Qualitative behavior of a higher-order fuzzy difference equation

1.   Introduction

Difference equations play an important role in modeling problems that arise in biology, physics, engineering, finance and many other areas ^[1,2]. In many cases, the obtained models are restricted in their ability to describe phenomena due to the incomplete or uncertain information about the variables, parameters and initial conditions available. To take into account these uncertainties or lack of precision, we may use a fuzzy environment in variables, parameters and initial conditions, by turning general difference equations into fuzzy difference equations. The most striking applications of the fuzzy notion can be seen in some branches of engineering. See ^[3,4] for a few applications from mechanical engineering. The fuzzy notion also has applications in differential equations, which are closely related to difference equations. See ^[5]. Most numerical methods convert differential equations to difference equations, and their relationship is discussed in ^[6].

In ^[7], Bajo and Liz investigated the behavior of the ordinary difference equation

where the parameters $a, b$ and the initial conditions $x_{-1}, x_{0}$ are real numbers.

Moreover, in ^[8], Rahman et al. investigated the behavior of the fuzzy difference equation

where $A, B$ and the initial conditions $x_{-1}, x_{0}$ are positive fuzzy numbers.

In ^[9], Shojaei et al. investigated the behavior of the ordinary difference equation

where the parameters $\alpha, \beta, \gamma$ and the initial conditions $x_{-2}, x_{-1}, x_{0}$ are real numbers.

Also, in ^[10], Atak et al. investigated the behavior of the fuzzy difference equation

where $(z_{n})$ is a sequence of positive fuzzy numbers, $C$ and the initial conditions $z_{-2}, z_{-1}, z_{0}\ $are positive fuzzy numbers.

Motivated by above mentioned studies, in this paper, we investigate the qualitative behavior of the fuzzy difference equation

where $n\in \mathbb{N}_{0} = \; \mathbb{N}\cup \left\{ 0\right\}, \; (z_{n})$ is a sequence of positive fuzzy numbers, $A, B, C$ and the initial conditions $z_{-j} \; (j = 0, 1, ..., s)$ are positive fuzzy numbers and $s$ is a positive integer. Moreover, some examples are given to verify the effectiveness of the results obtained.

Recently some researchers have studied fuzzy difference equations and properties of their solutions in different approaches. See, for example, ^{[11,12,13,14,15,16,17,18,19,20]}. We also refer to ^[21,22] for some fundamentals of ordinary difference equations.

2.   Preliminaries

In this section, we give some definitions which will be used in this paper. For more details see ^[23,24,25].

Definition 2.1. Consider a fuzzy set $A$ which is a function from the set of real numbers $\mathbb{R}$ into the interval $\left[ 0, 1\right]$. We say that $A$ is a fuzzy number if it satisfies the following properties

(a) $A$ is normal, i.e., $\exists \; x_{0}\in \mathbb{R}$ with $A(x_{0}) = 1$,

(b) $A$ is fuzzy convex, i.e., $A(tx_{1}+(1-t)x_{2})\geq \min \left\{ A(x_{1}), A(x_{2})\right\}, \; \forall t\in \lbrack 0, 1]$ and $x_{1}, x_{2}\in \mathbb{R},$

(c) $A$ is upper semicontinuous on $\mathbb{R},$

(d) $A$ is compactly supported, i.e., $\overline{\left\{ x\in \mathbb{R}:A(x) > 0\right\} }$ is compact.

Let us denote by $\mathbb{R}_{F}$ the space of all fuzzy numbers. For $0 < \alpha \leq 1$ and $A\in \mathbb{R}_{F},$ we denote $\alpha$-cuts of fuzzy number $A$ by $[A]^{\alpha } = \left\{ x\in \mathbb{R}, A(x)\geq \alpha \right\}$ and $[A]^{0} = \overline{\left\{ x\in \mathbb{R}, A(x)\geq 0\right\} }.$ We call $[A]^{0},$ the support of fuzzy number $A$ and denote it by $supp\, (A).$

The fuzzy number $A$ is called positive if $supp\, (A)\subset (0, \infty).$ We denote by $\mathbb{R}_{F}^{+}$, the space of all positive fuzzy numbers.

Definition 2.2. (a) Let $A, B\in \mathbb{R}_{F}$ with $[A]^{\alpha } = [A_{l}^{\alpha }, A_{r}^{\alpha }]$ and $[B]^{\alpha } = [B_{l}^{\alpha }, B_{r}^{\alpha }]$ for $\alpha \in \lbrack 0, 1]$. We define $\left\vert \left\vert A\right\vert \right\vert$ on the space of fuzzy numbers as follow;

where $\sup$ is taken for all $\alpha \in \lbrack 0, 1]$. We recall the following metric

where $\sup$ is taken for all $\alpha \in \lbrack 0, 1]$.

(b) Let $(x_{n})$ be a sequence of positive fuzzy numbers and $x\in \mathbb{R }_{F}$. Then, we say that

The following lemma and definition are given in ^[24].

Lemma 2.1. Let $X, Y \; \in \mathbb{R}_{F}$ and $[X]^{\alpha } = [X_{l}^{\alpha }, X_{r}^{\alpha }], \; [Y]^{\alpha } = [Y_{l}^{\alpha }, Y_{r}^{\alpha }]$ for $\alpha \in \lbrack 0, 1]$ be the $\alpha$-cuts of $X, Y,$ respectively. Let $Z$ be a fuzzy number such that $[Z]^{\alpha } = [Z_{l}^{\alpha }, Z_{r}^{\alpha }]$ for $\alpha \in \lbrack 0, 1].$ Then, MIN$\left\{ X, Y\right\} = Z$ (resp. MAX$\left\{ X, Y\right\} = Z)$ if and only if $\min \left\{ X_{l}^{\alpha }, Y_{l}^{\alpha }\right\} = Z_{l}^{\alpha }$ and $\min \left\{ X_{r}^{\alpha }, Y_{r}^{\alpha }\right\} = Z_{r}^{\alpha }$ (resp. $\max \left\{ X_{l}^{\alpha }, Y_{l}^{\alpha }\right\} = Z_{l}^{\alpha }$ and $\max \left\{ X_{r}^{\alpha }, Y_{r}^{\alpha }\right\} = Z_{r}^{\alpha }$).

Definition 2.3. (a) We say that a sequence of positive fuzzy numbers $(x_{n})$ is bounded and persistent if there exist $n_{0}\in \mathbb{N}$ and $C, D\in \mathbb{R} _{F}^{+}$ such that MIN$\left\{ x_{n}, C\right\} = C$ and MAX$\left\{ x_{n}, D\right\} = D$ for $n\geq n_{0}.$

(b) We say that $(x_{n})$ for $n\in \mathbb{N}_{0}$ is an unbounded sequence if the $\left\vert \left\vert x_{n}\right\vert \right\vert$ for $n\in \mathbb{N}_{0}$ is an unbounded sequence.

We need the following lemma which is given in ^[18] in the next section.

Lemma 2.2. Let $f$ be a continuous function from $\mathbb{R}^{+}\times \mathbb{R}^{+}\times...\times \mathbb{R}^{+}$ into $\mathbb{R}^{+}$ and $B_{0}, B_{1}, ..., B_{k}\in \mathbb{R}_{F}$. Then,

3.   Main results

In this section, we study the existence and properties of the positive solutions to (1.5). If $(z_{n})$ is a sequence of positive fuzzy numbers which satisfies (1.5), we say $(z_{n})$ is a positive solution of (1.5).

Theorem 3.1. Consider (1.5) where $A, B, C\in \mathbb{R}_{F}^{+}$. Then, for any positive fuzzy numbers $z_{-j} \; (j = 0, 1, ..., s)$ there exists a unique positive solution $(z_{n})$ of (1.5) with the initial conditions $z_{-j}$ $(j = 0, 1, ..., s).$

Proof. Suppose that there exists a sequence of fuzzy numbers $(z_{n})$ satisfying (1.5) with the initial conditions $z_{-j}\ (j = 0, 1, ..., s).$ Consider the $\alpha$-cuts

for $n = -s, -s+1, ...$ and all $\alpha \in (0, 1].$ Then, from (1.5)-(3.1) and Lemma 2.2, it follows that

from which we have

for $n\in \mathbb{N}_{0}$ and $\alpha \in (0, 1].$ Then, it is clear that for any $(L_{j}^{\alpha }, R_{j}^{\alpha }), \; j = -s, -s+1, ..., 0,$ there exists a unique solution $(L_{n}^{\alpha }, R_{n}^{\alpha })$ with the initial conditions $(L_{j}^{\alpha }, R_{j}^{\alpha }), \; j = -s, -s+1, ..., 0$ for all $\alpha \in (0, 1].$

Now, we prove that $[L_{n}^{\alpha }, R_{n}^{\alpha }]$ for all $\alpha \in (0, 1],$ where $(L_{n}^{\alpha }, R_{n}^{\alpha })$ is the solution of the system (3.2) with the initial conditions $(L_{j}^{\alpha }, R_{j}^{\alpha }), \; j = -s, -s+1, ..., 0,$ determines the solution $(z_{n})$ of (1.5) with the initial conditions $z_{-j}\ (j = 0, 1, ..., s)$ such that

Since $A, B, C, z_{-j}(j = 0, 1, ..., s)\in \mathbb{R}_{F}^{+}$ for any $\alpha _{1}, \alpha _{2}\in (0, 1]$ and $\alpha _{1}\leq \alpha _{2},$ we have

for $j = -s, -s+1, ..., 0.$ We prove by the induction that

From (3.4), we see that (3.5) holds for $n = -s, -s+1, ..., 0.$ Suppose that (3.5) is valid for $n\leq k, \; k\in \left\{ 1, 2, ...\right\}.$ Then, from (3.2), (3.4) and (3.5) for $n\leq k,$ it follows that

Therefore, (3.5) is satisfied. Moreover, from (3.2), we have

Then, since $A, B, C, z_{-j}\ (j = 0, 1, ..., s)\in \mathbb{R}_{F}^{+}$, we have that $A_{l}^{\alpha }, \; A_{r}^{\alpha }, \; B_{l}^{\alpha }, \; B_{r}^{\alpha }, \; C_{l}^{\alpha }, \; C_{r}^{\alpha }, \; L_{-j}^{\alpha }, \; R_{-j}^{\alpha } \; (j = 0, 1, ..., s)$ are left continuous. So, from (3.6) we see that $L_{1}^{\alpha }$ and $R_{1}^{\alpha }$ are also left continuous. Working inductively we can easily obtain that $L_{n}^{\alpha }$ and $R_{n}^{\alpha }$ are left continuous for $n\in \mathbb{N}.$

Now, we prove that $\overline{\cup _{\alpha \in (0, 1]}[L_{n}^{\alpha }, R_{n}^{\alpha }]}$ is compact. It is sufficient to prove that $\cup _{\alpha \in (0, 1]}[L_{n}^{\alpha }, R_{n}^{\alpha }]$ is bounded. Let $n = 1,$ since $A, B, C, z_{-j}\ (j = 0, 1, ..., s)\in \mathbb{R}_{F}^{+}$, there exist constants

for $j = 0, 1, ..., s$ such that

for $j = 0, 1, ..., s.$ Therefore, from (3.6) and (3.7) we can easily prove that

from which it is clear that

Also, (3.9) implies that $\overline{\cup _{\alpha \in (0, 1]}[L_{1}^{\alpha }, R_{1}^{\alpha }]}$ is compact and $\overline{\cup _{\alpha \in (0, 1]}[L_{1}^{\alpha }, R_{1}^{\alpha }]}\subset (0, \infty).$ Working inductively, we can easily see that $\overline{\cup _{\alpha \in (0, 1]}[L_{n}^{\alpha }, R_{n}^{\alpha }]}$ is compact and

Therefore, using (3.5), (3.10) and since $L_{n}^{\alpha }, \; R_{n}^{\alpha }$ are left continuous, we see that $[L_{n}^{\alpha }, R_{n}^{\alpha }]$ determines a sequence of positive fuzzy numbers $(z_{n})$ such that (3.3) holds.

We prove now that $(z_{n})$ is the solution of (1.5) with the initial conditions $z_{-j}\ (j = 0, 1, ..., s).$ Since

for all $\alpha \in (0, 1],$ we have that $(z_{n})$ is the solution of (1.5) with the initial conditions $z_{-j}\ (j = 0, 1, ..., s).$

Suppose that there exists another solution $(\widetilde{z}_{n})$ of (1.5) with the initial conditions $z_{-j}\ (j = 0, 1, ..., s).$ Then, we can easily show that

Then, from (3.3) and (3.11), $[z_{n}]^{\alpha } = [\widetilde{z} _{n}]^{\alpha }$ for $\alpha \in (0, 1]$ and $n = -s, -s+1, ...$ from which we get $z_{n} = \widetilde{z}_{n}$ for $n = -s, -s+1, ...$. Thus, the proof is completed.

In order to study the further dynamics of (1.5), we apply the results concerning the following system of ordinary difference equations

where the parameters $a_{1}, \; b_{1}, \; c_{1}, \; a_{2}, \; b_{2}, \; c_{2}$ and initial conditions $u_{-j}, v_{-j} \; (j = 0, 1, ..., s)$ are positive real numbers and $s$ is a positive integer. Note that system (3.12) can be written as

by the change of variables $u_{n} = \left(\frac{a_{2}}{c_{2}}\right) ^{\frac{1 }{s+1}}x_{n}, \; v_{n} = \left(\frac{a_{1}}{c_{1}}\right) ^{\frac{1}{s+1} }y_{n}$ with $q = \; \frac{b_{1}}{a_{1}}$ and $p = \; \frac{b_{2}}{a_{2}}.$ The equilibrium points of (3.13) are the solutions of the equations

We need the following results concerning the behavior of the solutions of the system (3.13) which has been presented in ^[26].

If $p < 1, \; q < 1$, then (3.13) has equilibrium points $(0, 0)$ and $(\sqrt[ s+1]{1-p}, \sqrt[s+1]{1-q}).$ In addition, if $p < 1, \; q = 1,$ then system (3.13) has an equilibrium $(\sqrt[s+1]{1-p}, 0)$ and if $p = 1, \; q < 1,$ then system (3.13) has an equilibrium $(0, \sqrt[s+1]{1-q}).$ Finally, if $p > 1, \; q > 1,$ then system (3.13) has the unique equilibrium $(0, 0)$.

Theorem 3.2. Let $(x_{n}, y_{n})$ be any positive solution of system (3.13), then the following statements are true:

$\mathbf{(1)}$ For every $m\geq 0,$ the following results hold:

and

$\mathbf{(2)}$ If $p < 1$ and $q < 1,$ then the following statements are true for $j = -s, -s+1, ..., 0$.

$\mathbf{(i)}$ If $(x_{j}, y_{j})\in (0, \sqrt[s+1]{1-p})\times (\sqrt[ s+]{1-q}, \infty),$ then $(x_{n}, y_{n})\in (0, \sqrt[s+1]{1-p})\times (\sqrt[ s+]{1-q}, \infty).$

$\mathbf{(ii)}$ If $(x_{j}, y_{j})\in (\sqrt[s+1]{1-p}, \infty)\times (0, \sqrt[s+1]{1-q}),$ then $(\sqrt[s+1]{1-p}, \infty)\times (0, \sqrt[s+1]{1-q }).$

$\mathbf{(3)}$ If $p > 1$ and $q > 1,$ then every positive solution $(x_{n}, y_{n})$ of system (3.13) converges to $(0, 0)$ as $n\rightarrow \infty$.

The following corollary can be obtained from Theorem 3.2.

Corollary 3.1. Let $(x_{n}, y_{n})$ be positive solution to system (3.13), then the following statements are true.

$\mathbf{(1)}$ If $p > 1$ and $q > 1,$ then every positive solution $(x_{n}, y_{n})$ of system (3.13) is bounded and persistent.

$\mathbf{(2)}$ If $p < 1$ and $q < 1,$ then the system (3.13) has unbounded solutions:

$\mathbf{(i)}$ If $x_{j}\in \left(0, \sqrt[s+1]{1-p}\right) \;$and$\; y_{j}\in \left(\sqrt[s+1]{1-q}, \infty \right)$ for $j = -s, -s+1, ..., 0,$ then $\underset{n\rightarrow \infty }{lim}x_{n} = 0$ and $\underset{ n\rightarrow \infty }{lim}y_{n} = \infty$.

$\mathbf{(ii)}$ If $x_{j}\in \left(\sqrt[s+1]{1-p}, \infty \right)$ and $y_{j}\in \left(0, \sqrt[s+1]{1-q}\right)$ for $j = -s, -s+1, ..., 0,$ then $\underset{n\rightarrow \infty }{lim}y_{n} = 0$ and $\underset{n\rightarrow \infty }{lim}x_{n} = \infty$.

Proof. $\mathbf{(1)}$ Let $\beta = \max \left\{ x_{-s}, x_{-s+1}, ..., x_{0}\right\}$ and $\gamma = \max \left\{ y_{-s}, y_{-s+1}, ..., y_{0}\right\},$ then it is easy to see from system (3.13) that

for all $n\in \mathbb{N}_{0}.$ Hence, every positive solution $(x_{n}, y_{n})$ of system (3.13) is bounded and persistent.

$\mathbf{(2)}$ We only prove the part (i), since the part (ii) can be proved similarly.

$\mathbf{(i)}$ Assume that $(x_{n}, y_{n})$ is a positive solution of system (3.13) such that $x_{j}\in \left(0, \sqrt[s+1]{1-p}\right) \;$and$\; y_{j}\in \left(\sqrt[s+]{1-q}, \infty \right)$ for $j = -s, -s+1, ..., 0$. Then, from system (3.13), we obtain the following inequalities

and

from which it follows that

So, the proof is completed.

Theorem 3.3. Consider the fuzzy difference Eq (1.5). The the following statements are true.

$\mathbf{(1)}$ If $A_{r}^{\alpha } < B_{l}^{\alpha }$ for all $\alpha \in (0, 1],$ then every positive solution of (1.5) is bounded and persistent.

$\mathbf{(2)}$ If there exists an $\overline{\alpha }\in (0, 1]$ such that $B_{r}^{\overline{\alpha }} < A_{l}^{\overline{\alpha }},$ then the Eq (1.5) has unbounded solutions.

Proof. $\mathbf{(1)}$ Consider the following system of ordinary difference equations

where

Let $(s_{n}, t_{n})$ be a solution of system (3.16) with the initial conditions $(s_{-j}, t_{-j}) = (\gamma _{-j}, \beta _{-j})$ for $j = 0, 1, ..., p$ where $\gamma _{-j}$ and $\beta _{-j}$ are given

Then, from (3.16) and (3.17) it folows that

and

Hence, by induction, we get $s_{n}\leq L_{n}^{\alpha }$ and $R_{n}^{\alpha }\leq t_{n}$ for $n\in < /p > < p > \mathbb{N} < /p > < p > .$ Assume that $A_{r}^{\alpha } < B_{l}^{\alpha }$ for all $\alpha \in (0, 1],$ then it follows that $\gamma _{A} < \beta _{B}$ and $\beta _{A} < \gamma _{B}.$ From (2) of Theorem 3.2, the solution $(s_{n}, t_{n})$ of system (3.15) is bounded and persistent, which is the solution $(z_{n})$ of (1.5). This completes the proof of $(1)$.

$\mathbf{(2)}$ Suppose that there exists an $\overline{\alpha }\in (0, 1]$ such that $B_{r}^{\overline{\alpha }} < A_{l}^{\overline{\alpha }}.$ If $A_{l}^{\overline{\alpha }} = a_{1}, \; A_{r}^{\overline{\alpha }} = a_{2}, \; B_{l}^{\overline{\alpha }} = b_{2}, \; B_{r}^{\overline{\alpha }} = b_{1}, \; L_{n}^{\overline{\alpha }} = x_{n}$ and $R_{n}^{\overline{\alpha }} = y_{n}$ for $n = -s, -s+1, ...,$ then we can apply (i) of (2) in Corollary 3.1 to system (3.2) (We can use (ii) of (2) in Corollary 3.1, too). If there exists an $\overline{\alpha }\in (0, 1]$ such that $B_{r}^{\overline{ \alpha }} < A_{l}^{\overline{\alpha }}$ and $x_{j}\in \left(0, \sqrt[s+1]{1-p} \right) \;$and$\; y_{j}\in \left(\sqrt[s+1]{1-q}, \infty \right)$ for $j = -s, -s+1, ..., 0,$ then there exist solutions $(x_{n}, y_{n})$ of system (3.13) where $\overline{\alpha } = \alpha$ with initial conditions $(x_{-j}, y_{-j})$ for $j = 0, 1, ..., s$ such that

Moreover, if $x_{-j} < y_{-j} \; (\; j = 0, 1, ..., s),$ we can find $z_{-j} \; \in \mathbb{R}_{F}^{+}$ such that

and

for $j = -s, -s+1, ..., 0.$ Let $(z_{n})$ be a positive solution of (1.5) with the initial conditions $z_{-j} \; \left(j = 0, 1, ..., s\right)$ and $[z_{n}]^{\alpha } = [L_{n}^{\alpha }, R_{n}^{\alpha }]$ for $\alpha \in (0, 1].$ Since (3.22) and (3.23) hold and $(L_{n}^{\alpha }, R_{n}^{\alpha })$ satisfies system (3.2), we have

Therefore, from (3.21), (3.24) and since

where sup is taken for all $\alpha \in (0, 1],$ it is clear that solution $(z_{n})$ is unbounded. This completes the proof of $(2)$.

Theorem 3.4. If $A_{r}^{\alpha } < B_{l}^{\alpha }$ for all $\alpha \in (0, 1],$ then every positive solution $(z_{n})$ of (1.5) converges to $0$ as $n\rightarrow \infty.$

Proof. Let $(z_{n})$ be a positive solution of (1.5) such that (3.3) holds with $A_{r}^{\alpha } < B_{l}^{\alpha }$ for all $\alpha \in (0, 1].$ Then, we can apply (3) of Theorem 3.2 to system (3.2). So, we get

Therefore, from (3.25), we get

This completes the proof.

4.   Numerical examples

In this section, to verify obtained results, we give two numerical examples for $s = 3$ with different values of $A, B, C$ where the initial conditions $z_{-3}, \; z_{-2}, \; z_{-1}, \; z_{0}\ $are satisfied

From (4.1), we get

for all $\alpha \in [0, 1].$

Example 4.1. Consider Eq (1.5) where the initial conditions are satisfied (4.1) and $A, B, C$ are satisfied

Then, from (4.2), we get $[A]^{\alpha } = \left[ \frac{\alpha +1}{4}, \frac{3-\alpha }{4}\right], \; [B]^{\alpha } = \left[ \alpha +1, 3-\alpha \right]$ and $[C]^{\alpha } = \left[ \frac{\alpha +1}{2}, \frac{3-\alpha }{2} \right]$ for all $\alpha \in (0, 1].$ By Theorem 3.1, there exists a unique solution. Since $A_{r}^{\alpha } < B_{l}^{\alpha }$ for all $\alpha \in \lbrack 0, 1]$, then by case (1) in Theorem 3.3, the positive solution $(z_{n})$ of fuzzy difference Eq (1.5) is bounded and persistent and by Theorem 3.4, it converges to $0$ as $n\rightarrow \infty.$ For $\alpha _{1} = 0.2$ and $\alpha _{2} = 0.8,$ the $\alpha -$cuts of the solution $L_{n}^{\alpha }$ and $R_{n}^{\alpha }$ are depicted in Figures 1 and 2, respectively.

Example 4.2. Consider Eq (1.5) where the initial conditions are satisfied (4.1) and $A, B, C$ are satisfied

Then, from (4.3), we get $[A]^{\alpha } = \left[ \alpha +2, 4-\alpha \right], \; [B]^{\alpha } = \left[ \frac{\alpha +1}{4}, \frac{3-\alpha }{4} \right]$ and $[C]^{\alpha } = [\alpha +1, 3-\alpha ]$ for all $\alpha \in (0, 1].$ By Theorem 3.1 there exists a unique positive solution. For any $\alpha \in \lbrack 0, 1]$, we have $A_{l}^{\alpha } > B_{r}^{\alpha }$. So, by case (2) in Theorem 3.3, the corresponding fuzzy difference equation has unbounded solutions. For $\alpha _{1} = 0.2$ and $\alpha _{2} = 0.8,$ the $\alpha$-cuts of the solution $L_{n}^{\alpha }$ and $R_{n}^{\alpha }$ are depicted in Figures 3 and 4, respectively.

5.   Conclusions

In this study, we investigated behavior of the fuzzy difference equation $z_{n+1} = Az_{n-s}/(B+C\prod\limits_{i = 0}^{s}z_{n-i})$. We have shown that, under certain conditions, the positive solutions of this equation converge to zero. We have also considered the case where the solutions are unbounded. Finally, we have supported our theoretical results via two numerical examples. This study extends the results in the references ^[8,10].

Conflict of interest

The authors declare that they have no conflicts of interest.