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 mathit{WS}_{theta}^{f} = mathit{WS}^{f}, where mathit{WS}_{theta}^{f} and mathit{WS}^{f} denote the sets of all f-Wijsman lacunary statistically convergent sequences and f-Wijsman statistically convergent sequences, respectively.
Highlights
Zygmund [ ] was the person behind the introduction of the idea of statistical convergence
Aizpuru et al [ ] have recently introduced a new concept of density by moduli and obtained a new concept of non-matrix convergence, namely, f -statistical convergence which is, a generalization of the concept of statistical convergence and intermediate between the ordinary convergence and the statistical convergence. This idea of replacing natural density with density by moduli has motivated us to look for some new generalizations of statistical convergence and we have introduced and studied the concepts of f -statistical convergence of order α [ ] and f -lacunary statistical convergence [ ]
We study a relationship between Wijsman lacunary strong convergence with respect to a modulus and Wijsman lacunary statistical convergence and characterize those θ for which
Summary
Zygmund [ ] was the person behind the introduction of the idea of statistical convergence. For any an unbounded modulus f , a sequence (Ek) of non-empty closed subsets of a metric space (M, ρ) is said to be f -Wijsman statistically convergent to a nonempty closed subset E of M, or WSf -convergent to E, if, for each x ∈ M, (d(x, Ek)) is f statistically convergent to d(x, E); i.e., for each x ∈ M and for each ε > , lim f n→∞ f (n). In this case, we write WSf – lim Ek = E or Ek → E(WSf ).
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