| Summary: | In this paper, the existence and uniqueness of globally stable fixed points of asymptotically contractive mappings in complete <i>b</i>-metric spaces were studied. Also, we investigated the existence of fixed points under the setting of a continuous mapping. Furthermore, we introduce a contraction mapping that generalizes that of Banach, Kanan, and Chatterjea. Using our new introduced contraction mapping, we establish some results on the existence and uniqueness of fixed points. In obtaining some of our results, we assume that the space is associated with a partial order, and the <i>b</i>-metric function has the regularity property. Our results improve, and generalize some current results in the literature.
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