A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

1.   Introduction

The Sturm-Liouville problem arises within many areas of science, engineering and applied mathematics. It has been studied for more than two decades. Many physical, biological and chemical processes are described using models based on it (see ^[1,2,3], ^[8], ^[9] and ^[11]).

For the homogeneous Sturm-Liouville problem with nonlocal conditions you can see ^[2], ^[9] and ^{[11,12,13,14,15]}. For the nonhomogeneous equation see ^[7]. In ^[7] the authors studied the nonhomogeneous Sturm-Liouville boundary value problem of the differential equation

with the conditions

Here, we are concerned, firstly, with the nonlocal problem of the nonlinear differential inclusion

with the nonlocal conditions ($\; \eta > \xi$)

For

we study the existence of multiple solutions (eignevalues and eignefunctions) of the nonhomogeneous Sturm-Liouville problem of the differential equation

with the conditions (1.2).

The special case of the nonlocal condition (1.2)

will be considered.

2.   Existence of solutions

Consider the nonlocal boundary value problem of the nonlinear differential inclusion (1.1)-(1.2) under the following assumptions.

(ⅰ) The set $F(t, x)$ is nonempty, closed and convex for all $(t, x)\in [0, 1]\times R\times R$.

(ⅱ) $F(t, x)$ is measurable in $t\in [0, 1]$ for every $x, y\in R$.

(ⅲ) $F(t, x)$ is upper semicontinuous in $x$ and $y$ for every $t\in [0, 1]$.

(ⅳ) There exist a bounded measurable function $m: [0, 1]\longrightarrow R$ and a constant $\lambda$, such that

Remark 1. From the assumptions (i)-(iv) we can deduce that (see ^[1], ^[5] and ^[6]) there exists $f\in F(t, x)$, such that

(v) $f:I\times R\longrightarrow R$ is measurable in $t$ for every $x, y\in R$ and continuous in $x$ for $t\in [0, 1]$ and there exist a bounded measurable function $m:[0, \pi] \rightarrow R$ and a constant $\lambda^2$ such that

and $f$ satisfies the nonlinear differential equation

So, any solution of (2.1) is a solution of (1.1).

(ⅵ) $\lambda \, (\eta-\xi) \neq -2, \; \lambda \in R$.

(ⅶ)

For the integral representation of the solution of (2.1) and (1.2) we have the following lemma.

Lemma 2.1. If the solution of the problem (2.1) and (1.2) exists, then it can be represented by the integral equation

where ${\mathcal{A}} = (\eta-\xi)[2+\lambda(\eta-\xi)]\neq 0$.

Proof. Integrating both sides of Eq (2.1) twice, we obtain

and using the assumption $x' (0)-\lambda x (0) = 0$, we obtain

The assumption $\; \int_\xi^\eta x(\tau)d\tau = 0$ implies that

and we can get

Substituting (2.5) into (2.4), we obtain

Now from (2.3), (2.5) and (2.6), we obtain

To complete the proof, differentiate equation (2.2) twice, we obtain

and

Now

and

From that, we get $x' (0)-\lambda x (0) = 0$.

Now, to ensure that $\; \int_\xi^\eta x(\tau) d\tau = 0$,

we have

from that, we obtain as before

This proves the equivalence between the integral equation (2.2) and the nonlocal boundary value problem (1.1)-(1.2).

Now, for the existence of at least one continuous solution for the problem of the integral equation (2.2), we have the following theorem.

Theorem 2.1. Let the assumptions (v)-(vii) be satisfied, then there exists at least one solution $x\in C[0, \pi]$ of the nonlocal boundary value problem (2.1) and (1.2). Moreover, from Remark 1, then there exists at least one solution $x\in C[0, \pi]$ of the nonlocal boundary value problem (1.1)-(1.2).

Proof. Define the set $Q_r\subset C[0, \pi]$ by

It is clear that the set $Q_r$ is nonempty, closed and convex.

Define the operator $T$ associated with (2.2) by

Let $x\in Q_r$, we have

and we have

Then $T:Q_r\rightarrow Q_r$ and the class $\{Tx\}\subset Q_r$ is uniformly bounded in $Q_r$.

In what follows we show that the class $\{Tx\}$, $x\in Q_r$ is equicontinuous. For $t_1, t_2\in [0, \pi], \; t_1 < t_2$ such that $|t_2-t_1| < \delta$, we have

Hence the class of function $\{Tx\}$, $x\in Q_r$ is equicontinuous. By Arzela-Ascolis ^[4] Theorem, we found that the class $\{Tx\}$ is relatively compact.

Now we prove that $T:Q_r\rightarrow Q_r$ is continuous.

Let $\{x_n\} \subset Q_r$, such that $x_n\rightarrow x_0\in Q_r$, then

and

Now, we have

and

then applying Lebesgue Dominated convergence theorem ^[4], we obtain

Then $Tx_n (t)\rightarrow Tx_0 (t)$. Which means that the operator $T$ is continuous.

Since all conditions of Schauder theorem ^[4] are hold, then $T$ has a fixed point in $Q_r$, then the integral equation (2.2) has at least one solution $x\in C[0, \pi]$.

Consequently the nonlocal boundary value problem (2.1)-(1.2) has at least one solution $x\in C[0, \pi].$ Moreover, from Remark 1, then there exists at least one solution $x\in C[0, \pi]$ of the nonlocal boundary value problem (1.1)-(1.2).

Now, we have the following corollaries

Corollary 1. Let $\; \lambda^2 x(t)\; = \; f(t, \lambda x(t))\; \in F(t, \lambda x(t)).\;$ Let the assumptions of Theorem 2.1 be satisfied. Then there exists at lease one solution $x \in C[0, \pi]$ of

with the nonlocal condition (1.2). Moreover, from Remark 1, there exists at lease one solution $x \in C[0, \pi]$ of the problem (1.1)-(1.2).

Corollary 2. Let the assumptions of Theorem 2.1 be satisfied. Then there exists a solution $x \in C[0, \pi]$ of the problem (2.1) and (1.4).

Proof. Putting $\xi = 0$ and $\eta = \pi$ and applying Theorem 2.1 we get the result.

3.   Maximal and minimal solutions

Taking $\; J = (0, \pi).\;$ Here, we study the existence of maximal and minimal solutions of the problem (2.1) and (1.2) which is equivalent to the integral equation (2.2).

Definition 3.1. ^[10] Let $\; q(t)\;$ be a solution $\; x(t)\;$ of (2.2) Then $\; q(t)\;$ is said to be a maximal solution of (2.2) if every solution of (2.2) on $\; J\;$ satisfies the inequality $\; x(t)\; \leq \; q(t), \; \; t\in J\;$. A minimal solution $\; s(t)\;$ can be defined in a similar way by reversing the above inequality i.e. $\; x(t)\; \geq\; s(t), \; t\in J.\;$

We need the following lemma to prove the existence of maximal and minimal solutions of (2.2).

Lemma 3.2. Let $\; f(t, x)\;$ satisfies the assumptionsin Theorem 2.1 and let $\; x(t), \; y(t)\;$ be continuous functions on $J$ satisfying

where one of them is strict.

Suppose $\; f(t, x)\;$ is nondecreasing function in$\; x.\;$ Then

Proof. Let the conclusion (3.1) be false; then there exists $\; t_1\;$ such that

and

From the monotonicity of the function $\; f\;$ in $\; x, \;$ we get

This contradicts the fact that $\; x(t_1)\; = \; y(t_1);\;$then

Theorem 3.2. Let the assumptions of Theorem 2.1 besatisfied. Furthermore, if $\; f(t, x)\;$ is nondecreasing function in$\; x, \;$ then there exist maximal and minimal solutions of (2.2).

Proof. Firstly, we shall prove the existence of maximal solution of (2.2). Let $\epsilon > 0$ be given. Now consider the integral equation

where

Clearly the function $\; f_{\epsilon} (t, x_\epsilon)\;$ satisfies assumption (v) and

Therefore, Equation (3.2) has a continuous solution $\; x_\epsilon(t)\;$ according to Theorem 2.1.

Let $\epsilon_1$ and $\epsilon_2$ be such that $0 < \epsilon_2 < \epsilon_1 < \epsilon.$ Then

Applying Lemma 3.2, then (3.3) and (3.4) imply that

As shown before in the proof of Theorem 2.1, the family of functions $\; x_{\epsilon}(t)\;$ defined by Eq (3.2) is uniformly bounded and of equi-continuous functions. Hence by the Arzela-Ascoli Theorem, there exists a decreasing sequence $\epsilon _n$ such that $\; \epsilon_n \rightarrow 0$ as $n \rightarrow \infty,$ and $\; \lim\limits_{n \rightarrow \infty} x_{\epsilon_n}(t)$ exists uniformly in $\; I.\;$ We denote this limit by $\; q(t).\;$ From the continuity of the function $f_{\epsilon_n}$ in the second argument, we get

which proves that $\; q(t)\;$ is a solution of (2.2).

Finally, we shall show that $\; q(t)\;$ is maximal solution of (2.2). To do this, let $\; x(t)\;$ be any solution of (2.2). Then

Applying Lemma 3.2, then (2.2) and (3.5 imply that

From the uniqueness of the maximal solution (see ^[10]), it is clear that $x_\epsilon (t)$ tends to $\; q(t)\;$ uniformly in $t\; \in\; J\; \; as\; \; \epsilon \rightarrow 0.$

In a similar way we can prove that there exists a minimal solution of (2.2).

4.   The homogeneous problem

Here, we study the existence and some general properties of the eigenvalues and eigenfunctions of the problem of the homogeneous equation

with the nonlocal condition (1.2).

Lemma 4.3. The eigenfunctions of the nonlocal boundary value problem (4.1) and (1.2) are in the form of

Proof. Firstly, we prove that the eigenvalues are

The general solution of the problem (4.1) and (1.2) is given by

Differentiating equation (4.4), we obtain

Using the first condition, when $t = 0$, we obtain

Integrating both sides of (4.4) from $\xi$ to $\eta$, we obtain

Substituting $c_1 = c_2$, we obtain

Multiplying (4.6) by $\frac{\lambda}{c_1}$, we obtain

From (4.7), we deduce that

Therefore, from (4.4) we can get

Corollary 3. The eigenfunctions of the nonlocal boundary value problem (4.1) and (1.4) are in the form of

Proof. Putting $\xi = 0$ and $\eta = \pi$ and applying Lemma 4.3 we obtain the result.

5.   The nonhomogeneous problem

Now, we study the existence of multiple solutions of the nonhomogeneous problem (1.3) and (1.2). Let $x_1, \; x_2$ be two solutions of the problem (1.3) and (1.2). Let $u(t) = x_1(t)-x_2(t)$, then the function $u$ satisfy the Sturm-Liouville problem

with the nonlocal conditions

So, the values of (eigenvalues) $\lambda_n$ for the non zero solution of (4.1) and (1.2) is the same values (eigenvalues) of $\lambda_n$ for the multiple solutions (eigenfunctions) of (1.3) and (1.2), i.e.

Theorem 5.3. The multiple solutions (eigenfunctions) $x_n(t)$ of the problem (1.3) and (1.2) are given by

Proof. Here we use the variation of parameter method to get the solution of (1.3) and (1.2). Assume that the solutions of (1.3) and (1.2) are given by

So, we have

Now, we can get $W (x_1, x_2) = \lambda$. Hence

thus

From (5.3) and (5.2), we obtain

By using the first condition $x' (0)- \lambda x (0) = 0$, we get

therefore the multiple solutions of the nonlocal problem (1.3) and (1.2) are given by

To complete the proof and to ensure that $x_n (t)$ is the solution of (1.3) and (1.2), we firstly prove that

Differentiating (5.4) twice, we get

and

and

Also we have $x' (0)- \lambda x (0) = 0$.

Example 1. Let $\; h(t, \lambda)\; = \; \lambda^2.\;$ Then we find that

and the multiple solutions of the nonlocal problem (1.3) and (1.2) are given by

Now consider the Riemann integral boundary condition (1.4).

Corollary 4. The multiple solutions (eigenfunctions) $x_n(t)$ of the problem (1.3)-(1.4) are given by

Proof. In this special case, we put $\xi = 0$ and $\eta = \pi$ and applying Theorem 5.3 we get the result.

Example 2. Let $\; h(t, \lambda)\; = \; \lambda^2.\;$ Then we find that

and the solution $\; x_n(t)\;$ of the problem (1.3)-(1.4) are given by

6.   Conclusions

Here, we proved the existence of solutions $\; x \in C[0, \pi]\;$ of the nonlocal boundary value problem of the differential inclusion (1.1) with the nonlocal condition (1.2).

The maximal and minimal solutions of the problem (1.1)-(1.2) have been proved. The eigenvalues and eigenfunctions of the homogeneous and nonhomogeneous equations (4.1) and (1.3) with the nonlocal condition (1.2) have been obtained. Two examples have been studied to illustrate our results.

Acknowledgments

We thank the referees for their constructive remarks and comments on our work which reasonably improved the presentation and the structure of the manuscript.

Conflict of interest

The authors declare no conflict of interest.