2010/05/26 by Hüseyi̇n Çakallı, Huseyin Cakalli, Cakalli, Huseyin · 2 citations
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Rings, Modules, and Algebras #math.CA #math.FA #math.GN #msc:46T20
paper · pdf · doi:10.48550/arxiv.1005.4940
I withdraw my paper due to the acceptance in the journal "Mathematical and Computer Modelling"
arxiv created 2010/09/23 · arxiv updated 2010/09/24
Recently, a concept of forward continuity and a concept of forward compactness are introduced in the senses that a function f is forward continuous if limn→∞ Δf(xn)=0 whenever limn→∞ Δxn=0, and a subset E of R is forward compact if any sequence x=(xn) of points in E has a subsequence z=(zk)=(x_nk) of the sequence x such that limk→ ∞ Δzk=0 where Δzk=zk+1-zk. These concepts suggest us to introduce a concept of second forward continuity in the sense that a function f is second forward continuous if limn→∞Δ2f(xn)=0 whenever limn→∞Δ2xn=0, and a subset E of R is second forward compact if whenever x=(xn) is a sequence of points in E there is a subsequence z=(zk)=(x_nk) of x with limk→ ∞ Δ2zk=0 where Δ2 yn=yn+2-2yn+1+yn. We investigate the impact of changing the definition of convergence of sequences on the structure of forward continuity in the sense of second forward continuity, and compactness of sets in the sense of second forward compactness, and prove related theorems.