Abstract
We show that the main result in the work by Mutlu et al. is not true. We explain point by point some of its main mistakes and we propose an alternative version to smooth away the defects of it.
Highlights
Following Matthews [1], a partial metric on a nonempty set X is a mapping p : X×X → [0, ∞) verifying, for all x, y, z ∈ X,(P1) p (x, x) ≤ p (x, y) ;(P2) p (x, x) = p (x, y) = p (y, y) ⇒ x = y; (1)(P3) p (x, y) = p (y, x) ;(P4) p (x, z) + p (y, y) ≤ p (x, y) + p (y, z) .In this case, (X, p) is called a partial metric space
It is not clear how we can show a counterexample of Theorem 2 because Definition 1 is not well posed
Many coupled/tripled/quadrupled/multidimensional fixed point theorems in various abstract metric spaces have come to be simple consequences of their corresponding unidimensional results. Following this line of research, we present here a correct version of Theorem 2 for three reasons mainly: (1) for the sake of completeness; (2) to describe how coupled results in partial metric spaces can be deduced from the unidimensional case; (3) to show some possible hypotheses to ensure the existence of common coupled fixed points when we work with two different mappings
Summary
Following Matthews [1], a partial metric on a nonempty set X is a mapping p : X×X → [0, ∞) verifying, for all x, y, z ∈ X,. (P4) p (x, z) + p (y, y) ≤ p (x, y) + p (y, z) In this case, (X, p) is called a partial metric space. The authors of [2] used the notation d for a partial metric space, we prefer using p in order to avoid confusion with the metric case. F and G mappings have the following properties: if n is even, F (xn, yn) ≽ G (xn−1, yn−1) and. Assume that F, G : X × X → X are satisfied by Definition 1 and are continuous mappings possessing the mixed monotone property on X.
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