Lacunary ℐ -Invariant Convergence of Sequence of Sets in Intuitionistic Fuzzy Metric Spaces

Abstract

The concepts of invariant convergence, invariant statistical convergence, lacunary invariant convergence, and lacunary invariant statistical convergence for set sequences were introduced by Pancaroğlu and Nuray (2013). We know that ideal convergence is more general than statistical convergence for sequences. This has motivated us to study the lacunary $\mathrm{ℐ}$ -invariant convergence of sequence of sets in intuitionistic fuzzy metric spaces (briefly, IFMS). In this study, we examine the notions of lacunary $\mathrm{ℐ}$ -invariant convergence $\left(W{\mathrm{ℐ}}_{\sigma \theta }^{\left(\eta ,\nu \right)}\right)$ (Wijsman sense), lacunary ${\mathrm{ℐ}}^{\ast }$ -invariant convergence $\left(W{\mathrm{ℐ}}_{\sigma \theta }^{\ast \left(\eta ,\nu \right)}\right)$ (Wijsman sense), and $q$ -strongly lacunary invariant convergence $\left({\left[W{N}_{\sigma \theta }^{\left(\eta ,\nu \right)}\right]}_q\right)$ (Wijsman sense) of sequences of sets in IFMS. Also, we give the relationships among Wijsman lacunary invariant convergence, ${\left[W{N}_{\sigma \theta }^{\left(\eta ,\nu \right)}\right]}_q$ , $W{\mathrm{ℐ}}_{\sigma \theta }^{\left(\eta ,\nu \right)}$ , and $W{\mathrm{ℐ}}_{\sigma \theta }^{\ast \left(\eta ,\nu \right)}$ in IFMS. Furthermore, we define the concepts of $W{\mathrm{ℐ}}_{\sigma \theta }^{\left(\eta ,\nu \right)}$ -Cauchy sequence and $W{\mathrm{ℐ}}_{\sigma \theta }^{\ast \left(\eta ,\nu \right)}$ -Cauchy sequence of sets in IFMS. Furthermore, we obtain some features of the new type of convergences in IFMS.