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.
Highlights
Several authors have studied invariant convergent sequences.Let σ be a mapping of the positive integers into themselves
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
Introducing lacunary I2∗-invariant convergence concept and lacunary I2-invariant Cauchy concepts, we give the relationships among these concepts and relationships with lacunary I2-invariant convergence concept
Summary
Several authors have studied invariant convergent sequences (see, [8,9,10, 13, 15,16,17, 19]).Let σ be a mapping of the positive integers into themselves. The concept of lacunary I2-invariant convergence for double sequences is given. L, A sequence x = (xk) is said to be strongly lacunary q-invariant convergent (0 < q < ∞) to L if lim r→∞
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