Hyers-Ulam Stability of Euler’s Equation in the Calculus of Variations

Abstract

In this paper we study Hyers-Ulam stability of Euler’s equation in the calculus of variations in two special cases: when $F=F(x,y^{\prime })$ and when $F=F(y,y^{\prime }).$ For the first case we use the direct method and for the second case we use the Laplace transform. In the first Theorem and in the first Example the corresponding estimations for $\left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|$ are given. We mention that it is the first time that the problem of Ulam-stability of extremals for functionals represented in integral form is studied.

1. Introduction

We emphasize that it was Ulam [1], in 1940, who stated the Ulam stability problem, putting an open problem concerning the approximate homomorphisms of groups. Hyers [2], in 1941, established a first result to the problem posed by Ulam [1]. It was Obloza [3] and Alsina and Ger [4] that began the study of the stability of differential equations. In the papers [5,6,7,8,9,10,11], the stability of first order linear differential equations and linear differential equations of higher order was studied. In [12] linear difference equations was investigated. The Hyers-Ulam stability of integral equations was also studied in [13,14,15,16,17,18,19,20,21] and systems of differential equations in [22,23,24]. A. Prastaro and Th.M. Rassias in [25] started the study of Hyers-Ulam stability of partial differential equations. These have also been studied in [26,27,28,29,30,31]. A summary of these issues can be found in the books [32,33]. There are several ways to study Ulam stability, for example the direct method [6,7,8], the fixed point method [34,35], the integral inequalities method [36], the changes of variables method [26,29,30], and the integral transforms method. Laplace transform was used, for instance, in [37,38,39].

In the following we study Hyers-Ulam stability of Euler’s equation in the calculus of variations, using the direct method and the Laplace transform. We use different methods due to the form of the considered functions. For this we consider a class of functions $A\subseteq C^2\left(\left[a,b\right],\mathbb{R}\right),a<b$. Let $y\in A,y=y\left(x\right)$ be an element in A. We consider the following problem in the calculus of variations (see [40]): let the function $F:M⟶\mathbb{R},$$M\subseteq {\mathbb{R}}^3, F\in C^2\left(M\right)$ be given and it is required to find the extremum of 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)

It is known (see [40]) that the necessary condition of extremum is given by Euler’s equation

$$F_y^{\prime }\left(x,y,y^{\prime }\right)-\frac{d}{dx}\left[F_{y^{\prime }}^{\prime }\left(x,y,y^{\prime }\right)\right]=0,$$

(2)

$$y\left(a\right)=y_a,y\left(b\right)=y_b,y_a,y_b\in \mathbb{R}\mathrm{being}\mathrm{given},$$

(3)

or with other conditions set on y. Equation (2) can be represented (by derivation) in the form:

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

(4)

The solutions of Equation (2) or (4) are called extremals. The problem of Ulam-stability of the extremals for functional represented in integral form has not been studied yet.

In the following we will study the Ulam stability of Euler Equation (2) when the function F has various forms.

We first recall some notions and results regarding the Laplace transform.

Let $y:\mathbb{R}\to \mathbb{R}$ be a function such that the following conditions are satisfied:

1.

$y\left(x\right)=0,x<0$;

2.

y is continuous;

3.

$\exists M>0$ and ${\sigma }_0\ge 0$ such that

$$\left|y\left(x\right)\right|\le M·e^{{\sigma }_{0}x},\forall x\in \mathbb{R}.$$

(5)

We denote by $M\left(y\right)$ the set of all numbers that satisfy the condition 3.

The number ${\sigma }_y=inf\left\{{\sigma }_0\mid {\sigma }_0\in M\left(y\right)\right\}$ is called abscissa of convergence of $y.$

We denote by $\mathcal{L}\left(y\right)$ the Laplace transform of the function y, defined by

$$\mathcal{L}\left(y\right)\left(s\right)=Y\left(s\right)={\int }_0^{\infty }y\left(x\right)e^{-sx}dx,$$

on $\{s\in \mathbb{R}|s>{\sigma }_y\}$. It is well known that the Laplace transform is linear and one-to-one if the involved functions are continuous. The inverse Laplace transform is denoted by ${\mathcal{L}}^{-1}\left(Y\right)$. We write $y^{\left(n\right)}\left(0\right)$ instead the lateral limit $y^{\left(n\right)}\left(0^{+}\right)$ for $n\ge 0$.

The following properties are used in the paper:

$$\mathcal{L}\left(y^{\prime }\right)=sY\left(s\right)-y\left(0\right),$$

(6)

$$\mathcal{L}\left(y^{″}\right)=s^{2}Y\left(s\right)-sy\left(0\right)-y^{\prime }\left(0\right),$$

(7)

$$\mathcal{L}\left(x^n\right)=\frac{n!}{s^{n+1}}\iff {\mathcal{L}}^{-1}\left(\frac{1}{s^{n+1}}\right)=\frac{x^n}{n!},s>0,n\in \mathbb{N},$$

(8)

$$\mathcal{L}\left(cosax\right)=\frac{s}{s^2+a^2}\iff {\mathcal{L}}^{-1}\left(\frac{s}{s^2+a^2}\right)=cosax,s>0,a\in \mathbb{R},$$

(9)

$$\mathcal{L}\left(sinax\right)=\frac{a}{s^2+a^2}\iff {\mathcal{L}}^{-1}\left(\frac{a}{s^2+a^2}\right)=sinax,s>0,a\in \mathbb{R},$$

(10)

$$\mathcal{L}\left(e^{\lambda x}cosax\right)=\frac{s-\lambda }{{\left(s-\lambda \right)}^2+a^2}\iff {\mathcal{L}}^{-1}\left(\frac{s-\lambda }{{\left(s-\lambda \right)}^2+a^2}\right)=e^{\lambda x}cosax,s>\lambda ,$$

(11)

$$\mathcal{L}\left(e^{\lambda x}sinax\right)=\frac{a}{{\left(s-\lambda \right)}^2+a^2}\iff {\mathcal{L}}^{-1}\left(\frac{a}{{\left(s-\lambda \right)}^2+a^2}\right)=e^{\lambda x}sinax,s>\lambda .$$

(12)

2. Hyers-Ulam Stability

Let $\epsilon >0$. In the following we consider Euler’s Equation (2) and the inequality

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

(13)

Definition 1.

The Equation (2) is called Hyers-Ulam stable if there is a real number$c>0$ so that for any solution $y\left(x\right)$ of the inequality (13) (named approximate solution), there is a solution $y_0\left(x\right)$ of the Equation (2) such that

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

(14)

Definition 2.

The Equation (2) is called Hyers-Ulam-Rassias stable if there is a real number $c>0$ and a function $\phi :[a,b]\to \left(0,\infty \right)$ so that for any solution $y\left(x\right)$ of the inequality (13) (named approximate solution), there is a solution $y_0\left(x\right)$ of the Equation (2) such that

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

(15)

We will study the cases:

($\alpha$)

$F=F(x,y^{\prime })$ and

($\beta$)

$F=F(y,y^{\prime }).$

2.1. The Case ($\alpha$) $F=F(x,y^{\prime })$

We consider the functional

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

(16)

Since F does not depend on $y,$ Euler’s equation has the form:

$$\frac{d}{dx}\left[F_{y^{\prime }}^{\prime }\left(x,y^{\prime }\right)\right]=0.$$

(17)

Next we study the case when $F_{y^{\prime }}^{\prime }\left(x,y^{\prime }\right)=f\left(x\right)·y^{\prime }\left(x\right)+c,f:[a,b]⟶\left(0,\infty \right),c\in \mathbb{R},f$ continuous. Let $\epsilon >0$. We consider the equation

$$\frac{d}{dx}\left[f\left(x\right)·y^{\prime }\left(x\right)+c\right]=0$$

(18)

and the inequality

$$\left|\frac{d}{dx}\left[f\left(x\right)·y^{\prime }\left(x\right)+c\right]\right|\le \epsilon .$$

(19)

Theorem 1.

If

1. 

$F\in C^2\left(M\right),M\subset {\mathbb{R}}^2$ such that $F_{y^{\prime }}^{\prime }\left(x,y^{\prime }\right)=f\left(x\right)·y^{\prime }\left(x\right)+c,f:[a,b]⟶\left(0,\infty \right);$

2. 

$y_a,y_a^{\prime }\in \mathbb{R}$ are given and $y\left(a\right)=y_a,y^{\prime }\left(a\right)=y_a^{\prime };$

3. 

$\left|F\left(x,y^{\prime }\right)-F\left(x,y_0^{\prime }\right)\right|\le L·\left|y^{\prime }-y_0^{\prime }\right|,L\in {\mathbb{R}}_{+},\forall x\in [a,b],y^{\prime },y_0^{\prime }\in \mathbb{R},$

then

(i) 

the Cauchy problem

$$\left(2\right)+\left\{\begin{array}{c}y\left(a\right)=y_a\\ y^{\prime }\left(a\right)=y_a^{\prime }\end{array}\right.$$

(20)

has a unique solution.

(ii) 

the Equation (18) is Hyers-Ulam-Rassias stable.

(iii) 

$$\left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|\le L·\epsilon ·\left(h\left(b\right)-h\left(a\right)\right),$$

(21)

where y is a solution of (19) and $y_0$ is a solution of (18).

Proof. 

(i)

This results from Cauchy-Picard’s Theorem of existence and uniqueness (see [41]).

(ii)

We consider the inequality (19) which can be written

$$-\epsilon \le \frac{d}{dx}\left[f\left(x\right)·y^{\prime }\left(x\right)+c\right]\le \epsilon .$$

(22)

Integrating from a to x we obtain

$$-\epsilon \left(x-a\right)\le f\left(x\right)·y^{\prime }\left(x\right)-f\left(a\right)·y^{\prime }\left(a\right)\le \epsilon \left(x-a\right).$$

(23)

Dividing by $f\left(x\right)$ we have

$$-\epsilon \frac{x-a}{f\left(x\right)}\le y^{\prime }\left(x\right)-f\left(a\right)·y^{\prime }\left(a\right)\frac{1}{f\left(x\right)}\le \epsilon \frac{x-a}{f\left(x\right)}.$$

(24)

We denote by $g\left(x\right)$ an antiderivative of $f\left(a\right)·y^{\prime }\left(a\right)\frac{1}{f\left(x\right)}$ and by $h\left(x\right)$ an antiderivative of $\frac{x-a}{f\left(x\right)}.$ Integrating now the above relation from a to x we obtain

$$-\epsilon \left(h\left(x\right)-h\left(a\right)\right)\le y\left(x\right)-y\left(a\right)-g\left(x\right)+g\left(a\right)\le \epsilon \left(h\left(x\right)-h\left(a\right)\right).$$

(25)

Let

$$y_0\left(x\right)=y\left(a\right)+g\left(x\right)-g\left(a\right)$$

(26)

a solution of Equation (18). Therefore

$$\left|y\left(x\right)-y_0\left(x\right)\right|\le ϵ\left(h\left(x\right)-h\left(a\right)\right),\forall x\in [a,b],$$

(27)

hence the Equation (18) is Hyers-Ulam-Rassias stable.

(iii)

If y is a solution of (19) and $y_0$ is a solution of (18), then

$$\begin{array}{cc}\hfill & \left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|\le {\int }_a^{b}L·\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|dx\hfill \\ \hfill & \stackrel{\left(24\right)}{\le }{\int }_a^{b}L·\epsilon ·\frac{x-a}{f\left(x\right)}dx=L·\epsilon ·\left(h\left(b\right)-h\left(a\right)\right).\hfill \end{array}$$

(28)

□

Example 1.

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

$$J\left[y\left(x\right)\right]={\int }_1^2{y}^{\prime }\left(x\right)\left(1+x^2{y}^{\prime }\left(x\right)\right)dx,y\left(1\right)=3,y^{\prime }\left(1\right)=4.$$

(29)

Euler’s equation becames

$$\frac{d}{dx}\left(1+2x^2{y}^{\prime }\right)=0.$$

(30)

We consider the inequality

$$\left|\frac{d}{dx}\left(1+2x^2{y}^{\prime }\right)\right|\le \epsilon ,$$

(31)

that is

$$-\epsilon \le \frac{d}{dx}\left(1+2x^2{y}^{\prime }\right)\le \epsilon .$$

(32)

Integrating from 1 to x we obtain

$$-\epsilon \left(x-1\right)\le 2x^2{y}^{\prime }-8\le \epsilon \left(x-1\right).$$

(33)

Dividing by $2x^2\ne 0$ we have

$$-\frac{\epsilon }{2}\frac{x-1}{x^2}\le y^{\prime }\left(x\right)-\frac{4}{x^2}\le \frac{\epsilon }{2}\frac{x-1}{x^2}.$$

(34)

Integrating from 1 to x we obtain

$$-\frac{\epsilon }{2}\left(lnx+\frac{1}{x}-1\right)\le y\left(x\right)+\frac{4}{x}-7\le \frac{\epsilon }{2}\left(lnx+\frac{1}{x}-1\right).$$

(35)

Let

$$y_0\left(x\right)=-\frac{4}{x}+7$$

(36)

a solution of Equation (30).

Therefore

$$\left|y\left(x\right)-y_0\left(x\right)\right|\le \frac{\epsilon }{2}\left(lnx+\frac{1}{x}-1\right)\le \frac{\epsilon }{2}\left(ln2-\frac{1}{2}\right),\forall x\in [1,2],$$

(37)

hence the Equation (30) is Hyers-Ulam stable.

More, if y is a solution of (31) and $y_0$ is a solution of (30), then

$$\begin{array}{cc}\hfill \left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|& \le {\int }_1^2\left|y^{\prime }\left(x\right)\left(1+x^2{y}^{\prime }\left(x\right)\right)-y_0^{\prime }\left(x\right)\left(1+x^2{y}_0^{\prime }\left(x\right)\right)\right|dx\hfill \\ \hfill & \le {\int }_1^2\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|+x^2\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|\left|y^{\prime }\left(x\right)+y_0^{\prime }\left(x\right)\right|dx\hfill \\ \hfill & ={\int }_1^2\left|y^{\prime }\left(x\right)-y_0^{\prime }\left(x\right)\right|\left[1+x^2\left|y^{\prime }\left(x\right)+y_0^{\prime }\left(x\right)\right|\right]dx\hfill \\ \hfill & \stackrel{\left(34\right)}{\le }{\int }_1^2\frac{\epsilon }{2}\frac{x-1}{x^2}\left[1+x^2\left|y^{\prime }\left(x\right)+y_0^{\prime }\left(x\right)\right|\right]dx\hfill \\ \hfill & \stackrel{\left(34\right)}{=}{\int }_1^2\frac{\epsilon }{2}\frac{x-1}{x^2}\left[1+x^2\left(\frac{\epsilon }{2}\frac{x-1}{x^2}+\frac{8}{x^2}\right)\right]dx\hfill \\ \hfill & ={\int }_1^2\frac{\epsilon }{2}\frac{x-1}{x^2}\left[9+\frac{\epsilon }{2}\left(x-1\right)\right]dx={\int }_1^2\left(\frac{9\epsilon }{2}\frac{x-1}{x^2}+\frac{{\epsilon }^2}{4}\frac{{\left(x-1\right)}^2}{x^2}\right)dx\hfill \\ \hfill & =\frac{9\epsilon }{2}\left(ln2-\frac{1}{2}\right)+\frac{{\epsilon }^2}{4}\left(\frac{3}{2}-2ln2\right).\hfill \end{array}$$

(38)

2.2. The Case ($\beta$) $F=F(y,y^{\prime })$

We consider the functional

$$J\left[y\left(x\right)\right]={\int }_0^{b}F\left(y,y^{\prime }\right)dx,J:A\to \mathbb{R},y\in A,A\subseteq C^2\left(\left[0,b\right],\mathbb{R}\right).$$

(39)

Since F does not depend on $x,$ Euler’s equation has the form:

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

(40)

Next we study the case when Euler’s equation has the form

$$y^{″}+c_1·y^{\prime }+c_2·y=0,c_1,c_2\in \mathbb{R},c_1^2-4c_2<0.$$

(41)

We also consider the inequality

$$\left|y^{″}+c_1·y^{\prime }+c_2·y\right|\le \epsilon .$$

(42)

Theorem 2.

If

1.

$F\in C^2\left(M\right),M\subset {\mathbb{R}}^2$ such that Euler’s equation has the form $y^{″}+c_1·y^{\prime }+c_2·y=0,c_1,c_2\in \mathbb{R},c_1^2-4c_2<0,c_2\ne 0;$

2.

$y_1,y_2\in \mathbb{R}$ are given and $y\left(0\right)=y_1,y^{\prime }\left(0\right)=y_2;$

then

(i)

the Cauchy problem

$$\left(2\right)+\left\{\begin{array}{c}y\left(0\right)=y_1\\ y^{\prime }\left(0\right)=y_2\end{array}\right.$$

(43)

has a unique solution.

(ii)

the Equation (41) is Hyers-Ulam-Rassias stable.

Proof. 

(i)

This results from Cauchy-Picard’s Theorem of existence and uniqueness (see [41]).

(ii)

We consider the inequality (42) which can be written

$$-\epsilon \le y^{″}+c_1·y^{\prime }+c_2·y=0\le \epsilon .$$

(44)

We apply Laplace transform and we obtain

$$-\frac{\epsilon }{s}\le s^{2}Y\left(s\right)-sy\left(0\right)-y^{\prime }\left(0\right)+c_{1}sY\left(s\right)-c_{1}y\left(0\right)+c_{2}Y\left(s\right)\le \frac{\epsilon }{s},$$

(45)

that is

$$-\frac{\epsilon }{s}\le \left(s^2+c_{1}s+c_2\right)Y\left(s\right)-sy\left(0\right)-y^{\prime }\left(0\right)-c_{1}y\left(0\right)\le \frac{\epsilon }{s}.$$

(46)

Dividing by $s^2+c_{1}s+c_2>0$ we get

$$-\frac{\epsilon }{s\left(s^2+c_{1}s+c_2\right)}\le Y\left(s\right)-y\left(0\right)\frac{s}{\left(s^2+c_{1}s+c_2\right)}-\left(y^{\prime }\left(0\right)+c_{1}y\left(0\right)\right)\frac{1}{\left(s^2+c_{1}s+c_2\right)}\le \frac{\epsilon }{s\left(s^2+c_{1}s+c_2\right)}.$$

(47)

We apply now inverse Laplace transform and we obtain

$$\begin{array}{cc}\hfill -\epsilon {\mathcal{L}}^{-1}\left(\frac{1}{s\left(s^2+c_{1}s+c_2\right)}\right)& \le y\left(x\right)-y\left(0\right){\mathcal{L}}^{-1}\left(\frac{s}{\left(s^2+c_{1}s+c_2\right)}\right)-\left(y^{\prime }\left(0\right)+c_{1}y\left(0\right)\right){\mathcal{L}}^{-1}\left(\frac{1}{\left(s^2+c_{1}s+c_2\right)}\right)\hfill \\ \hfill & \le \epsilon {\mathcal{L}}^{-1}\left(\frac{1}{s\left(s^2+c_{1}s+c_2\right)}\right).\hfill \end{array}$$

We remark that

$${\mathcal{L}}^{-1}\left(\frac{1}{s\left(s^2+c_{1}s+c_2\right)}\right)=\frac{1}{c_2}{\mathcal{L}}^{-1}\left(\frac{1}{s}-\frac{s+c_1}{s^2+c_{1}s+c_2}\right).$$

(48)

But

$$\begin{array}{cc}\hfill & {\mathcal{L}}^{-1}\left(\frac{s+c_1}{s^2+c_{1}s+c_2}\right)={\mathcal{L}}^{-1}\left(\frac{s+\frac{C_1}{2}}{{\left(s+\frac{C_1}{2}\right)}^2+\frac{-c_1^2+4c_2}{4}}+\frac{\frac{C_1}{2}}{{\left(s+\frac{C_1}{2}\right)}^2+\frac{-c_1^2+4c_2}{4}}\right)\hfill \\ \hfill & =e^{-\frac{C_1}{2}x}cos\frac{\sqrt{-c_1^2+4c_2}}{2}x+\frac{c_1}{\sqrt{-c_1^2+4c_2^2}}·e^{-\frac{C_1}{2}x}sin\frac{\sqrt{-c_1^2+4c_2}}{2}x.\hfill \end{array}$$

We denote by

$$y_0\left(x\right)=y\left(0\right){\mathcal{L}}^{-1}\left(\frac{s}{s^2+c_{1}s+c_2}\right)+\left(y^{\prime }\left(0\right)+c_{1}y\left(0\right)\right){\mathcal{L}}^{-1}\left(\frac{1}{s^2+c_{1}s+c_2}\right).$$

(49)

Hence we have

$$\left|y\left(x\right)-y_0\left(x\right)\right|\le \frac{\epsilon }{c_2}\left(1-e^{-\frac{C_1}{2}x}cos\frac{\sqrt{-c_1^2+4c_2}}{2}x-e^{-\frac{C_1}{2}x}sin\frac{\sqrt{-c_1^2+4c_2}}{2}x\right),\forall x\in [0,b],$$

(50)

so the Equation (41) is Hyers-Ulam-Rassias stable.

□

Example 2.

We consider $J:A⟶R,y\in A,A\subseteq C^2\left([0,\frac{\pi }{2}],R\right),$

$$J\left[y\left(x\right)\right]={\int }_0^{\frac{\pi }{2}}\left(y^{\prime 2}-y^2\right)dx,y\left(0\right)=1,y^{\prime }\left(0\right)=1.$$

(51)

Euler’s equation is of the form

$$y^{″}+y=0.$$

(52)

We consider the inequality

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

(53)

that is

$$-\epsilon \le y^{″}+y\le \epsilon .$$

(54)

We apply Laplace transform and we obtain

$$-\frac{\epsilon }{s}\le s^{2}Y\left(s\right)-sy\left(0\right)-y^{\prime }\left(0\right)+Y\left(s\right)\le \frac{\epsilon }{s},$$

(55)

hence

$$-\frac{\epsilon }{s}\le \left(s^2+1\right)Y\left(s\right)-s-1\le \frac{\epsilon }{s}.$$

(56)

Dividing by $s^2+1>0$ we get

$$-\frac{\epsilon }{s\left(s^2+1\right)}\le Y\left(s\right)-\frac{s}{s^2+1}-\frac{1}{s^2+1}\le \frac{\epsilon }{s\left(s^2+1\right)}.$$

(57)

We apply now inverse Laplace transform and we obtain

$$-\epsilon {\mathcal{L}}^{-1}\left(\frac{1}{s\left(s^2+1\right)}\right)\le y\left(x\right)-{\mathcal{L}}^{-1}\left(\frac{s}{s^2+1}\right)-{\mathcal{L}}^{-1}\left(\frac{1}{s^2+1}\right)\le \epsilon {\mathcal{L}}^{-1}\left(\frac{1}{s\left(s^2+1\right)}\right),$$

(58)

or

$$-\epsilon {\mathcal{L}}^{-1}\left(\frac{1}{s\left(s^2+1\right)}\right)\le y\left(x\right)-cosx-sinx\le \epsilon {\mathcal{L}}^{-1}\left(\frac{1}{s\left(s^2+1\right)}\right).$$

(59)

But

$$L^{-1}\left(\frac{1}{s\left(s^2+1\right)}\right)={\mathcal{L}}^{-1}\left(\frac{1}{s}-\frac{s}{s^2+1}\right)=1-cosx.$$

(60)

We denote by

$$y_0\left(x\right)=cosx+sinx.$$

(61)

Hence we have

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

(62)

so the Equation (52) is Hyers-Ulam-Rassias stable.

3. Conclusions

In this paper we have studied the Hyers-Ulam stability of Euler’s equation in the calculus of variations in some special cases, that is, when $F=F(x,y^{\prime }),$ using the direct method, and when $F=F(y,y^{\prime }),$ using the Laplace transform. The problem of Ulam-stability of the extremals for functional representation in integral form has not been studied yet. Our estimations (50) and (62) regarding the difference between an aproximate solution y and exact solution $y_0$ improve on those obtained by Rezaei, Jung, and Rassias in 2013 [37] and by Alqifiary and Jung in 2014 [38] for the linear differential Equations (41) and (52), using the Laplace transform. In Theorem 1 and Example 1, the corresponding estimations for $\left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|$ are given. We intend to study other special cases of the Euler equation. We will also focus on the stability of the extremals for functionals, represented in integral form, that depend on higher order derivatives, in which case Poisson’s equation intervenes. Among our future concerns will also be studying the stability of the extremals for functionals depending on the functions of several variables where the Euler-Ostrogradsky equation is involved.