Some Fixed Point Results in Partial Rectangular Metric-Like Spaces

Abstract

In this paper, we introduce a new concept of partial rectangular metric-like space and prove some results on the existence and uniqueness of a fixed point of a function T : X ⟶ X , defined on a partial rectangular metric-like space X , which fulfills a nonlinear contractive condition using a comparison function and the diameter of the orbits. The obtained results generalize some previously acknowledged results in partial metric spaces, partial rectangular metric spaces, and rectangular metric-like spaces. The examples presented prove the usefulness of the introduced generalizations.

Unique common fixed point results for two maps in incomplete spaces

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In this paper, in the first part, we will improve and extend the main results 1 and 2 of Mishra et al [6] in a b-metric space, and in the second and last part, we will improve and extend Theorem 5 and Theorem 6 of Shah [7] in a b-metric-like space, in both sections, we will use our new notion of occasionally weakly biased maps of type (A). Some illustrative examples are given in order to clarify and support our findings.

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“…In this second section, we will improve and extend Theorem 5 and Theorem 6 of Shah [7] using our new notion of occasionally weakly biased maps of type (A) in a b-metric-like space.…”

“…Definition 4 ([1]) A b-metric-like on a nonempty set X is a function D : X × X → [0, +∞) such that for all r, s, t ∈ X and a constant K ≥ 1 the following three conditions hold true: In his paper [7], Shah proved the below results.…”