Rough ideal convergence in a partial metric space
Sukila Khatun, Amar Kumar Banerjee, Rahul Mondal
TL;DR
The paper extends rough convergence to the setting of partial metric spaces using an ideal $\mathcal{I}$, by defining rough $\mathcal{I}$-convergence and the associated limit sets $\mathcal{I}-LIM^{r}x_n$. It establishes fundamental properties, including the nonuniqueness of rough $\mathcal{I}$-limits for $r>0$, a diameter bound $diam(\mathcal{I}-LIM^{r}x_n) \le 2r+2a$ when $p(x,x)=a$, and the closedness and nonemptiness criteria tied to $\mathcal{I}$-boundedness. The work also links rough $\mathcal{I}$-limit points to $\mathcal{I}$-cluster points and examines stability under subsequences and perturbations, thereby providing a comprehensive framework for approximate convergence in partial metric spaces under ideal filtering.
Abstract
In this paper, using the concept of ideal, we study the idea of rough ideal convergence of sequences which is an extension of the notion of rough convergence of sequences in a partial metric space. We define the set of rough $\mathcal{I}$-limit points and the set of rough $\mathcal{I}$-cluster points and then we prove some relevant results associated with these sets.