# Periodicity of the Positive Solutions of a Fuzzy Max-Difference Equation

Authors: Qiuli He; Chunyan Tao; Taixiang Sun; Xinhe Liu; Dongwei Su
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We investigate the periodic nature of the positive solutions of the fuzzy max-difference equationxn+1=maxAn/xn-m,xn-k,n=0,1,…, where k,m∈{1,2,…}, An is a periodic sequence of fuzzy numbers, and x-d,x-d+1,…,x0 are positive fuzzy numbers with d=m,k. We show that every positive solution of this equation is eventually periodic with period k+1.


---

## Body

## 1. Introduction

The max operator arises naturally in certain models in automatic control theory (see [1, 2]). In recent years, the discrete case involving difference equations with maximum has been receiving increasing attention (see [3–8]). Elsayed and Stević [9] considered the max-difference equation
(1)xn+1=max⁡⁡{Bxn,xn-2},n=0,1,…,
where B∈R≡(-∞,+∞) and the initial conditions x-2,x-1,x0∈R and showed that every well-defined solution of this equation is eventually periodic with period 3.In [10], Iričanin and Elsayed investigated the max-difference equation
(2)xn+1=max⁡⁡{Bxn,xn-3},n=0,1,…,
where B∈R and the initial conditions x-3,x-2,x-1,x0∈R and showed that every well-defined solution of this equation is eventually periodic with period 4.Recently Xiao and Shi [11] studied the max-difference equation
(3)xn+1=max⁡⁡{Bxn,xn-1},n=0,1,…,
where B∈R and the initial conditions x-1,x0∈R and showed that every well-defined solution of the above equation is eventually periodic with period 2.In [12], we dealt with the max-difference equation
(4)xn+1=max⁡⁡{Bxn,xn-k},n=0,1,…,
where B∈R, k∈{1,2,…} and the initial conditions x-k,x-k+1,…,x0∈R and showed that every well-defined solution of the above equation is eventually periodic with period k+1, which extended the results of [9–11] to the general case.Recently there has been an increase in interest in the study of fuzzy difference equations (see [13–15]). In [16], Stefanidou and Papaschinopoulos studied the periodicity of the positive solutions of the following fuzzy max-difference equation
(5)xn=max⁡⁡{Axn-k,Bxn-m},n=0,1,…,
where A, B, and the initial conditions x-d,x-d+1,…,x0 with d=max⁡⁡{k,m} are positive fuzzy numbers.In [17], Zhang et al. dealt with the existence, the boundedness, and the asymptotic behavior of the positive solutions to a first order fuzzy Ricatti difference equation
(6)xn+1=A+xnB+xn,n=0,1,…,
where A,B, and the initial condition x0 are positive fuzzy numbers.In this note, our goal is to investigate the periodicity of the positive solutions of the fuzzy max-difference equation(7)xn+1=max⁡⁡{Anxn-m,xn-k},n=0,1,…,
where k,m∈{1,2,…}, An is a periodic sequence of fuzzy numbers, and x-d,x-d+1,…,x0 are positive fuzzy numbers with d={m,k}. Our main result is the following theorem.Theorem 1.
Letk,m∈{1,2,…} and An be a periodic sequence of fuzzy numbers. Then every positive solution of (7) is eventually periodic with period k+1.


## 2. Preliminaries

We need the following definitions. A functionU from R+=(0,+∞) into the interval [0,1] is called a fuzzy number if the following statements hold (see [18]).(1)
U is normal (i.e., U(x)=1 for some x∈R+).(2)
U is a convex fuzzy set (i.e., U(λx+(1-λ)y)≥min⁡⁡{U(x),U(y)} for any λ∈[0,1] and any x,y∈R+).(3)
U is upper semicontinuous.(4)
The supportsupp⁡U=⋃a∈(0,1]‍[U]a¯={x:U(x)>0}¯ is compact,where [U]a={x∈R+:U(x)≥a} (for any a∈(0,1]) (which are said to be the a-cuts of the fuzzy number U) and M¯ is the closure of set M. We see from [19, Theorem 3.1.5 and Theorem 3.1.8] the a-cuts of the fuzzy number U are closed intervals.A fuzzy numberU is said to be positive if min⁡(supp⁡U)>0. If U∈R+, then U is a positive fuzzy number (it is called a trivial fuzzy number also) with [U]a=[U,U] for any a∈(0,1].For some positive integerk, let U1,U2,…,Uk be fuzzy numbers and a∈(0,1] with
(8)[Ui]a=[Ui,l,a,Ui,r,a]for0≤i≤k.
Write
(9)Vl,a=max⁡⁡{Ui,l,a:0≤i≤k},Vr,a=max⁡⁡{Ui,r,a:0≤i≤k}.
Then we know from [20, Theorem 2.1] that there exists a fuzzy number V such that
(10)[V]a=[Vl,a,Vr,a]foranya∈(0,1].
By [21] and [22, Lemma 2.3] one can define
(11)V=max⁡⁡{Ui:0≤i≤k}.A sequence of positive fuzzy numbers{xn}n=-d∞ is said to be a solution of (7) if it satisfies (7). If there exists a positive integer M and p such that, for all n≥M,
(12)xn+p=xn,
then {xn}n=-d∞ is said to be eventually periodic with period p.Proposition 2.
Letx-d,x-d+1,…,x0 be a sequence of positive fuzzy numbers. Then there exists a unique positive solution {xn}n=-d∞ of (7) with initial values x-d,x-d+1,…,x0.Proof.
Assume that[An]a=[An,l,a,An,r,a] (for any a∈(0,1]) and n≥0. Let x-d,x-d+1,…,x0 be positive fuzzy numbers such that
(13)[xi]a=[Pi,a,Qi,a]for-d≤i≤0,a∈(0,1],
and let {(Pn,a,Qn,a)}n=-d∞(a∈(0,1]) be the unique positive solution of the following system of difference equations:
(14)Pn+1,a=max⁡⁡{An,l,aQn-m,a,Pn-k,a},Qn+1,a=max⁡⁡{An,r,aPn-m,a,Qn-k,a},
with initial values (Pi,a,Qi,a)(-d≤i≤0). Arguing as in Proposition 3.1 of [23] we may show that {(Pn,a,Qn,a)}n=-d∞(a∈(0,1]) determines a sequence of positive fuzzy numbers {xn}n=-d∞ with
(15)[xn]a=[Pn,a,Qn,a],n≥-d,a∈(0,1],
and that {xn}n=-d∞ is the unique positive solution of (7) with initial values x-d,x-d+1,…,x0. This completes the proof of the proposition.


## 3. Proof of Theorem1

Lemma 3.
Consider the system of difference equations(16)yn+1=max⁡⁡{Cnzn-m,yn-k},zn+1=max⁡⁡{Bnyn-m,zn-k},n=0,1,…,
where Bn,Cn are two periodic sequences of positive real numbers and the initial values y-d,z-d,…,y0,z0 are positive real numbers. Then every positive solution of (16) is eventually periodic of period k+1.Proof.
Let{(yn,zn)}n=-d∞ be a positive solution of (16). We have from (16) that, for any n≥0 and any i≥0,
(17)y(n+1)(k+1)+i=max⁡{C(n+1)(k+1)+i-1z(n+1)(k+1)+i-m-1,yn(k+1)+i}≥yn(k+1)+i,(18)z(n+1)(k+1)+i=max⁡⁡{B(n+1)(k+1)+i-1y(n+1)(k+1)+i-m-1,zn(k+1)+i}≥zn(k+1)+i.
Then {yn(k+1)+i}n=0∞ and {zn(k+1)+i}n=0∞ are increasing for every 0≤i≤k.
Now we show that{yn(k+1)+i}n=0+∞ is a constant sequence eventually for every 0≤i≤k. Indeed, if {yn(k+1)+r}n=0+∞ is not constant sequence eventually for some 0≤r≤k, then there exist km<n1<n2<⋯ such that yni(k+1)+r>y(ni-1)(k+1)+r and Cni(k+1)+r-1 is a constant sequence for all i≥1 since Cn is a periodic sequence. Thus we have
(19)yni+1(k+1)+r=max⁡{Cni+1(k+1)+r-1zni+1(k+1)+r-m-1,y(ni+1-1)(k+1)+r}=Cni+1(k+1)+r-1zni+1(k+1)+r-m-1>y(ni+1-1)(k+1)+r≥yni(k+1)+r=max⁡{Cni(k+1)+r-1zni(k+1)+r-m-1,y(ni-1)(k+1)+r}=Cni(k+1)+r-1zni(k+1)+r-m-1.
From this we obtain that, for all i≥1,
(20)zni(k+1)+r-m-1>zni+1(k+1)+r-m-1.
This is a contradiction.
In a similar fashion, we can show that{zn(k+1)+i}n=0+∞ is also a constant sequence eventually for every 0≤i≤k.
From the above we see that{(yn,zn)}n=-d∞ is eventually periodic with period k+1. This completes the proof of Lemma 3.Proof of Theorem1.
Let{xn}n=-d∞ be a positive solution of (7) with initial values x-d,x-d+1,…,x0 satisfying (13) and let (15) hold. We see from Proposition 2 that {(Pn,a,Qn,a)}n=-d∞(a∈(0,1]) satisfies system (14). Using Lemma 3 we know that {(Pn,a,Qn,a)}n=-d∞ is eventually periodic with period k+1. Therefore, it follows from (14) and Lemma 3 that {xn}n=-d∞ is eventually periodic of period k+1. This completes the proof of Theorem 1.



---

*Source: 760247-2014-05-22.xml*