipschitz condition in $u$ and $u''$ with constants $k_{i} > 0, \ i = 0, 1$ such that

and

Thus, we have

Theorem 2.5. If $f$ is Lipschitz in $u$ and $u'',$ where the constants $k_{i} > 0, \ i = 0, 1$ satisfy (2.46) and if (1.7) and (1.8) hold, then the system (2.1) has a unique solution $(u, \ v).$

Proof. Suppose there are two solutions $u$ and $v$ such that $u\neq v.$ Then from Pr.(1.5)-(1.6), we have

subject to

where $w = u-v.$

Thus

Proceeding as before, we obtain

and

where

and

Applying the Lipschitz condition to $f (x, \ u, \ u'')-f (x, \ v, \ v'')$ to obtain

Combining these inequalities, we obtain

where $\gamma_{1} = 1-\left(\int_{0}^{1}q^{2}(x)dx+ (\frac{ A_{1}\sqrt{C_{1}}}{4}+ \frac{\lambda k_{0}\sqrt{C_{1}}}{4})+ \frac{\lambda k_{1}}{4}\right) > 0.$ This is a contradiction.

This completes the proof.

Example 1. Consider

under the boundary conditions (1.6) with $p(x) = q(x) = \frac{1}{2}x^{2}.$ Here $r:[0, 1]\rightarrow \mathbb{R}$ is a continuous function. Let

It is clear that

So that $a(x) = b(x) = 1$ and $c(x) = r(x).$ However, $\lambda = \frac{1}{2} < \frac{16 \alpha\beta -A_{1}}{5d}$ since $\underset{0\leq x\leq1}{\sup}A(x) = A_{1} = \pi^{2}$ and $d = \max\{a, b\} = 1.$

Hence by Theorem 2.4, the solution exists.

To prove the uniqueness, note that $f (x, \ u, \ u'')$ is Lipschitz:

Indeed,

So $k_{0} = 1$ and $k_{1} = 1.$ But the condition (2.46) implies that $\lambda < \frac{16\alpha\beta-A_{1}}{1+4\alpha}.$ A simple substitution gives $\frac{16\alpha\beta-A_{1}}{1+4\alpha}\approx 0.95 > \frac{1}{2},$ this means that the condition (2.46) is satisfied. So by Theorem 2.5 the solution is unique.

Acknowledgments

The authors would like to acknowledge the support of Prince Sultan University, Saudi Arabia for paying the Article Processing Charges (APC) of this publication. The authors would like to thank Prince Sultan University for their support.

The authors would like to thank the reviewers for their valuable comments.

Conflict of interest

The authors declare no conflict of interest.