Fuzzy Difference Equations: The Initial Value Problem

2001 ◽  
Vol 5 (6) ◽  
pp. 315-325 ◽  
Author(s):  
James J. Buckley ◽  
◽  
Thomas Feuring ◽  
Yoichi Hayashi ◽  
◽  
...  
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
National Income ◽  
Linear Difference Equation ◽  
Second Order ◽  
Solution Methods ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
The Difference

In this paper we study fuzzy solutions to the second order, linear, difference equation with constant coefficients but having fuzzy initial conditions. We look at two methods of solution: (1) in the first method we fuzzify the crisp solution and then check to see if it solves the difference equation; and (2) in the second method we first solve the fuzzy difference equation and then check to see if the solution defines a fuzzy number. Relationships between these two solution methods are also presented. Two applications are given: (1) the first is about a second order difference equation, having fuzzy initial conditions, modeling national income; and (2) the second is from information theory modeling the transmission of information.

scholarly journals Note on some representations of general solutions to homogeneous linear difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Fibonacci Sequence ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
Homogeneous Linear

Abstract It is known that every solution to the second-order difference equation $x_{n}=x_{n-1}+x_{n-2}=0$ x n = x n − 1 + x n − 2 = 0 , $n\ge 2$ n ≥ 2 , can be written in the following form $x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ x n = x 0 f n − 1 + x 1 f n , where $f_{n}$ f n is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

scholarly journals Boundedness of solutions and stability of certain second-order difference equation with quadratic term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Emin Bešo ◽  
Senada Kalabušić ◽  
Naida Mujić ◽  
Esmir Pilav
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Second Order ◽  
Quadratic Term ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractWe consider the second-order rational difference equation $$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$xn+1=γ+δxnxn−12, where γ, δ are positive real numbers and the initial conditions $x_{-1}$x−1 and $x_{0}$x0 are positive real numbers. Boundedness along with global attractivity and Neimark–Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.

scholarly journals The global behavior of a quadratic difference equation

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6203-6210
Author(s):  
Vahidin Hadziabdic ◽  
Midhat Mehuljic ◽  
Jasmin Bektesevic ◽  
Naida Mujic
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Julia Set ◽  
Second Order ◽  
Global Behavior ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

In this paper we will present the Julia set and the global behavior of a quadratic second order difference equation of type xn+1 = axnxn-1 + ax2n-1 + bxn-1 where a > 0 and 0 ? b < 1 with non-negative initial conditions.

scholarly journals Global Dynamics of Delayed Sigmoid Beverton–Holt Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Toufik Khyat ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Global Dynamics ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Difference

In this paper, certain dynamic scenarios for general competitive maps in the plane are presented and applied to some cases of second-order difference equation xn+1=fxn,xn−1, n=0,1,…, where f is decreasing in the variable xn and increasing in the variable xn−1. As a case study, we use the difference equation xn+1=xn−12/cxn−12+dxn+f, n=0,1,…, where the initial conditions x−1,x0≥0 and the parameters satisfy c,d,f>0. In this special case, we characterize completely the global dynamics of this equation by finding the basins of attraction of its equilibria and periodic solutions. We describe the global dynamics as a sequence of global transcritical or period-doubling bifurcations.

scholarly journals On the Behaviour of the Solutions of a Second-Order Difference Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-14 ◽  
Author(s):  
Leonid Gutnik ◽  
Stevo Stevic
Keyword(s):  
Difference Equation ◽  
Explicit Formula ◽  
Second Order ◽  
Order Difference ◽  
Monotone Solution ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Difference

We study the difference equationxn+1=α−xn/xn−1,n∈ℕ0, whereα∈ℝand wherex−1andx0are so chosen that the corresponding solution(xn)of the equation is defined for everyn∈ℕ. We prove that whenα=3the equilibriumx¯=2of the equation is not stable, which corrects a result due to X. X. Yan, W. T. Li, and Z. Zhao. For the caseα=1, we show that there is a strictly monotone solution of the equation, and we also find its asymptotics. An explicit formula for the solutions of the equation are given for the caseα=0.

scholarly journals TRANSFORMATION OF THE LINEAR DIFFERENCE EQUATION INTO A SYSTEM OF THE FIRST ORDER DIFFERENCE EQUATIONS

2019 ◽  
pp. 76-80
Author(s):  
M.I. Ayzatsky
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Linear Difference Equation ◽  
Second Order ◽  
Dimensional System ◽  
Order Difference ◽  
Order Linear ◽  
First Order ◽  
Second Order Difference Equation ◽  
Order Difference Equation

The transformation of the N-th-order linear difference equation into a system of the first order difference equations is presented. The proposed transformation opens possibility to obtain new forms of the N-dimensional system of the first order equations that can be useful for the analysis of solutions of the N-th-order difference equations. In particular for the third-order linear difference equation the nonlinear second-order difference equation that plays the same role as the Riccati equation for second-order linear difference equation is obtained. The new form of the Ndimensional system of first order equations can also be used to find the WKB solutions of the linear difference equation with coefficients that vary slowly with index.

scholarly journals Local Dynamics and Global Stability of Certain Second-Order Rational Difference Equation with Quadratic Terms

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
S. Jašarević Hrustić ◽  
M. R. S. Kulenović ◽  
M. Nurkanović
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Initial Conditions ◽  
Second Order ◽  
Global Dynamics ◽  
Order Difference ◽  
Local Dynamics ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

We present a complete local dynamics and investigate the global dynamics of the following second-order difference equation:xn+1=Axn2+Exn-1+F/axn2+exn-1+f,  n=0,1,2,…, where the parametersA,E,F,a,e, andfare nonnegative numbers with conditionA+E+F>0,a+e+f>0, and the initial conditionsx-1,x0are arbitrary nonnegative numbers such thataxn2+exn-1+f>0,  n=0,1,2,….

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

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