On Ulam stability of a second order linear difference equation

1.   Introduction

The problem of Ulam stability can be formulated for various functional equations. The starting point of Ulam stability theory was a problem formulated by Ulam in 1940 in a talk at the University of Wisconsin-Madison concerning the approximate solutions of group homomorphisms. Generally, we say that an equation is Ulam stable if for every approximate solution of the equation there exists an exact solution of the equation near it (see the papers ^{[10,16,18,23,25]}). For more details on the Ulam stability on the functional equations, see the monographs ^[17,19]. The problem can be also formulated for difference equations. Since a discrete dynamical system is described by a difference equation, this type of stability is related to the notion of perturbation of such a system. The Ulam stability of difference equations was intensively studied in the recent years (see for more details ^[10]).

Recently many papers on Hyers-Ulam stability of difference equations are devoted to the relation of it with hyperbolicity and the exponential dichotomy. Remark here the papers by D. Dragičević concerning some nonautonomous and nonlinear difference equations ^[12,13,14]. D. R. Anderson and M. Onitsuka gave interesting results on the influence of stepsize in Hyers-Ulam stability of the first order difference equations and on the best constant of some second order linear difference equations with constant coefficients ^[1,20]. Recall also the results given by A. R. Baias, J. Brzdek, D. Popa et al. in the characterization of Ulam stability and on the best constant for various linear and nonlinear difference equations ^{[2,3,4,5,7,11,15]}.

Let $\mathbb{K}$ be either the field $\mathbb{R}$ of real numbers or the field $\mathbb{C}$ of complex numbers and $X$ be a Banach space over $\mathbb{K}$. Consider the difference equation

where $f_n:X^p \rightarrow X$ and $x_0, x_1, \ldots, x_{p-1} \in X$.

Definition 1.1. The Eq (1.1) is called Ulam stable if there exists a constant $L\geq0$ such that for every $\varepsilon > 0$ and every sequence $(x_n)_{n\geq 0}$ in $X$ satisfying

there exists a sequence $(y_n)_{n\geq 0}$ in $X$ such that

and

Remark 1.2. If in Definition 1.1, $\varepsilon$ is replaced by a sequence of positive numbers $(\varepsilon_n)_{n \geq 0}$ and $L\varepsilon$ by a sequence $(\delta_n)_{n \geq 0}$ depending on $(\varepsilon_n)_{n \geq 0}$, then we get the notion of generalized Ulam stability. The number $L$ is called an Ulam constant of the Eq (1.1). Denote by $L_R$ the infimum of all Ulam constants of the Eq (1.1).

In particular, for $p = 1$, we consider the linear difference equation

with $(a_n)_{n \geq 0}$ a sequence in $\mathbb{K}$ and $(b_n)_{n \geq 0}$ a sequence in $X$.

The following result concerning the generalized Ulam stability of (1.5) can be found in ^[21].

Theorem 1.3. Let $(\varepsilon_n)_{n \geq 0}$ be a sequence of positive numbers such that

Then there exists $L \geq 0$ such that for every sequence $(x_n)_{n\geq 0}$ in $X$ satisfying

there exists a sequence $(y_n)_{n\geq 0}$ in $X$ such that

and

Remark 1.4. If in Theorem 1.3, we take $\varepsilon_n = \varepsilon, \ n \geq 0$, then the condition (1.6) becomes

On the other hand it was proved that if $\lim\limits_{n \rightarrow \infty} |a_n| = 1$, then (1.5) is not stable (see ^[8]).

So, we obtain the following result.

Theorem 1.5. Suppose that $\lim\limits_{n \rightarrow \infty} |a_n|$ exists. Then the Eq (1.5) is Ulam stable if and only if $\lim\limits_{n \rightarrow \infty} |a_n| \neq 1$.

2.   Second order equations

Let $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$ be sequences in $\mathbb{K}$ and $(c_n)_{n \geq 0}$ a sequence in $X$. In what follows we deal with Ulam stability of the second order linear difference equation

where $x_0, x_1 \in X$.

The following results will be useful in the sequel.

Lemma 2.1. Suppose that $(x_n)_{n\geq0}$ satisfies (2.1) and let $(u_n)_{n\geq 0}$ be a sequence in $\mathbb{K}$ defined by the Riccati difference equation

If $(z_n)_{n \geq 0}$ is given by the relation

then

Proof. From (2.1) it follows that

□

Lemma 2.2. Let $(A_n)_{n \geq 0}$, $A_n = \left(\begin{array}{ccccc} p_n & q_n \\ r_n & s_n \end{array}\right)$, be a sequence of matrices with entries in $\mathbb{K}$ and $(v_n)_{n \geq 0}$ be the sequence defined by the difference equation

Then

where

Proof. The proof may be established by using mathematical induction on $n$. □

Remark 2.3. In particular, for the sequence $(u_n)_{n \geq 0}$ given by (2.1) we get

where $\label{matrix1203} A_n = \left(\begin{array}{ccccc} a_n & b_n \\ 1 & 0 \end{array}\right)$, $n \geq 0$, and

Lemma 2.4. If the equation $x_{n+1}-u_nx_n-z_n = 0$ is Ulam stable with the constant $L_1$ and the Eq (2.4) is Ulam stable with the constant $L_2$, then the Eq (2.1) is Ulam stable with the constant $L_1L_2$.

Proof. Let $\varepsilon > 0$ and let $(x_n)_{n \geq 0}$ be a sequence in $X$ such that

Put $z_n = x_{n+1}-u_nx_n$, where $(u_n)_{n \geq 0}$ satisfies relation (2.2). Then

so according to the stability of the Eq (2.4), there exists $(w_n)_{n \geq 0}$,

such that

Taking account of (2.3) we get

Now, since the Eq (2.3) is Ulam stable, it follows that there exists a sequence $(y_n)_{n \geq 0}$, satisfying the relation

such that

To complete the proof it remains to show that $(w_n)_{n \geq 0}$ satisfies the Eq (2.1). For this we replace $(y_n)_{n \geq 0}$ from (2.6) to (2.5) and we obtain

Finally, taking account of the relation (2.2), we get

The theorem is proved. □

The main result on the stability of the Eq (2.1) is given in the next theorem.

Theorem 2.5. Suppose that for the sequence $(u_n)_{n \geq 0}$ given by (2.2), $\lim\limits_{n \rightarrow \infty} |u_n|$, $\lim\limits_{n \rightarrow \infty} |a_n-u_{n+1}|$ exist and:

1) $\lim\limits_{n \rightarrow \infty} |u_n| \neq 1$;

2) $\lim\limits_{n \rightarrow \infty} |a_n-u_{n+1}| \neq 1$.

Then the Eq (2.1) is Ulam stable.

In the following theorem we present a nonstability result for the Eq (2.1).

Theorem 2.6. If there exists $u_0 \in \mathbb{K}$ such that $\lim\limits_{n \rightarrow \infty} |a_n-u_{n+1}| = 1$ and $(u_n)_{n \geq 0}$ is bounded, then the Eq (2.1) is not Ulam stable.

Proof. Let $\varepsilon > 0$. Since $\lim\limits_{n \rightarrow \infty} |a_n-u_{n+1}| = 1$, from Theorem 1.5 it follows that the equation

is not Ulam stable, i.e., there exists a sequence $(\bar{z}_n)_{n \geq 0}$ in $X$, satisfying the inequality

such that for every sequence $(\bar{y}_n)_{n \geq 0}$ with

we have

Let $(x_n)_{n \geq 0}$ be a sequence in $X$ defined by the relation

(it suffices to take $x_0 = 0$ in order to determine $(x_n)_{n \geq 0}$ step by step).

The inequality (2.8) implies that the sequence $(x_n)_{n \geq 0}$ satisfies

Let now $(y_n)_{n \geq 0}$ be an arbitrary sequence defined by

and $(\bar{y}_n)_{n \geq 0}$ be the sequence given by

Then the relations (2.9) and (2.10) hold.

Finally, we have to prove that $\sup_{n \geq 0} \|\bar{y}_n-\bar{z}_n\| = \infty$. Suppose the contrary. Then there exists $M > 0$ such that

From (2.11) and (2.13) it follows that

for every $n \geq 0$, which contradicts relation (2.10), if we take also into account that $(u_n)_{n \geq 0}$ is bounded. □

The following examples illustrate our theoretical results.

Example 2.7. Suppose that $X$ is a Banach space over $\mathbb{C}$. The linear recurrence

is not Ulam stable.

Indeed, in this case $u_{n+1} = -2e^{in\frac{\pi}{2}}+\frac{i(-1)^n}{u_n}$. Further, taking $u_0 = i$, one can show that $(u_n)_{n \geq 0}$ is bounded and $\lim\limits_{n \rightarrow \infty} |a_n-u_{n+1}| = 1$, since $u_{n} = ie^{in\frac{\pi}{2}}$, $n \geq 0$ (induction on n).

Example 2.8. Let $X$ be a Banach space over $\mathbb{R}$. The linear recurrence

is not Ulam stable.

Indeed, if we take $u_0 = -1$, then $u_n = \frac{1}{2n-1}$, $n\geq 1$ and hence $(u_n)_{n\geq 0}$ is bounded with $\lim\limits_{n \rightarrow \infty} |a_n-u_{n+1}| = 1$.

Example 2.9. Suppose that $X$ is a Banach space over $\mathbb{C}$. The linear recurrence

is not Ulam stable.

Indeed, if we take $u_0 = e^{\frac{\pi i}{4}}$, then $u_n = e^{\frac{2n+1}{4}\pi i}$, $n \geq 1$ and hence $(u_n)_{n\geq 0}$ is bounded with $\lim\limits_{n \rightarrow \infty} |a_n-u_{n+1}| = 1$.

We present and discuss finally some particular cases and we give an example of Ulam stability for (2.1). We also mention that the following result contains in particular the case of stability for the linear difference equation with constant coefficients proved in ^[9,10].

Corollary 2.10. Let $a, b \in \mathbb{K}$ and $c \in X$. The linear difference equation of the second order with constant coefficients

is Ulam stable if and only if none of the roots of the characteristic equation $\lambda^2-a\lambda-b = 0$ lie on the unit circle.

Proof. If for all $n \geq 0$, $a_n = a$ and $b_n = b$ in (2.1), then $u_{n+1} = a + \frac{b}{u_n}$, $n \geq 0$, and by Remark 2.3,

where $\label{matrix1205} A = \left(\begin{array}{ccccc} a & b \\ 1 & 0 \end{array}\right)$ and

Suppose in what follows that the eigenvalues $\lambda_1$ and $\lambda_2$ of $A$, i.e. the roots of the characteristic equation $\lambda^2-a\lambda-b = 0$, are distinct. Then $a-\lambda_1 = \lambda_2$ and

where $\label{matrix1207} B = \frac{1}{\lambda_2-\lambda_1} \left(\begin{array}{ccccc} \lambda_2-a & -b \\ -1 & \lambda_2 \end{array}\right)$ and $\label{matrix1208} C = \frac{1}{\lambda_2-\lambda_1} \left(\begin{array}{ccccc} a-\lambda_1 & b \\ 1 & -\lambda_1 \end{array}\right)$.

Hence,

and consequently,

Next we show that the Eq (2.14) is Ulam stable if and only if $|\lambda_1|\neq 1$ and $|\lambda_2|\neq 1$. Indeed, to prove the necessity, let us suppose without loss of generality that $|\lambda_1| = 1$. Taking $u_0 = \lambda_2$, we get $u_n = \lambda_2$ and $\lim_{n \rightarrow \infty} |a_n-u_{n+1}| = |\lambda_1| = 1$. Hence, according to Theorem 2.6, the Eq (2.14) is not Ulam stable and we get a contradiction. Conversely, if $|\lambda_1|\neq 1$ and $|\lambda_2|\neq 1$, it is sufficient to consider $u_0 = \lambda_1$. Hence $u_n = \lambda_1$, $\lim_{n \rightarrow \infty} |u_n|\neq 1$ and $\lim_{n \rightarrow \infty} |a_n-u_{n+1}| = |\lambda_2| \neq 1$. It follows that the Eq (2.14) is Ulam stable and the proof is complete.

Similarly, one can easily remark that the previous assertion remains also true for $\lambda_1 = \lambda_2$. In this case, $\lambda_1 = \lambda_2 = \frac{a}{2}$, $b = -\frac{a^2}{4}$ and

where $B = I_2$ and $\label{matrix1209} C = \left(\begin{array}{ccccc} 1 & -\frac{a}{2} \\ \frac{2}{a} & -1 \end{array}\right)$, for $a \neq 0$.

Hence,

and consequently,

Furthermore, $\lim\limits_{n \rightarrow \infty} |u_n| = \lim\limits_{n \rightarrow \infty} |a_n-u_{n+1}| = |\frac{a}{2}|$ and the Eq (2.14) is Ulam stable if and only if $|a| \neq 2$. □

Corollary 2.11. Let $(b_n)_{n \geq 0}$ be a sequence in $K\setminus \{0\}$ and $(c_n)_{n \geq 0}$ a sequence in X. The linear difference equation

is Ulam stable if the following two limits exist and

Proof. Indeed, for $a_n = 0$ in (2.1), we get $u_{n+1} = \frac{b_n}{u_n}$, hence $\label{matrix1210} A_n = \left(\begin{array}{ccccc} 0 & b_n \\ 1 & 0 \end{array}\right)$, $n \geq 0$. Further, applying Remark 2.3, we get

for all $n \geq 1$. Finally, taking into account Theorem 2.5, we obtain the desired conclusion. □

Corollary 2.12. The linear difference equation

is Ulam stable if for some $\delta_0 \in \mathbb{K}$, the sequence

satisfies the relations $\lim\limits_{n \rightarrow \infty} |\frac{\delta_n}{\delta_{n-1}}|\neq 1$ and $\lim\limits_{n \rightarrow \infty} |a_n-\frac{\delta_{n+1}}{\delta_n}|\neq 1$.

Proof. By Remark 2.3,

where

and $\alpha_0 = 1, \beta_0 = 0, \alpha_1 = a_0, \beta_1 = \alpha^2-\alpha a_0$. Observe now that

Denoting further $\gamma_n = \alpha_n-\alpha \alpha_{n-1}$ ($n \geq 1$), where $\gamma_1 = a_0-\alpha$, one obtains $\gamma_n: = (a_{n-1}-\alpha)\gamma_{n-1}$ ($n \geq 2$) and consequently,

Thus, $\alpha_n = \alpha \alpha_{n-1}+\prod_{k = 0}^{n-1} (a_k-\alpha)$, $n \geq 1$. Further, putting $\frac{y_{n-1}}{y_n}$ ($n \geq 1$) with $y_0 = 1$ instead of the first $\alpha$ below, we may deduce that $y_n = \frac{1}{\alpha^n}$, and finally

Similarly, one may show that

Hence, if we take $\delta_0 = u_0$ and we substitute the above expressions in (2.23), one obtains an explicit formula for $u_n$, more precisely $u_n = \frac{\delta_n}{\delta_{n-1}}, \ n \geq 1$. Finally, applying Theorem 2.5 we finish the proof. □

Example 2.13. The linear recurrence

is Ulam stable.

Indeed, since the Eq (2.24) is a particular case of (2.21) with $\alpha = 2$, $a_n = 2\frac{2n+3}{n+1}$ and $c_n = 0$, we deduce that

and, consequently,

Moreover, $\lim\limits_{n \rightarrow \infty} |\frac{\delta_n}{\delta_{n-1}}|$ = $\lim\limits_{n \rightarrow \infty} |a_n-\frac{\delta_{n+1}}{\delta_n}| = 2$. Hence, using Corollary 2.12, we conclude that the linear difference equation (2.1) is Ulam stable.

3.   Third order equations

Consider in the following the third order linear difference equation

where $(a_n)_{n \geq 0}$, $(b_n)_{n \geq 0}$, $(c_n)_{n \geq 0}$ are sequences in $\mathbb{K}$ and $(d_n)_{n \geq 0}$ is a sequence in $X$.

Lemma 3.1. Suppose that $(x_n)_{n\geq0}$ satisfies (3.1). Consider

and

where $(u_n)_{n\geq 0}$ and $(u'_n)_{n\geq 0}$ are sequences in $\mathbb{K}$ defined as follows:

and

Then

and

Proof. From (3.4) it follows that

and hence, using (3.1) and (3.6) one gets

Finally, proceeding similarly to the proof of 2.1 one can show that $(z'_n)_{n \geq 0}$ satisfies (3.7). □

Remark 3.2. Observe that the sequence $(u_n)_{n \geq 0}$ defined above can be written in the following form

and, consequently

where $\label{matrix1211} A_n = \left(\begin{array}{ccccc} a_n & b_n & c_n \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right)$ with $\label{matrix1212} A_0 = \left(\begin{array}{ccccc} \alpha_2 & \beta_2 & \gamma_2 \\ \alpha_1 & \beta_1 & \gamma_1 \\ \alpha_0 & \beta_0 & \gamma_0 \end{array}\right)$ and

Lemma 3.3. If the equation $x_{n+1}-u_nx_n-z_n = 0$ is Ulam stable with the constant $L_1$ and (3.6) is Ulam stable with the constant $L_2$, then the Eq (3.1) is Ulam stable with the constant $L_1L_2$.

Proof. Let $\varepsilon > 0$ and let $(x_n)_{n \geq 0}$ be a sequence in $X$ such that

Put $z_n = x_{n+1}-u_nx_n$, where $(u_n)_{n \geq 0}$ satisfies relation (3.4). Then

and there exists $(w_n)_{n \geq 0}$,

such that

Taking account of (3.2) we get

Now, since the Eq (3.2) is Ulam stable and its stability does not depend on $z_n$, it follows that there exists a sequence $(y_n)_{n \geq 0}$,

such that

To complete the proof it remains for us to show that $(y_n)_{n \geq 0}$ satisfies the Eq (3.1). For this we replace $(w_n)_{n \geq 0}$ from (3.9) to (3.8) and we obtain

Finally, taking account of relation (3.4), we obtain

□

The following result holds as a direct consequence of Lemma 2.4 and Lemma 3.3.

Lemma 3.4. If the equation $x_{n+1}-u_nx_n-z_n = 0$ is Ulam stable with the constant $L_1$, $z_{n+1}-u'_nz_n-z'_n = 0$ is Ulam stable with the constant $L_2$ and (3.7) is Ulam stable with the constant $L_3$, then (3.1) is Ulam stable with the constant $L_1L_2L_3$.

As a consequence of Theorem 1.5 and Lemma 3.4, we get the following result on Ulam stability for the Eq (3.1).

Theorem 3.5. Suppose that for the sequences $(u_n)_{n \geq 0}$ and $(u'_n)_{n \geq 0}$ given by (3.4) and (3.5) , respectively, $\lim\limits_{n \rightarrow \infty} |u_n|$, $\lim\limits_{n \rightarrow \infty} |u'_n|$, $\lim\limits_{n \rightarrow \infty} |a_n-u_{n+2}-u'_{n+1}|$ exist and:

1) $\lim\limits_{n \rightarrow \infty} |u_n| \neq 1$;

2) $\lim\limits_{n \rightarrow \infty} |u'_n| \neq 1$;

3) $\lim\limits_{n \rightarrow \infty} |a_n-u_{n+2}-u'_{n+1}| \neq 1$.

Then the Eq (3.1) is Ulam stable.

Example 3.6. The following linear recurrence

is Ulam stable.

Indeed, in this case, if we take $u_0 = u_1 = 2$ and $u'_0 = 1$, one can see that $u_n = 2$ and $u'_n = \frac{1}{n+1}$ for all $n \geq 0$. Thus, all the assumptions of Theorem 3.5 are fulfilled, which means that we have stability.

A nonstability result for the Eq (3.1), similar to the one obtained for the second order linear difference equation (2.1), holds.

Theorem 3.7. If the Eq (3.6) is not Ulam stable and there exist $u_0, u_1 \in \mathbb{K}$ such that $(u_n)_{n \geq 0}$ is bounded, then the Eq (3.1) is not Ulam stable.

Proof. Let $\varepsilon > 0$. Since (3.6) is not Ulam stable, i.e. there exists a sequence $(\bar{z}_n)_{n \geq 0}$ in $X$, satisfying the inequality

such that for every sequence $(\bar{y}_n)_{n \geq 0}$ with

we have

Let $(x_n)_{n \geq 0}$ be such that

(it suffices to take $x_0 = 0$ in order to determine $(x_n)_{n \geq 0}$ step by step). Inequality 3.11 implies that the sequence $(x_n)_{n \geq 0}$ satisfies

Let now $(y_n)_{n \geq 0}$ be an arbitrary sequence defined by

and $(\bar{y}_n)_{n \geq 0}$ be the sequence given by

Then the relations (3.12) and (3.13) hold.

Finally, we have to prove that $\sup_{n \geq 0} \|\bar{y}_n-\bar{z}_n\| = \infty$. Suppose the contrary. Then there exists $M > 0$ such that

From (3.14) and (3.16) it follows that

for every $n \geq 0$, which contradicts relation (3.13), if we take also into account that $(u_n)_{n \geq 0}$ is bounded. □

The next corollary gives sufficient conditions for Eq (3.1) to be not Ulam stable. To prove this, just take into account Lemma 3.1, Theorem 2.6 and Theorem 3.7.

Corollary 3.8. If there exist $u_0, u_1, u'_0 \in \mathbb{K}$ such that $(u_n)_{n \geq 0}$ and $(u'_n)_{n \geq 0}$ are bounded and $\lim\limits_{n \rightarrow \infty} |a_n-u_{n+2}-u'_{n+1}| = 1$, then the Eq (3.1) is not Ulam stable.

Example 3.9. The following linear recurrence

is not Ulam stable. Indeed, letting $u_0 = u_1 = -1$ and $u'_0 = 1$, we get $u_n = -1$ and $u'_n = \frac{1}{n+1}$ for all $n \geq 0$ and consequently $\lim\limits_{n \rightarrow \infty} |a_n-u_{n+2}-u'_{n+1}| = 1$, which means that we do not have stability.

For the case of constant coefficients in the Eq (3.1), we get the result given in ^[22].

Corollary 3.10. The third order linear difference equation with constant coefficients

is Ulam stable if and only if none of the roots of the characteristic equation $\lambda^3-a\lambda^2-b\lambda-c = 0$ lie on the unit circle.

Proof. Indeed, let $\lambda_1$, $\lambda_2$ and $\lambda_3$ be the roots of the characteristic equation and suppose without loss of generality that $|\lambda_1| = 1$ and take $u_0 = u_1 = \lambda_2$, $u'_0 = \lambda_3$. Then, using Corollary 3.8 and Vieta's formulas, i.e. the relations $\lambda_1+\lambda_2+\lambda_3 = a$, $\lambda_1\lambda_2+\lambda_1\lambda_3+\lambda_2\lambda_3 = -b$ and $\lambda_1\lambda_2\lambda_3 = c$, one can easily check that $u_n = \lambda_2, n \geq 2$ and $u'_n = \lambda_3, n \geq 1$. Moreover, $\lim_{n \rightarrow \infty} |a-u_{n+2}-u'_{n+1}| = |a-\lambda_2-\lambda_3| = |\lambda_1|$, which means that (3.17) is not Ulam stable, a contradiction. Conversely, if $|\lambda_1| \neq 1$, $|\lambda_2| \neq 1$ and $|\lambda_3| \neq 1$, it is sufficient to consider $u_0 = u_1 = \lambda_1$ and $u'_0 = \lambda_2$. Indeed, due to the choice of the initial values it follows that $\lim_{n \rightarrow \infty} |u_n| = |\lambda_1| \neq 1$, $\lim_{n \rightarrow \infty} |u'_n| = |\lambda_2| \neq 1$ and $\lim_{n \rightarrow \infty} |a-u_{n+2}-u'_{n+1}| = |a-\lambda_1-\lambda_2| = |\lambda_3| \neq 1$. Finally, the Eq (3.17) is Ulam stable, by Theorem 3.5. □

Finally, it is worth mentioning here that the result on nonstability of (3.1) contains in particular the case of nonstability for constant coefficients proved in ^{[6,9,10,22,24]}.

4.   Conclusions

In this paper we give some results on Ulam stability for the second order and for the third order linear difference equation with nonconstant coefficients in a Banach space. As far as we know there are few results on Ulam stability for such equations (see, e.g., ^[11,12,13]). The novelty of this approach consists in the fact that we decompose the second order linear difference equation in a Riccati difference equation and a linear difference equation. In this way we can use the results on Ulam stability of the first order linear difference equation.

The importance of these results consists in the fact that they are related to the theory of perturbation of a discrete dynamical system (see ^[15]). Remark that for difference equations with constant coefficients we get the results obtained in the paper ^[3].

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

This research was partially supported by the project 38 PFE in the frame of the programme PDI-PFE-CDI 2021.

Conflict of interest

The authors declare no conflicts of interest.