Hardy-Type Tauberian Conditions on Time Scales

Abstract

Hardy’s well-known Tauberian theorem for number sequences states that if a sequence $$x=\left( x_{k}\right) $$ satisfies $$\lim Cx=L$$ and $$\Delta x_{k}:=x_{k+1}-x_{k}=O\left( 1/k\right) $$ , then $$\lim x=L$$ , where Cx denotes the Cesaro mean (arithmetic mean) of x. In this study, we extend this result to the theory of time scales. We also discuss the theory on some special time scales.