The asymptotically regularity and sequences in partial cone b-metric spaces with application

The aim of this paper is to propose a new space called partial cone b-metric space by using both the notions of cone b-metric spaces and partial metric spaces and by defining asymptotically regular maps and sequences. We also prove some fixed point theorems for such maps and sequences. Our results extend and generalize some interesting results of [11] and [21] in partial cone b-metric space. An example is also given to support the validity of our results.