ϕ-statistically quasi Cauchy sequences

Let P denote the space whose elements are finite sets of distinct positive integers. Given any element σ of P, we denote by p(σ) the sequence {pn(σ  )} such that pn(σ)=1pn(σ)=1 for n ∈ σ   and pn(σ)=0pn(σ)=0 otherwise. Further Ps={σ∈P:∑n=1∞pn(σ)≤s}, i.e. Ps is the set of those σ whose support has cardinality at most s. Let (ϕn  ) be a non-decreasing sequence of positive integers such that nϕn+1≤(n+1)ϕnnϕn+1≤(n+1)ϕn for all n∈Nn∈N and the class of all sequences (ϕn) is denoted by Φ  . Let E⊆N.E⊆N. The number δϕ(E)=lims→∞1ϕs|{k∈σ,σ∈Ps:k∈E}| is said to be the ϕ-density of E. A sequence (xn  ) of points in RR is ϕ-statistically convergent (or Sϕ  -convergent) to a real number ℓ for every ε > 0 if the set {n∈N:|xn−ℓ|≥ɛ}{n∈N:|xn−ℓ|≥ɛ} has ϕ-density zero. We introduce ϕ-statistically ward continuity of a real function. A real function is ϕ-statistically ward continuous if it preserves ϕ-statistically quasi Cauchy sequences where a sequence (xn) is called to be ϕ-statistically quasi Cauchy (or Sϕ  -quasi Cauchy) when (Δxn)=(xn+1−xn)(Δxn)=(xn+1−xn) is ϕ-statistically convergent to 0. i.e. a sequence (xn  ) of points in RR is called ϕ-statistically quasi Cauchy (or Sϕ  -quasi Cauchy) for every ε > 0 if {n∈N:|xn+1−xn|≥ɛ}{n∈N:|xn+1−xn|≥ɛ} has ϕ-density zero. Also we introduce the concept of ϕ-statistically ward compactness and obtain results related to ϕ-statistically ward continuity, ϕ-statistically ward compactness, statistically ward continuity, ward continuity, ward compactness, ordinary compactness, uniform continuity, ordinary continuity, δ-ward continuity, and slowly oscillating continuity.