ORDER CONVERGENCE OF ORDER BOUNDED SEQUENCES IN RIESZ SPACES

Abstract

We consider sequences  in a Dedekind $\sigma$-complete Riese space, satisfying a

recursive relation

\[ x_{n+p}\ge \sum_{j=1}^p \alpha_{n,j} x_{n+p-j} \qquad \text{for } n=1, 2, \cdots\]

where $p$ is a given natural number and $\alpha_{n,j}$ are nonnegative real numbers satisfying $\sum_{j=1}^p\alpha_{n,j}=1$. We obtain a sufficient condition on coefficients $\alpha_{n,j}$ for which order boundedness of such a sequence $(x_n)_{n=1}^\infty$ implies its order convergence. In a particular case when $\alpha_{n,j}=\alpha_{j}$ for all $n$ and $j$, it is shown that every order bounded sequence satisfying the above recursive relation order converges if and only if natural numbers $j \le p$ for which $\alpha_{j}>0$, are relative prime.