Lacunary I_2-Invariant Convergence and Some Properties

Abstract

In this paper, the concept of lacunary invariant uniform density of any subset $A$ of the set $\mathbb{N}\times\mathbb{N}$ is defined. Associate with this, the concept of lacunary $\mathcal{I}_2$-invariant convergence for double sequences is given. Also, we examine relationships between this new type convergence concept and the concepts of lacunary invariant convergence and $p$-strongly lacunary invariant convergence of double sequences. Finally, introducing lacunary $\mathcal{I}_2^*$-invariant convergence concept and lacunary $\mathcal{I}_2$-invariant Cauchy concepts, we give the relationships among these concepts and relationships with lacunary $\mathcal{I}_2$-invariant convergence concept.