scholarly journals Periodicity of the Positive Solutions of a Fuzzy Max-Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Qiuli He ◽  
Chunyan Tao ◽  
Taixiang Sun ◽  
Xinhe Liu ◽  
Dongwei Su
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Fuzzy Numbers ◽  
Periodic Sequence

We investigate the periodic nature of the positive solutions of the fuzzy max-difference equationxn+1=maxAn/xn-m,xn-k,n=0,1,…, wherek,m∈{1,2,…},Anis a periodic sequence of fuzzy numbers, andx-d,x-d+1,…,x0are positive fuzzy numbers withd=m,k. We show that every positive solution of this equation is eventually periodic with periodk+1.

scholarly journals ON EXISTENCE OF POSITIVE SOLUTIONS OF NEUTRAL DIFFERENCE EQUATIONS

1994 ◽  
Vol 25 (3) ◽  
pp. 257-265
Author(s):  
J. H. SHEN ◽  
Z. C. WANG ◽  
X. Z. QIAN
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Real Numbers ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Existence Of Positive Solutions

Consider the neutral difference equation \[\Delta(x_n- cx_{n-m})+p_nx_{n-k}=0, n\ge N\qquad (*) \] where $c$ and $p_n$ are real numbers, $k$ and $N$ are nonnegative integers, and $m$ is positive integer. We show that if \[\sum_{n=N}^\infty |p_n|<\infty \qquad (**) \] then Eq.(*) has a positive solution when $c \neq 1$. However, an interesting example is also given which shows that (**) does not imply that (*) has a positive solution when $c =1$.

scholarly journals Global Behavior of the Difference Equationxn+1=xn-1g(xn)

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Hongjian Xi ◽  
Taixiang Sun ◽  
Bin Qin ◽  
Hui Wu
Keyword(s):  
Continuous Function ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Global Behavior ◽  
Initial Values ◽  
The Difference ◽  
Surjective Function

We consider the following difference equationxn+1=xn-1g(xn),n=0,1,…,where initial valuesx-1,x0∈[0,+∞)andg:[0,+∞)→(0,1]is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges toa,0,a,0,…,or0,a,0,a,…for somea∈[0,+∞). (2) Assumea∈(0,+∞). Then the set of initial conditions(x-1,x0)∈(0,+∞)×(0,+∞)such that the positive solutions of this equation converge toa,0,a,0,…,or0,a,0,a,…is a unique strictly increasing continuous function or an empty set.

scholarly journals Asymptotic Behavior of a Second-Order Fuzzy Rational Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Q. Din
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Fuzzy Numbers ◽  
Initial Conditions ◽  
Second Order ◽  
Qualitative Behavior ◽  
Existence Of Positive Solutions ◽  
Rational Difference Equation ◽  
Boundedness And Persistence ◽  
Theoretical Results

We study the qualitative behavior of the positive solutions of a second-order rational fuzzy difference equation with initial conditions being positive fuzzy numbers, and parameters are positive fuzzy numbers. More precisely, we investigate existence of positive solutions, boundedness and persistence, and stability analysis of a second-order fuzzy rational difference equation. Some numerical examples are given to verify our theoretical results.

scholarly journals On the Asymptotic Behavior of a Difference Equation with Maximum

2008 ◽  
Vol 2008 ◽  
pp. 1-6 ◽  
Author(s):  
Fangkuan Sun
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
The Difference

We study the asymptotic behavior of positive solutions to the difference equationxn=max{A/xn-1α,B/xn−2β},n=0,1,…,where0<α, β<1, A,B>0. We prove that every positive solution to this equation converges tox∗=max{A1/(α+1),B1/(β+1)}.

scholarly journals On Dynamical Behavior of Discrete Time Fuzzy Logistic Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Qianhong Zhang ◽  
Fubiao Lin
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Existence And Uniqueness ◽  
Fuzzy Numbers ◽  
Dynamical Behavior ◽  
Logistic Equation ◽  
State Variables ◽  
Initial Value ◽  
Numerical Examples ◽  
Logistic Difference Equation

The aim of this paper is to investigate the dynamical behavior of the following model which describes the logistic difference equation taking into account the subjectivity in the state variables and in the parameters. xn+1=Axn(1~-xn),  n=0,1,2,⋯, where {xn} is a sequence of positive fuzzy numbers. A,1~ and the initial value x0 are positive fuzzy numbers. The existence and uniqueness of the positive solution and global asymptotic behavior of all positive solution of the fuzzy logistic difference equation are obtained. Moreover, some numerical examples are presented to show the effectiveness of results obtained.

scholarly journals On a Class of Nonautonomous Max-Type Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Wanping Liu ◽  
Xiaofan Yang ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Positive Sequence ◽  
Finite Limit

This paper addresses the max-type difference equationxn=max{fn/xn−kα,B/xn−mβ},n∈ℕ0, wherek,m∈ℕ,B>0, and(fn)n∈ℕ0is a positive sequence with a finite limit. We prove that every positive solution to the equation converges tomax{(limn→∞fn)1/(α+1),B1/(β+1)}under some conditions. Explicit positive solutions to two particular cases are also presented.

scholarly journals On the Global Attractivity of a Max-Type Difference Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-5 ◽  
Author(s):  
Ali Gelişken ◽  
Cengiz Çinar
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Global Attractivity ◽  
Period 2 ◽  
The Difference

We investigate asymptotic behavior and periodic nature of positive solutions of the difference equation , where and . We prove that every positive solution of this difference equation approaches or is eventually periodic with period 2.

scholarly journals On the Recursive Sequencexn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
Stevo Stevic
Keyword(s):  
Periodic Solution ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Complete Picture ◽  
Positive Equilibrium ◽  
Natural Numbers

We give a complete picture regarding the behavior of positive solutions of the following important difference equation:xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj,n∈ℕ0, whereαi, i∈{1,…,k}, andβj, j∈{1,…,m}, are positive numbers such that∑i=1kαi=∑j=1mβj=1,andpi, i∈{1,…,k}, andqj, j∈{1,…,m}, are natural numbers such thatp1<p2<⋯<pkandq1<q2<⋯<qm. The case whengcd(p1,…,pk,q1,…,qm)=1is the most important. For the case we prove that if allpi, i∈{1,…,k}, are even and allqj, j∈{1,…,m}, are odd, then every positive solution of this equation converges to a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium.

scholarly journals Eventual periodicity of the fuzzy max-difference equation $x_{n} = \max \{ C, \frac{x_{n-m-k}}{x_{n-m}}\}$

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Caihong Han ◽  
Guangwang Su ◽  
Lue Li ◽  
Guoen Xia ◽  
Taixiang Sun
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Fuzzy Number ◽  
Fuzzy Numbers ◽  
Initial Values ◽  
Positive Integers

AbstractIn this paper, we study the eventual periodicity of the fuzzy max-type difference equation $x_{n} =\max \{C , \frac{x_{n-m-k}}{x_{n-m} }\}, n\in \{0,1,\ldots \} $ x n = max { C , x n − m − k x n − m } , n ∈ { 0 , 1 , … } , where m and k are positive integers, C and the initial values are positive fuzzy numbers. Let the support $\operatorname{supp} C=\overline{\{t : C(t) > 0\}}=[C_{1},C_{2}]$ supp C = { t : C ( t ) > 0 } ‾ = [ C 1 , C 2 ] of C. We show that: (1) if $C_{1}>1$ C 1 > 1 , then every positive solution of this equation equals C eventually; (2) there exists a positive fuzzy number C with $C_{1}=1$ C 1 = 1 such that this equation has a positive solution which is not eventually periodic; (3) if $C_{2}\leq 1$ C 2 ≤ 1 , then this equation has a positive solution which is not eventually periodic; (4) if $C_{1}<1<C_{2}$ C 1 < 1 < C 2 , then every positive solution of the above equation is not eventually periodic.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

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