Qualitative behavior of a second-order fuzzy difference equation

1 Introduction

The theory of discrete dynamical systems and difference equations developed greatly during the last twenty-five years of the twentieth century. Applications of discrete dynamical systems and difference equations have appeared recently in many areas. The theory of difference equations occupies a central position in applied analysis. There is no doubt that the theory of difference equations continue to play an important role in mathematics as a whole. Nonlinear difference equations of order greater than one are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology physics, engineering, economics, probability theory, genetics, psychology and resource management. It is very interesting to investigate the behavior of solutions of higher-order rational difference equations and to discuss the local asymptotic stability of its equilibrium points. Rational difference equations have been studied by several authors. Especially there has been a great interest in the study of the attractivity of the solutions of such equations. Rational difference equations are special type of nonlinear difference equations. When we divide two difference equations (may be linear or nonlinear), we obtain a rational difference equation (which is always nonlinear). The solutions of rational difference equations can be found rarely in compact or closed forms. For basic theory and applications of nonlinear difference equations we refer the interested reader to [1, 2]. For applications of nonlinear difference equations in population dynamics we refer to [3–7, 18–21].

In case of fuzzy difference equations, parameters, initial conditions and solutions of these equations are fuzzy numbers, while in case of difference equations all these are usually real numbers. Hence study of fuzzy difference equations is more interesting as well as complicated. Using difference equation approach we can investigate dynamical aspects of advanced models in epidemiology, finance and other branches of science. In difference equations assuming the transmission coefficients and other parameter as positive fuzzy numbers, we can explore several dynamics of humans health related issues. These issues include cardiology, hydrology, health services and vital status ascertainment in clinical trials. The readers are referred to see the monographs [22, 29]. For basic theory of fuzzy difference equations see [8]. In [9], embedding problem of non-compact fuzzy number space is given. It is very interesting to investigate the dynamics of nonlinear fuzzy difference equations [10–15]. Bajo and Liz [16] investigated the global behavior of difference equation: x n + 1 = x n - 1 a + bx n - 1 x n , for all values of real parameters a,  b.

Our aim in this paper is to investigate the qualitative behavior of following second-order fuzzy rational difference equation x n + 1 = x n - 1 A + Bx n - 1 x n ,(1) where A,  B and initial conditions x_-1,  x₀ are positive fuzzy numbers.

Definition 1. Let X be a normed space. We will denote by 𝔼 ( X ) the family of fuzzy sets u  :  X → [0, 1] which has the following properties:

u is normal, i . e ., there exist an x in X such that u (x) = 1,

An element u ∈ 𝔼 ( X ) is called a fuzzy vector. If X = ℝ , then an element u ∈ 𝔼 ( ℝ ) is called fuzzy number.

Remark 1. In particular, if { [a _α ,  b _α ] } _α∈(0,1] is a family of compact intervals in ℝ such that

[a _β ,  b _β ] ⊂ [a _α ,  b _α ] for all 0 < α ≤ β

Moreover, α-cuts of u are given by [ u ] α = { x ∈ ℝ : u ( x ) ≥ α } for α ∈ [0, 1]. We say a fuzzy number u is positive if supp u ⊂ (0, ∞). Let u,  v be positive fuzzy numbers with [u]  _α  = [u_l,α,  u_r,α] and [v]  _α  = [v_l,α,  v_r,α] for α ∈ (0, 1], then we consider the following metric: D ( u , v ) = sup α ∈ ( 0 , 1 ] max { | u l , α - v l , α | , | u r , α - v r , α | } .

A sequence (x _n ) of fuzzy number is said to be bounded and persists if there exist positive real numbers β,  γ such that supp x _n  ⊂ [β,  γ] for n = 1,  2, ⋯. Moreover, if (x _n ) be a sequence of positive fuzzy numbers and x be any positive fuzzy number such that [x _n ]  _α  = [L_l,α,  R_r,α] and [x]  _α  = [L _α ,  R _α ] for α ∈ (0, 1] and n = 0,  1,  2, ⋯ then (x _n ) converges to x with respect to metric D, if lim n → ∞ D ( x n , x ) = 0 .

Definition 2. Let (x _n ) be a positive solution of the Equation (1) and x ¯ be its equilibrium point, i . e ., x ¯ = x ¯ A + B x ¯ 2 , then

x ¯ is stable, if for given ɛ > 0 there exist a δ = δ (ɛ) > 0 such that if D (x_-i,  x) ≤ δ for i∈ { - 1, 0 }, then D ( x n , x ¯ ) ≤ ɛ for all n > 0.

3 Main results

This section is devoted to the main results, which are associated with the proposed problem. Arguing as in [15], we have the following result.

Theorem 1. Let A,  B are positive fuzzy numbers, then Equation (1) has unique positive solution {x _n } with initial conditions x_-1,  x₀ are positive fuzzy numbers.

Proof. Let {x _n } be a positive solution of Equation (1). The α-cuts of positive fuzzy numbers x _n ,  A,  B are given by [ x n ] α = [ L l , α , R r , α ] , [ A ] α = [ A l , α , A r , α ] , [ B ] α = [ B l , α , B r , α ] .

Then from Equation (1), one has [ x n + 1 ] α = [ L l + 1 , α , R r + 1 , α ] = [ x n - 1 A + Bx n - 1 x n ] α = [ x n - 1 ] α [ A ] α + [ B ] α [ x n - 1 ] α [ x n ] α = [ L n - 1 , α , R n - 1 , α ] [ A l , α , A r , α ] + [ B l , α , B r , α ] [ L n - 1 , α , R n - 1 , α ] [ L n , α , R n , α ] = [ L n - 1 , α A r , α + B r , α R n - 1 , α R n , α , R n - 1 , α A l , α + B l , α L n - 1 , α L n , α ] .

Hence, we obtain the following system

L n + 1 , α = L n - 1 , α A r , α + B r , α R n - 1 , α R n , α R n + 1 , α = R n - 1 , α A l , α + B l , α L n - 1 , α L n , α(2) for n = 0,  1,  2, ⋯ , and α ∈ (0, 1]. Let (L_n,α,  R_n,α) be a solution of the System (2) with initial conditions L_-1,α,  L_0,α,  R_-1,α,  R_0,α for every α ∈ (0, 1]. Then, it is sufficient to prove that (x _n ) is a solution of original System (1) with initial conditions x_-1,  x₀ and [x _n ]  _α  = [L_n,α,  R_n,α] for every n = 0,  1,  2, ⋯ and α ∈ (0, 1]. Moreover, for any fuzzy number u ∈ 𝔼 ( ℝ ) with [ u ] α = [ u - ( α ) , u ¯ ( α ) ] , one has u - ( α ) is non-decreasing and left continuous, while u ¯ ( α ) is non-increasing and left continuous for 0 < α ≤ 1. As A,  B,  x_-1,  x₀ are positive fuzzy numbers, so for α₁,  α₂ ∈ (0, 1] with α₁ ≤ α₂, one has 0 < A l , α 1 ≤ A l , α 2 ≤ A r , α 2 ≤ A r , α 1 , 0 < B l , α 1 ≤ B l , α 2 ≤ B r , α 2 ≤ B r , α 1 , 0 < L - 1 , α 1 ≤ L - 1 , α 2 ≤ R - 1 , α 2 ≤ R - 1 , α 1 , 0 < L 0 , α 1 ≤ L 0 , α 2 ≤ R 0 , α 2 ≤ R 0 , α 1 .

Now, it is easy to show by induction on n that 0 < L_n,α1 ≤ L_n,α2 ≤ R_n,α2 ≤ R_n,α1, n = 0,  1,  2, ⋯.

Similarly, by induction on n, one can show that L_n,α, R_n,α are left continuous for all n = 0,  1,  2, ⋯, and 0 < α ≤ 1. Furthermore, it can be inductively proved that ⋃ α ∈ [ 0 , 1 ] [ L n , α , R n , α ] ¯ ⊂ ( 0 , ∞ ) , i . e . , ∪ α ∈ [ 0 , 1 ] [ L n , α , R n , α ] ¯ is compact. Hence [L_n,α,  R_n,α] determines a sequence (x _n ) of positive fuzzy numbers such that [x _n ]  _α  = [L_n,α,  R_n,α] for every n = 0,  1,  2, ⋯, and α ∈ (0, 1] . ■

In order to study the further dynamics of the fuzzy difference Equation (1) we will use the results concerning the behavior of the solutions of the corresponding system of two parametric ordinary difference equations given by:

u n + 1 = u n - 1 a 1 + b 1 v n - 1 v n , v n + 1 = v n - 1 a 2 + b 2 u n - 1 u n ,(3) where the parameters a₁,  a₂,  b₁,  b₂ as well as the initial conditions u_-1,  u₀,  v_-1,  v₀ are positive real numbers.

Let a₁ < 1,  a₂ < 1, then O = (0,  0) and E = ( 1 - a 2 b 2 , 1 - a 1 b 1 ) be equilibrium points of the System (3).

Theorem 2. Let { ( u n , v n ) } n = - 1 ∞ be any solution of the System (3), then the following statements are true:

For every m ≥ 0 the following results hold: 0 ≤ u n ≤ ( 1 a 1 ) m + 1 u - 1 , if n = 2 m + 1 , 0 ≤ u n ≤ ( 1 a 1 ) m + 1 u 0 , if n = 2 m + 2 , 0 ≤ v n ≤ ( 1 a 2 ) m + 1 v - 1 , if n = 2 m + 1 , 0 ≤ v n ≤ ( 1 a 2 ) m + 1 v 0 , if n = 2 m + 2 .

Proof. (1) The proof follows from induction.

(2) Let α = max {u_-1,  u₀} and β = max{ v_-1,  v₀ }, then it is easy to see from (1) that 0 ≤ u _n  < α,  0 ≤ v _n  < β for all n = 0,  1,  2, ⋯.

Hence, every solution { ( u n , v n ) } n = - 1 ∞ of the System (3) is bounded.

(4) To construct corresponding linearized form of System (3), we consider the following transformations: ( u n , u n - 1 , v n , v n - 1 ) → ( f , f 1 , g , g 1 ) where f = u_n+1,  g = v_n+1,  f₁ = u _n ,  g₁ = v _n . The linearized system of (3) about the equilibrium point (0, 0) is given by: X n + 1 = F J ( 0 , 0 ) X n , Where ( u n u n - 1 v n v n - 1 ) and F J ( 0 , 0 ) = ( 0 1 a 1 0 0 1 0 0 0 0 0 0 0 0 1 1 a 2 0 ) .

The characteristic polynomial of F _J  (0, 0) is given by P ( λ ) = λ 4 - ( 1 a 1 + 1 a 2 ) λ 2 - 1 a 1 a 2 .(4)

Since all eigenvalues of Jacobian matrix F _J  (0, 0) about (0, 0) lie in open unit disk |λ| < 1, if 1 < a₁ and 1 < a₂. Hence, the equilibrium point (0, 0) is locally asymptotically stable.

(4) The linearized system of (2) about the equilibrium point E = ( 1 - a 2 b 2 , 1 - a 1 b 1 ) is given by:

X_n+1 = F _J  (E) X _n , where X n = ( u n u n - 1 v n v n - 1 ) and F J ( E ) ( 0 1 - 1 - a 1 1 - a 2 b 1 b 2 - 1 - a 1 1 - a 2 b 1 b 2 1 0 0 0 - 1 - a 1 1 - a 2 b 2 b 1 0 - 1 - a 1 1 - a 2 b 2 b 1 0 0 1 0 0 ) .

The characteristic polynomial of F _J  (E) is given by

P ( λ ) = λ 4 + ( a 1 + a 2 - a 1 a 2 - 3 ) λ 2 + 2 ( a 1 + a 2 - a 1 a 2 - 1 ) λ + a 1 + a 2 - a 1 a 2 .(5)

The roots of characteristic polynomial P (λ) are given by - 1 , 1 ± a 1 a 2 - a 1 - a 2 + 1 .

It is sufficient to prove that any one of these roots has absolute value greater than one. For this consider | 1 + a 1 a 2 - a 1 - a 2 + 1 | = | 1 + ( 1 - a 1 ) ( 1 - a 2 ) | > 1 .

(5) The proof of 5(i) and 5(ii) follow from induction.

(6) For 1 < a₁ and 1 < a₂, from (i) of (3) of Theorem 2 (0, 0) is locally asymptotically stable. Moreover, from (2) the solution { (u _n ,  v _n )} of the System (3) is bounded if 1 < a₁ and 1 < a₂. Now, it is sufficient to prove that { (u _n ,  v _n )} is decreasing. From the System (3) one has u n + 1 = u n - 1 a 1 + b 1 v n - 1 v n ≥ u n - 1 a 1 < u n - 1 .

This implies that u_2n+1 < u_2n-1 and u_2n+3 < u_2n+1. Hence the subsequences {u_2n+1} ,   { u_2n+2 } are decreasing, i . e ., the sequence {u _n } is decreasing. Similarly, one has v n + 1 = v n - 1 a 2 + b 2 u n u n - 1 ≤ v n - 1 a 2 < v n - 1 .

This implies that v_2n+1 < v_2n-1 and v_2n+3 < v_2n+1. Hence the subsequences {v_2n+1} ,   { v_2n+2 } are decreasing, i . e ., the sequence {v _n } is decreasing. Hence lim n → ∞ x n = lim n → ∞ y n = 0 ■

The following results give the rate of convergence of solutions of a system of difference equations X n + 1 = ( A + B ( n ) ) X n ,(6) where X _n is an m-dimensional vector, A ∈ C^m×m is a constant matrix, and B  :  Z⁺ → C^m×m is a matrix function satisfying ∥ B ( n ) ∥ → 0(7) as n→ ∞, where ∥· ∥ denotes any matrix norm which is associated with the vector norm ∥ B ( n ) ∥ = x 2 + y 2 .

Proposition 1. (Perron’s theorem) [17] Suppose that condition (8) holds. If X _n is a solution of (7), then either X _n  = 0 for all large n or ρ = lim n → ∞ ( ∥ X n ∥ ) 1 n(8) exist and is equal to the modulus of one of the eigen values of matrix A.

Proposition 2. [17] Suppose that condition (8) holds. If X _n is a solution of (7), then either X _n  = 0 for all large n or ρ = lim n → ∞ ∥ X n + 1 ∥ ∥ X n ∥(9) exist and is equal to the modulus of one of the eigen values of matrix A.

Let {(u _n ,  v _n )} be any solution of the System (3) such that lim n → ∞ u n = u ¯ , lim n → ∞ v n = v ¯ , where ( u ¯ , v ¯ ) = ( 0 , 0 ) . To find the error terms, from the System (3) we have u n + 1 - u ¯ = u n - 1 a 1 + b 1 v n - 1 v n - u ¯ a 1 + b 1 v ¯ 2 = ( a 1 + b 1 v n v ¯ ) ( u n - 1 - u ¯ ) - b 1 u n - 1 v ¯ ( v n - v ¯ ) - b 1 v n u ¯ ( v n - 1 - v ¯ ) ( a 1 + b 1 v n - 1 v n ) ( a 1 + b 1 v ¯ 2 ) = a n ( u n - u ¯ ) + b n ( u n - 1 - u ¯ ) + c n ( v n - v ¯ ) + d n ( v n - 1 - v ¯ ) and v n + 1 - v ¯ = v n - 1 a 2 + b 2 u n u n - 1 - v ¯ a 2 + b 2 u ¯ 2 = ( a 2 + b 2 u n u ¯ ) ( v n - 1 - v ¯ ) - b 2 v n - 1 u ¯ ( u n - u ¯ ) - b 2 u n v ¯ ( u n - 1 - u ¯ ) ( a 2 + b 2 u n u n - 1 ) ( a 2 + b 2 u ¯ 2 ) = a n 1 ( u n - u ¯ ) + b n 1 ( u n - 1 - u ¯ ) + c n 1 ( v n - v ) + d n 1 ( v n - 1 - v ¯ )

Let e n 1 = u n - u ¯ and e n 2 = v n - v ¯ , one has e n + 1 1 = a n e n 1 + b n e n - 1 1 + c n e n 2 + d n e n - 1 2 , and e n + 1 2 = a n 1 e n 1 + b n 1 e n - 1 1 + c n 1 e n 2 + d n 1 e n - 1 2 , where a n = 0 , b n = a 1 + b 1 v n v ¯ ( a 1 + b 1 v n - 1 v n ) ( a 1 + b 1 v ¯ 2 ) , c n = - b 1 u n - 1 v ¯ ( a 1 + b 1 v n - 1 v n ) ( a 1 + b 1 v ¯ 2 ) dn = - b 1 v n u ¯ ( a 1 + b 1 v n - 1 v n ) ( a 1 + b 1 v ¯ 2 ) a n 1 = - b 2 u n v ¯ ( a 2 + b 2 u n u n - 1 ) ( a 2 + b 2 u ¯ 2 ) b n 1 = - b 2 u n v ¯ ( a 2 + b 2 u n u n - 1 ) ( a 2 + b 2 u ¯ 2 ) c n 1 = 0 d n 1 = a 2 + b 2 u n u ¯ ( a 2 + b 2 u n u n - 1 ) ( a 2 + b 2 u ¯ 2 ) .

Moreover, lim n → ∞ a n = lim n → ∞ c n = lim n → ∞ d n = 0 , lim n → ∞ b n = 1 a 1 , lim n → ∞ a n 1 = lim n → ∞ b n 1 = lim n → ∞ c n 1 = 0 lim n → ∞ d n 1 = 1 a 2 .

Now the limiting system of error terms can be written as ( e n + 1 1 e n 1 e n + 1 2 e n 2 ) = ( 0 1 a 1 0 0 1 0 0 0 0 0 0 0 0 1 1 a 2 0 ) ( e n 1 e n - 1 1 e n 2 e n - 1 2 ) , which is similar to linearized system of (3) about the equilibrium point (0, 0).

Using Proposition 1, one has following result.

Theorem 3. Assume that { (u _n ,  v _n )} be a positive solution of the System (2) such that lim n → ∞ u n = 0 and lim n → ∞ v n = 0 . Then, the error term E n = ( e n 1 e n - 1 1 e n 2 e n - 1 2 ) of every solution of (3) satisfies both of the following asymptotic relations, lim n → ∞ ( ∥ E n ∥ ) 1 n = | λ F J ( 0 , 0 ) | lim n → ∞ ∥ E n + 1 ∥ ∥ E n ∥ = | λ F J ( 0 , 0 ) | , where λF _J  (0, 0) are the characteristic roots of Jacobian matrix F _J  (0, 0).

Theorem 4. Assume that (x _n ) be a positive solution of (1) and 1 < A_l,α, then (x _n ) is bounded and persists. Moreover, (x _n ) converges to 0.

Proof. Consider the following system s n + 1 = s n - 1 γ A + γ B t n - 1 t n , t n + 1 = t n - 1 β A + β B s n - 1 s n(10)

Where [ A l , α , A r , α ] ⊂ ∪ α ∈ [ 0 , 1 ] [ A l , α , A r , α ] ¯ ⊂ [ β A , γ A ] , and [ B l , α , B r , α ] ⊂ ∪ α ∈ [ 0 , 1 ] [ B l , α , B r , α ] ¯ ⊂ [ β B , γ B ] .

Let (s _n ,  t _n ) be a solution of the System (11) with initial conditions s_-1 = β_-1, s₀ = β₀, t_-1 = γ_-1, t₀ = γ₀, where β _i ,  γ _i for i∈ { - 1, 0 } are given as [ L - 1 , α , R - 1 , α ] ⊂ ∪ α ∈ [ 0 , 1 ] [ L - 1 , α , R - 1 , α ] ¯ ⊂ [ β - 1 , γ - 1 ] , and [ L 0 , α , R 0 , α ] ⊂ ∪ α ∈ [ 0 , 1 ] [ L 0 , α , R 0 , α ] ¯ ⊂ [ β 0 , γ 0 ] .

Then, it follows that s 1 = s - 1 γ A + γ B t - 1 t 0 ≤ L 1 , α , t 1 = t - 1 β A + β B s - 1 s 0 ≥ R 1 , α .

Hence by induction one has s _n  ≤ L_n,α,  t _n  ≥ R_n,α for all n = 1,  2, ⋯. Assume that 1 < A_l,α, then it follows that 1 < γ _A ,  1 < β _A . Hence the solution (s _n ,  t _n ) of the System (11) is bounded and persist, which is the solution (x _n ) of the Equation (1). Next, from (6) of the Theorem 2, it is easy to see that lim n → ∞ L n , α = lim n → ∞ R n , α = 0 . Then it follows that lim n → ∞ D ( x n , 0 ) = lim n → ∞ sup α ∈ ( 0 , 1 ] { max { | L n , α - 0 | , | R r , α - 0 | } } = 0 . ■

Theorem 5. Assume that A,  B are positive real numbers and A < 1, then unique positive equilibrium point x ¯ = x ¯ A + B x ¯ 2 of the Equation (1) is unstable.

Proof. Let (x _n ) be a positive solution of the Equation (1) and A,  B are positive real numbers with A < 1. Then, from (5)(i) and (5)(ii) of Theorem 2, either lim n → ∞ R n , α = 0 and lim n → ∞ L n , α = ∞ , or lim n → ∞ R n , α = ∞ and lim n → ∞ L n , α = 0 . Hence, there are no positive numbers β,  γ such that ∪_α∈[0,1] [L_n,α,  R_n,α] ⊂ [β,  γ]. This completes the proof. ■

Example 1. Consider the System (3) with initial conditions u₀ = 0.05,  v₀ = 0.1,  u_-1 = 0.07, v_-1 = 0.06. Moreover, choosing the parameters a₁ = a₂ = 1.006,  b₁ = b₂ = 2.8.

Then, the System (3) can be written as

u n + 1 = u n - 1 1.006 + 2.8 v n - 1 v n , v n + 1 = v n - 1 1.006 + 2.8 u n - 1 u n ,(11) with initial condition u₀ = 0.05,  v₀ = 0.1,  u_-1 = 0.07,  v_-1 = 0.06. The plot of solutions for the System (12) is shown in Fig. 1 and its global attractor is shown in Fig. 2.

OPEN IN VIEWER

Fig.1

Plot of solutions for the System (12).

OPEN IN VIEWER

Fig.2

An attractor for the System (12).

Example 2. Consider System (3) with initial conditions u₀ = 0.2,  v₀ = 0.4,  u_-1 = 0.5,  v_-1 = 0.05. Moreover, choose the parameters a₁ = 1.05,  a₂ = 1.03,  b₁ = 10,  b₂ = 2.

Then, the System (3) can be written as

u n + 1 = u n - 1 1.05 + 10 v n - 1 v n , v n + 1 = v n - 1 1.03 + 2 u n - 1 u n ,(12) with initial condition u₀ = 0.2,  v₀ = 0.4,  u_-1 = 0.5,  v_-1 = 0.05. The plot of u _n for the System (13) is shown in Fig. 3, the plot of v _n is shown in Fig. 4 and an attractor of the System (3) is shown in Fig. 5.

OPEN IN VIEWER

Fig.3

Plot of u _n for the System (13).

OPEN IN VIEWER

Fig.4

Plot of v _n for the System (13).

OPEN IN VIEWER

Fig.5

An attractor of System (13).

The present work is related to the qualitative nature of the positive solutions of a second-order rational fuzzy difference equations with initial conditions as well as the parameters are positive fuzzy numbers. Furthermore, the existence of positive solutions, boundedness, persistence, and the parametric conditions for stability analysis are investigated. Due to the diverse applications of fuzzy difference equations in the applied sciences, the researchers are more interested to overcome the problems faced in this numerical technique. Since it is well known that, in fuzzy difference equations parameters are positive fuzzy numbers and problems due to these parameters may be handle through a strong and precise parameterized fuzzy mathematical tools. Therefore, well-known parameterized mathematical tools known as fuzzy soft set [25] and soft fuzzy rough sets (tool used for reduction of parameters) [24, 26–28] can be used to overcome the upcoming difficulties faced in the aforementioned numerical technique.