scholarly journals Qualitative Research of First Order Linear Difference Equations

2019 ◽  
Vol 8 (6S4) ◽  
pp. 1021-1025
Keyword(s):  
Qualitative Research ◽  
Difference Equation ◽  
Difference Equations ◽  
Iterative Technique ◽  
Linear Difference Equations ◽  
Order Linear ◽  
First Order ◽  
Nonnegative Coefficients

In this article the oscillation of difference equation with deviating argumentand nonnegative coefficients is considered. Sufficient oscillation conditions involving limit inf are given. The iterative technique is used to improve the results

Linear Stochastic Difference Equations

2013 ◽  
Author(s):  
Lars Peter Hansen ◽  
Thomas J. Sargent
Keyword(s):  
Time Series ◽  
Difference Equation ◽  
Difference Equations ◽  
Martingale Difference ◽  
Stochastic Difference Equation ◽  
Stochastic Difference Equations ◽  
Order Linear ◽  
Economic Agents ◽  
First Order ◽  
Competitive Equilibria

This chapter describes the vector first-order linear stochastic difference equation. It is first used to represent information flowing to economic agents, then again to represent competitive equilibria. The vector first-order linear stochastic difference equation is associated with a tidy theory of prediction and a host of procedures for econometric application. Ease of analysis has prompted the adoption of economic specifications that cause competitive equilibria to have representations as vector first-order linear stochastic difference equations. Because it expresses next period's vector of state variables as a linear function of this period's state vector and a vector of random disturbances, a vector first-order vector stochastic difference equation is recursive. Disturbances that form a “martingale difference sequence” are basic building blocks used to construct time series. Martingale difference sequences are easy to forecast, a fact that delivers convenient recursive formulas for optimal predictions of time series.

scholarly journals Special cases of critical linear difference equations

2021 ◽  
pp. 1-17
Author(s):  
Jan Jekl
Keyword(s):  
Difference Equations ◽  
Adjoint Equation ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Linear ◽  
Special Cases ◽  
Even Order ◽  
Nonnegative Coefficients ◽  
Sturm Liouville ◽  
Second Order Equations

In this paper, we investigate even-order linear difference equations and their criticality. However, we restrict our attention only to several special cases of the general Sturm–Liouville equation. We wish to investigate on such cases a possible converse of a known theorem. This theorem holds for second-order equations as an equivalence; however, only one implication is known for even-order equations. First, we show the converse in a sense for one term equations. Later, we show an upper bound on criticality for equations with nonnegative coefficients as well. Finally, we extend the criticality of the second-order linear self-adjoint equation for the class of equations with interlacing indices. In this way, we can obtain concrete examples aiding us with our investigation.

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.

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