Hyers–Ulam Stability of Order k for Euler Equation and Euler–Poisson Equation in the Calculus of Variations

Abstract

In this paper, we define and study Hyers–Ulam stability of order 1 for Euler’s equation and Hyers–Ulam stability of order $m-1$ for the Euler–Poisson equation in the calculus of variations in two special cases, when these equations have the form $y^{″}\left(x\right)=f\left(x\right)$ and $y^{\left(m\right)}\left(x\right)=f\left(x\right),$ respectively. We prove some estimations for $\left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|$, where y is an approximate solution and $y_0$ is an exact solution of the corresponding Euler and Euler-Poisson equations, respectively. We also give two examples.

1. Introduction

Hyers–Ulam stability has been the subject of many papers. Ulam stability was proposed by Ulam in [1] in 1940. The first result in this direction was given in 1941 by Hyers [2]. The first authors which started the study of Hyers–Ulam stability of differential equations was Obloza [3] and Alsina and Ger [4]. After that, many types of differential equations were studied. First-order linear differential equations and linear differential equations of higher order were investigated, for example, in [5,6,7,8,9,10,11,12,13]. Integral equations in [14,15,16,17,18,19,20,21,22,23,24] and partial differential equations in [25,26,27,28,29,30] have also been studied. The books [31,32] can be consulted for more details. The Hyers–Ulam stability of fractional differential equations and of fractional integral equations has recently begun to be studied (see [33,34,35,36,37,38,39,40,41,42]).

In what follows, we define and study Hyers–Ulam stability of order 1 for Euler’s equation and Hyers–Ulam stability of order $m-1$ for the Euler–Poisson equation in the calculus of variations. Section 2 is dedicated to the study of Euler’s equation, and Section 3 is dedicated to the Euler–Poisson equation. We also establish an estimation for $\left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|$, where y is an approximate solution and $y_0$ is an exact solution for the considered equations.

This paper is a continuation of the paper [43]. In [43], Hyers–Ulam stability of the Euler equation in two special cases was studied when $F=F(x,y^{\prime })$ and when $F=F(y,y^{\prime }).$ An estimation for $\left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|$ is also given in [43] for the case $F=F(x,y^{\prime })$. This was the first time, in [43], that the problem of Ulam stability of extremals for functionals represented in integral form was studied. In [43], a direct method and the Laplace transform method were used. Here, we will use Taylor’s formula.

2. Hyers–Ulam Stability of Euler’s Equation

We consider a class of functions $A\subseteq C^2\left(I,\mathbb{R}\right),I\subseteq \mathbb{R}$ an open interval. Let $\left[a,b\right]\subset I,a<b,a\ge 0$. Let $y\in A,y=y\left(x\right)$ be an element in A. Let the function $F:M⟶\mathbb{R},$$M=I\times {\mathbb{R}}^2,F\in C^2\left(M\right)$. We consider the functional

$$J\left[y\left(x\right)\right]={\int }_a^{b}F\left(x,y,y^{\prime }\right)dx,J:A\to \mathbb{R},y\in A,J\left[y\left(x\right)\right]\in \mathbb{R},$$

(1)

and the conditions

$$y\left(a\right)=y_a,y^{\prime }\left(a\right)=y_a^{\prime },$$

(2)

where $y_a,y_a^{\prime }\in \mathbb{R}$ are given.

We consider the following problem in the calculus of variations (see [44]): find the extremum of this functional. The necessary condition of extremum (see [44]) is given by Euler’s equation

$$F_y^{\prime }\left(x,y,y^{\prime }\right)-\frac{d}{dx}\left[F_{y^{\prime }}^{\prime }\left(x,y,y^{\prime }\right)\right]=0,y\in C^2\left[a,b\right].$$

(3)

Equation (3) can be represented (by derivation) in the form:

$$\frac{{\partial }^{2}F}{\partial y^{\prime 2}}·y^{″}+\frac{{\partial }^{2}F}{\partial y^{\prime }\partial y}·y^{\prime }+\frac{{\partial }^{2}F}{\partial y^{\prime }\partial x}-\frac{\partial F}{\partial y}=0,y\in C^2\left[a,b\right].$$

(4)

The solutions of Equation (3) or (4) are called extremals.

We will study Hyers–Ulam stability of Equation (3) (or (4)).

Let $\epsilon >0$ and $a,b\in (0,\infty )$.

We consider the following inequalities:

$$\left|F_y^{\prime }\left(x,y,y^{\prime }\right)-\frac{d}{dx}\left[F_{y^{\prime }}^{\prime }\left(x,y,y^{\prime }\right)\right]\right|\le \epsilon ,y\in C^2\left[a,b\right],$$

(5)

or

$$\left|\frac{{\partial }^{2}F}{\partial y^{\prime 2}}·y^{″}+\frac{{\partial }^{2}F}{\partial y^{\prime }\partial y}·y^{\prime }+\frac{{\partial }^{2}F}{\partial y^{\prime }\partial x}-\frac{\partial F}{\partial y}\right|\le \epsilon ,y\in C^2\left[a,b\right].$$

(6)

Definition 1.

Equation (3) (or (4)) is called Hyers–Ulam stable if there is a real number $c>0$ such that for any solution $y\left(x\right)$ of the inequality (5) (or (6)), there is a solution $y_0\left(x\right)$ of the Equation (3) (or (4)) such that

$$\left|y\left(x\right)-y_0\left(x\right)\right|\le c·\epsilon ,\forall x\in [a,b].$$

In the following definition, we give a new notion of stability, named stability of order 1.

Definition 2.

Equation (3) (or (4)) is called Hyers–Ulam stable of order 1 if there are real numbers $c_1>0,c_2>0$ so that for any solution $y\left(x\right)$ of the inequality (5) (or (6)), there is a solution $y_0\left(x\right)$ of Equation (3) (or (4)) such that

$$\left|y\left(x\right)-y_0\left(x\right)\right|\le c_1·\epsilon ,\forall x\in [a,b],$$

and

$$\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|\le c_2·\epsilon ,\forall x\in [a,b].$$

y is called an approximate solution and $y_0$ is called an exact solution for Equation (3) (or (4)).

In the following, we will study the case where Euler’s equation is

$$y^{″}\left(x\right)=f\left(x\right),f\in C[a,b],x\in [a,b].$$

(7)

Remark 1.

If $y=y\left(x\right)$ is a solution of (7), $x\in [a,b]$, then (see [45]),

$$y\left(x\right)=y\left(a\right)+\frac{x-a}{1!}{y}^{\prime }\left(a\right)+{\int }_a^x\frac{x-s}{1!}f\left(s\right)ds.$$

(8)

Let $\epsilon >0.$ We consider the inequality

$$\left|y^{″}\left(x\right)-f\left(x\right)\right|\le \epsilon ,y\in C^2\left[a,b\right].$$

(9)

Remark 2.

A function $y=y\left(x\right)$ is a solution of (9) if and only if there exists a function $g\in C\left[a,b\right]$ such that

(1)

$\left|g\left(x\right)\right|\le \epsilon ,\forall x\in \left[a,b\right],$

(2)

$y^{″}\left(x\right)-f\left(x\right)=g\left(x\right),\forall x\in \left[a,b\right].$

Remark 3.

If $y=y\left(x\right)$ is a solution of (9), using Remark 2 and Remark 1, we have

$$y\left(x\right)=y\left(a\right)+\frac{x-a}{1!}{y}^{\prime }\left(a\right)+{\int }_a^x\frac{x-s}{1!}\left(f\left(s\right)+g\left(s\right)\right)ds.$$

(10)

Theorem 1.

 (i) 

For each solution $y=y\left(x\right)$ of (9), there exists a unique solution $y_0=y_0\left(x\right)$ of (7) such that

$$\left\{\begin{array}{c}{y}_0\left(a\right)=y\left(a\right)\hfill \\ y_0^{\prime }\left(a\right)=y^{\prime }\left(a\right).\hfill \end{array}\right.$$

(11)

 (ii) 

The Equation (7) is Hyers–Ulam stable of order 1. If y is a solution of (9) and $y_0$ is a solution of (7) satisfying conditions (11), then

$$\left|y\left(x\right)-y_0\left(x\right)\right|\le \epsilon \frac{{(b-a)}^2}{2},\forall x\in [a,b]$$

(12)

and

$$\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|\le \epsilon \left(b-a\right).$$

(13)

 (iii) 

If there exists $l_1,l_2:[a,b]⟶\left[0,\infty \right)$ continuous, such that

$$\begin{array}{cc}\hfill & \left|F\left(x,y_1\left(x\right),y_1^{\prime }\left(x\right)\right)-F\left(x,y_2\left(x\right),y_2^{\prime }\left(x\right)\right)\right|\hfill \\ \hfill & \le l_1\left(x\right)·\left|y_1\left(x\right)-y_2\left(x\right)\right|+l_2\left(x\right)\left|y_1^{\prime }\left(x\right)-y_2^{\prime }\left(x\right)\right|,\forall x\in [a,b],y_1,y_2\in A,\hfill \end{array}$$

then

$$\left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|\le \epsilon ·\frac{{(b-a)}^2}{2}{\int }_a^b{l}_1\left(x\right)dx+\epsilon \left(b-a\right){\int }_a^b{l}_2\left(x\right)dx,$$

(14)

where y is a solution of (9) and $y_0$ is a solution of (7), both satisfying the conditions of (2).

Proof 

(i) This results from Cauchy–Picard’s theorem of existence and uniqueness (see [46]).

(ii) Let $y=y\left(x\right)$ be a solution of (9). Let $y_0=y_0\left(x\right)$ be the unique solution of (7) which verifies the corresponding Cauchy conditions of (11). We have

$$\begin{array}{cc}\hfill & \left|y\left(x\right)-y_0\left(x\right)\right|=\hfill \\ \hfill & \left|y\left(a\right)+\frac{x-a}{1!}{y}^{\prime }\left(a\right)+{\int }_a^x\frac{x-s}{1!}\left(f\left(s\right)+g\left(s\right)\right)ds-y\left(a\right)-\frac{x-a}{1!}{y}^{\prime }\left(a\right)-{\int }_a^x\frac{x-s}{1!}f\left(s\right)ds\right|,\hfill \end{array}$$

hence

$$\left|y\left(x\right)-y_0\left(x\right)\right|=\left|{\int }_a^x\frac{x-s}{1!}g\left(s\right)ds\right|\le {\int }_a^x\left|\frac{x-s}{1!}g\left(s\right)\right|ds,$$

so

$$\left|y\left(x\right)-y_0\left(x\right)\right|\le \epsilon \frac{{(b-a)}^2}{2},\forall x\in [a,b].$$

We also have

$$\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|\le {\int }_a^{x}g\left(s\right)ds\le \epsilon \left(b-a\right).$$

Thus, Equation (7) is Hyers–Ulam stable of order 1.

(iii) If y is a solution of (9) and $y_0$ is a solution of (7), both satisfying conditions (2), then

$$\begin{array}{cc}\hfill & \left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|\le {\int }_a^b\left[l_1\left(x\right)·\left|y\left(x\right)-y_0\left(x\right)\right|+l_2\left(x\right)\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|\right]dx\hfill \\ \hfill & \stackrel{\left(\right),\left(\right)}{\le }\epsilon ·\frac{{(b-a)}^2}{2}{\int }_a^b{l}_1\left(x\right)dx+\epsilon \left(b-a\right){\int }_a^b{l}_2\left(x\right)dx.\hfill \end{array}$$

(15)

□

Example 1.

We consider $J:A⟶R,A\subseteq C^2\left(I\right),[1,2]\subset I,$

$$J\left[y\left(x\right)\right]={\int }_1^2\left(y^{{\prime }^2}-2xy\right)dx,$$

(16)

and the conditions

$$y\left(1\right)=0,y^{\prime }\left(1\right)=-\frac{1}{3}.$$

(17)

The Euler equation becomes

$$y^{″}+x=0.$$

(18)

Let $\epsilon >0.$ We consider the inequality

$$\left|y^{″}+x\right|\le \epsilon .$$

(19)

We remark that

$$y_0\left(x\right)=-\frac{x^3}{6}+\frac{x}{6}$$

(20)

is a solution of Equation (18), satisfying (17).

If y is a solution of (19) and $y_0$ is a solution of (18), both satisfying (17), then applying Theorem 1, we get

$$\left|y\left(x\right)-y_0\left(x\right)\right|\le \epsilon \frac{{(x-1)}^2}{2}\le \frac{\epsilon }{2},\forall x\in [1,2],$$

(21)

and

$$\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|\le \epsilon (x-1)\le \epsilon ,\forall x\in [1,2],$$

(22)

hence, Equation (18) is Hyers–Ulam stable of order 1.

Moreover,

$$\begin{array}{cc}\hfill \left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|& \le {\int }_1^2\left|y^{{\prime }^2}\left(x\right)-2xy\left(x\right)-y_0^{{\prime }^2}\left(x\right)+2x{y}_0\left(x\right)\right|dx\hfill \\ \hfill & \le {\int }_1^2\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|\left|y^{\prime }\left(x\right)+y_0^{\prime }\left(x\right)\right|+2x\left|y\left(x\right)-y_0\left(x\right)\right|dx\hfill \\ \hfill & \le {\int }_1^2\left[\epsilon \left(x-1\right)\left(\epsilon \left(x-1\right)+x^2+\frac{1}{3}\right)+\epsilon x{\left(x-1\right)}^2\right]dx=\frac{\epsilon \left(2\epsilon +13\right)}{4}.\hfill \end{array}$$

3. Hyers–Ulam Stability of the Euler–Poisson Equation

Now, we consider functionals dependent on higher derivatives.

Let $n\in \mathbb{N},n\ge 2,$$A\subseteq C^{2n}\left(I,\mathbb{R}\right),I\subseteq \mathbb{R}$ be an open interval. Let $\left[a,b\right]\subset I,a<b$. Let $y\in A,y=y\left(x\right)$ be an element in A. Let the function $F:M⟶\mathbb{R},$$M=I\times {\mathbb{R}}^{n+1}$. We suppose that F is $n+2$ times differentiable with respect to all arguments.

Let

$$J\left[y\left(x\right)\right]={\int }_a^{b}F\left(x,y\left(x\right),y^{\prime }\left(x\right),\dots ,y^{\left(n\right)}\left(x\right)\right)dx,$$

(23)

and the conditions

$$y\left(a\right)=y_a,y^{\prime }\left(a\right)=y_a^{\prime },\dots ,y^{(2n-1)}\left(a\right)=y_a^{(2n-1)},$$

(24)

where $y_a,y_a^{\prime },\dots ,y_a^{(2n-1)}\in \mathbb{R}$ are given.

The extremals of the functional (23), given conditions (24), are the integral curves of the Euler–Poisson equation (see [44]):

$$F_y^{\prime }-\frac{d}{dx}\left[F_{y^{\prime }}^{\prime }\right]+\frac{d^2}{d{x}^2}\left[F_{y^{\prime \prime }}^{\prime }\right]-\dots +{\left(-1\right)}^n\frac{d^n}{d{x}^n}\left[F_{y^{\left(n\right)}}^{\prime }\right]=0.$$

(25)

Let $\epsilon >0$. We consider the inequality

$$\left|F_y^{\prime }-\frac{d}{dx}\left[F_{y^{\prime }}^{\prime }\right]+\frac{d^2}{d{x}^2}\left[F_{y^{″}}^{\prime }\right]-\dots +{\left(-1\right)}^n\frac{d^n}{d{x}^n}\left[F_{y^{\left(n\right)}}^{\prime }\right]\right|\le \epsilon .$$

(26)

We give a new notion of stability, named stability of order $k,k\ge 1,k\in \mathbb{N}$.

Definition 3.

Equation (25) is called Hyers–Ulam stable of order k if there are real numbers $C_1>0,C_2>0,\cdots ,C_{k+1}>0$ such that for any solution $y\left(x\right)$ of the inequality (26) there is a solution $y_0\left(x\right)$ of the Equation (25) such that

$$\left|y\left(x\right)-y_0\left(x\right)\right|\le C_1·\epsilon ,\forall x\in [a,b],$$

and

$$\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|\le C_2·\epsilon ,\forall x\in [a,b],$$

⋯

$$\left|y^{\left(k\right)}\left(x\right)-y_0^{\left(k\right)}\left(x\right)\right|\le C_{k+1}·\epsilon ,\forall x\in [a,b].$$

y is called an approximate solution and $y_0$ is called an exact solution for Equation (25).

In the following, we will study the case where the Euler–Poisson equation is

$$y^{\left(m\right)}\left(x\right)=f\left(x\right),m=2n,f\in C[a,b],x\in [a,b].$$

(27)

Remark 4.

If $y=y\left(x\right)$ is a solution of (27), then (see [45])

$$y\left(x\right)=y\left(a\right)+\frac{x-a}{1!}{y}^{\prime }\left(a\right)+\cdots +\frac{{(x-a)}^{m-1}}{(m-1)!}{y}^{(m-1)}\left(a\right)+{\int }_a^x\frac{{\left(x-s\right)}^{m-1}}{(m-1)!}f\left(s\right)ds.$$

(28)

Let $\epsilon >0$. We also consider the inequality

$$\left|y^{\left(m\right)}\left(x\right)-f\left(x\right)\right|\le \epsilon ,\forall x\in [a,b],y\in C^m[a,b].$$

(29)

Remark 5.

A function $y=y\left(x\right)$ is a solution of (29) if and only if there exists a function $g\in C\left[a,b\right]$ such that

 (1) 

$\left|g\left(x\right)\right|\le \epsilon ,\forall x\in \left[a,b\right],$

 (2) 

$y^{\left(m\right)}\left(x\right)-f\left(x\right)=g\left(x\right),\forall x\in \left[a,b\right].$

Remark 6.

If $y=y\left(x\right)$ is a solution of (29), using Remark 5 and Remark 4, we have

$$y\left(x\right)=y\left(a\right)+\frac{x-a}{1!}{y}^{\prime }\left(a\right)+\cdots +\frac{{(x-a)}^{m-1}}{(m-1)!}{y}^{(m-1)}\left(a\right)+{\int }_a^x\frac{{\left(x-s\right)}^{m-1}}{(m-1)!}\left(f\left(s\right)+g\left(s\right)\right)ds.$$

(30)

Theorem 2.

 (i) 

For each solution $y=y\left(x\right)$ of (29), there exists a unique solution $y_0=y_0\left(x\right)$ of (27) such that

$$\left\{\begin{array}{c}{y}_0\left(a\right)=y\left(a\right)\hfill \\ y_0^{\prime }\left(a\right)=y^{\prime }\left(a\right)\hfill \\ \cdots \hfill \\ y_0^{(m-1)}\left(a\right)=y^{(m-1)}\left(a\right).\hfill \end{array}\right.$$

(31)

 (ii) 

The Equation (27) is Hyers–Ulam stable of order $m-1$, and if y is a solution of (29) and $y_0$ is a solution of (27) satisfying conditions (31), then

$$\left|y\left(x\right)-y_0\left(x\right)\right|\le \epsilon \frac{{(b-a)}^m}{m!},\forall x\in [a,b],$$

(32)

$$\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|\le \epsilon \frac{{(b-a)}^{m-1}}{(m-1)!},\forall x\in [a,b],$$

(33)

$$\left|y^{″}\left(x\right)-y_0^{″}\left(x\right)\right|\le \epsilon \frac{{(b-a)}^{m-2}}{(m-2)!},\forall x\in [a,b],$$

(34)

⋯

$$\left|y^{(m-1)}\left(x\right)-y_0^{(m-1)}\left(x\right)\right|\le \epsilon (b-a),\forall x\in [a,b].$$

(35)

 (iii) 

If there exists $l_1,l_2,\cdots l_n:[a,b]⟶\left[0,\infty \right)$ continuous, such that

$$\begin{array}{cc}\hfill & \left|F\left(x,y_1\left(x\right),y_1^{\prime }\left(x\right),\cdots ,y_1^{\left(n\right)}\left(x\right)\right)-F\left(x,y_2\left(x\right),y_2^{\prime }\left(x\right),\cdots ,y_2^{\left(n\right)}\left(x\right)\right)\right|\hfill \\ \hfill & \le l_1\left(x\right)\left|y_1\left(x\right)-y_2\left(x\right)\right|+l_2\left(x\right)\left|y_1^{\prime }\left(x\right)-y_2^{\prime }\left(x\right)\right|+\cdots +l_n\left(x\right)\left|y_1^{\left(n\right)}\left(x\right)-y_2^{\left(n\right)}\left(x\right)\right|,\hfill \end{array}$$

$\forall x\in [a,b],\forall y_1,y_2\in A,$ then

$$\begin{array}{cc}\hfill & \left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|\hfill \\ \hfill & \le \epsilon ·\frac{{(b-a)}^m}{m!}{\int }_a^b{l}_1\left(x\right)dx+\epsilon ·\frac{{(b-a)}^{m-1}}{(m-1)!}{\int }_a^b{l}_2\left(x\right)dx+\cdots +\epsilon ·\frac{{(b-a)}^n}{n!}{\int }_a^b{l}_n\left(x\right),\hfill \end{array}$$

(36)

where y is a solution of (29) and $y_0$ is a solution of (27), both satisfying the conditions of (24).

Proof

(i)

This results from Cauchy–Picard’s theorem of existence and uniqueness (see [46]).

(ii)

Let $y=y\left(x\right)$ be a solution of (29). Let $y_0=y_0\left(x\right)$ the unique solution of (27) satisfying the conditions of (31). Using Remark 6, we have

$$\begin{array}{cc}\hfill & \left|y\left(x\right)-y_0\left(x\right)\right|=\hfill \\ \hfill & \left|y\left(a\right)+\frac{x-a}{1!}{y}^{\prime }\left(a\right)+\cdots +{\int }_a^x\frac{{\left(x-s\right)}^{m-1}}{(m-1)!}\left(f\left(s\right)+g\left(s\right)\right)ds\right.\hfill \\ \hfill & \left.-y\left(a\right)-\frac{x-a}{1!}{y}^{\prime }\left(a\right)-\cdots -{\int }_a^x\frac{{\left(x-s\right)}^{m-1}}{(m-1)!}f\left(s\right)ds\right|,\hfill \end{array}$$

hence

$$\left|y\left(x\right)-y_0\left(x\right)\right|=\left|{\int }_a^x\frac{{\left(x-s\right)}^{m-1}}{(m-1)!}g\left(s\right)ds\right|\le {\int }_a^x\left|\frac{{\left(x-s\right)}^{m-1}}{(m-1)!}g\left(s\right)\right|ds,$$

so

$$\left|y\left(x\right)-y_0\left(x\right)\right|\le \epsilon \frac{{(b-a)}^m}{m!},\forall x\in [a,b].$$

(37)

We also have

$$\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|=\left|{\int }_a^x\frac{{\left(x-s\right)}^{m-2}}{(m-2)!}g\left(s\right)ds\right|\le {\int }_a^x\left|\frac{{\left(x-s\right)}^{m-2}}{(m-2)!}g\left(s\right)\right|ds\le \epsilon \frac{{(b-a)}^{m-1}}{(m-1)!},$$

$$\left|y^{″}\left(x\right)-y_0^{″}\left(x\right)\right|=\left|{\int }_a^x\frac{{\left(x-s\right)}^{m-3}}{(m-3)!}g\left(s\right)ds\right|\le {\int }_a^x\left|\frac{{\left(x-s\right)}^{m-3}}{(m-3)!}g\left(s\right)\right|ds\le \epsilon \frac{{(b-a)}^{m-2}}{(m-2)!},$$

⋯

$$\left|y^{(m-1)}\left(x\right)-y_0^{(m-1)}\left(x\right)\right|=\left|{\int }_a^{x}g\left(s\right)ds\right|\le {\int }_a^x\left|g\left(s\right)\right|ds\le \epsilon (b-a),\forall x\in [a,b],$$

thus the Equation (27) is Hyers–Ulam stable of order $m-1$.

(iii)

If y is a solution of (29), and $y_0$ is a solution of (27), both satisfying the conditions of (24), then

$$\begin{array}{cc}\hfill & \left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|\hfill \\ \hfill & \le {\int }_a^b\left[l_1\left(x\right)\left|y\left(x\right)-y_0\left(x\right)\right|+l_2\left(x\right)\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|+\cdots +l_n\left(x\right)\left|y^{\left(n\right)}\left(x\right)-y_0^{\left(n\right)}\left(x\right)\right|\right]dx\hfill \\ \hfill & \stackrel{\left(ii\right)}{\le }\epsilon ·\frac{{(b-a)}^m}{m!}{\int }_a^b{l}_1\left(x\right)dx+\epsilon ·\frac{{(b-a)}^{m-1}}{(m-1)!}{\int }_a^b{l}_2\left(x\right)dx+\cdots +\epsilon ·\frac{{(b-a)}^n}{n!}{\int }_a^b{l}_n\left(x\right).\hfill \end{array}$$

□

Example 2.

We consider $J:A⟶R,A\subseteq C^4\left(I,\mathbb{R}\right),[0,1]\subset I,$

$$J\left[y\left(x\right)\right]={\int }_0^1\left(360x^{2}y-y^{\prime {\prime }^2}\right)dx,$$

(38)

$$y\left(0\right)=0,y^{\prime }\left(0\right)=1,y^{″}\left(0\right)=-6,y^{″\prime }\left(0\right)=9.$$

(39)

The Euler–Poisson equation becomes

$$y^{IV}\left(x\right)-180x^2=0.$$

(40)

Let $\epsilon >0.$ We consider the inequality

$$\left|y^{IV}\left(x\right)-180x^2\right|\le \epsilon .$$

(41)

We apply Theorem 2; therefore, for each solution $y=y\left(x\right)$ of (41) satisfying (39), there exists a unique solution $y_0=y_0\left(x\right)$ of (40) satisfying (39) such that

$$\left|y\left(x\right)-y_0\left(x\right)\right|\le \epsilon \frac{1}{4!},\forall x\in [0,1],$$

(42)

$$\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|\le \epsilon \frac{1}{3!},\forall x\in [0,1],$$

$$\left|y^{″}\left(x\right)-y_0^{″}\left(x\right)\right|\le \epsilon \frac{1}{2!},\forall x\in [0,1],$$

$$\left|y^{″\prime }\left(x\right)-y_0^{″\prime }\left(x\right)\right|\le \epsilon \frac{1}{1!},\forall x\in [0,1];$$

hence, Equation (40) is Hyers–Ulam stable of order 3.

Let

$$y_0\left(x\right)=\frac{1}{2}{x}^6+\frac{3}{2}{X}^3-3X^2+X,$$

(43)

be the solution of Equation (40) satisfying the conditions of (39). We remark that $-6\le y^{″}\left(x\right)\le 18,\forall x\in [0,1].$.

If y is a solution of (41)and $y_0$ is a solution of (40), both satisfying the conditions of (39), then

$$\begin{array}{cc}\hfill \left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|& \le {\int }_0^1\left|360x^{2}y-{y^{″}}^2-360x^2{y}_0+{y_0^{″}}^2\right|dx\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & \le {\int }_0^1\left(360x^2\left|y-y_0\right|+\left|y^{″}-y_0^{″}\right|\left|y^{″}+y_0^{″}\right|\right)dx\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & \le {\int }_0^1\left[360x^2·\frac{\epsilon }{24}+\frac{\epsilon }{2}\left(\frac{\epsilon }{2}+36\right)\right]dx=\frac{\epsilon \left(\epsilon +92\right)}{4}.\hfill \end{array}$$

(46)

4. Conclusions

In this paper, we have defined and studied Hyers–Ulam stability of order 1 for Euler’s equation $y^{″}\left(x\right)=f\left(x\right)$ and Hyers–Ulam stability of order $m-1$ for the Euler–Poisson equation $y^{\left(m\right)}\left(x\right)=f\left(x\right),$ in the calculus of variations. An example is considered for each case. Some estimations for $\left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|$, where y is a solution of (9) and $y_0$ is a solution of (7), both satisfying the conditions of (2) and where y is a solution of (29) and $y_0$ is a solution of (27), both satisfying the conditions of (24), have been established. This paper is a continuation of the paper [43]. In [43], the Hyers–Ulam stability of Euler’s equation in two special cases was studied when $F=F(x,y^{\prime })$ and when $F=F(y,y^{\prime }).$ The general case will be the subject of future works.