Oscillation Properties for a Scalar Linear Mixed Type Difference Equation

Oscillation Properties for a Scalar Linear Mixed Type Difference Equation, 

by Sandra Pinelas,  Academia Militar e CEMAT/IST/UL.

Data: 19/3/2018

Hora: 15h30

Local: Sala  de Seminários,  Edifício VII, FCT-UNL

Abstract:

The aim of this work is to study the oscillation and asymptotic properties for a scalar linear difference equation of mixed type

\begin{eqnarray}

\triangle x\left(n\right)+\sum_{k=-p}^{q}a_{k}\left(n\right)x\left(n+k\right)=0, &  & n>n_{0} \end{eqnarray}

where $\triangle x\left(n\right)=x\left(n+1\right)-x\left(n\right)$ is the difference operator and $\left\{ a_{k}\left(n\right)\right\} $ are sequences of real numbers, for $k=-p,...q$, and $p>0,q\geq0$.

Difference equations with delayed and advanced arguments (also called mixed difference equations or equations with mixed arguments) occur in many problems of economy, biology and physics because difference equations with mixed arguments are much more suitable than delay difference equations for an adequate treatment of dynamic phenomena. The concept of delay is related to a memory of system, the past events are important for the current behavior, and the concept of advance is related to potential future events which can be known at the current time which could be useful for decision making.

It is well known that the solutions of these types of equations cannot be obtained in closed form. In the absence of closed form solutions a rewarding alternative is to resort to the qualitative study of the solutions of these types of difference equations. But it is not quite clear how to formulate an initial value problem for such equations and existence and uniqueness of solutions becomes a complicated issue.

To study the oscillation of solutions of difference equations, we need to assume that there exists a solution of such equation on the half line.