Density by moduli and Wijsman lacunary statistical convergence of sequences of sets

Abstract

Abstract The main object of this paper is to introduce and study a new concept of f-Wijsman lacunary statistical convergence of sequences of sets, where f is an unbounded modulus. The definition of Wijsman lacunary strong convergence of sequences of sets is extended to a definition of Wijsman lacunary strong convergence with respect to a modulus for sequences of sets and it is shown that, under certain conditions on a modulus f, the concepts of Wijsman lacunary strong convergence with respect to a modulus f and f-Wijsman lacunary statistical convergence are equivalent on bounded sequences. We further characterize those θ for which WS θ f = WS f $\mathit{WS}_{\theta}^{f} = \mathit{WS}^{f}$ , where WS θ f $\mathit{WS}_{\theta}^{f}$ and WS f $\mathit{WS}^{f}$ denote the sets of all f-Wijsman lacunary statistically convergent sequences and f-Wijsman statistically convergent sequences, respectively.

Keywords

• modulus function
• density
• lacunary sequence
• statistical convergence
• lacunary strong convergence
• Wijsman convergence