Abstract
Recently, Samet et al. (2012) introduced the notion ofα-ψ-contractive mappings and established some fixed point results in the setting of complete metric spaces. In this paper, we introduce the notion of weakα-ψ-contractive mappings and give fixed point results for this class of mappings in the setting of partial metric spaces. Also, we deduce fixed point results in ordered partial metric spaces. Our results extend and generalize the results of Samet et al.
Highlights
The notion of partial metric is one of the most useful and interesting generalizations of the classical concept of metric
We introduce the notion of weak α-ψ-contractive mappings and give fixed point results for this class of mappings in the setting of partial metric spaces
We deduce fixed point results in ordered partial metric spaces
Summary
We recall some definitions and some properties of partial metric spaces that can be found in [1, 5, 10, 16, 17]. A partial metric on a nonempty set X is a function p : X × X → [0, +∞) such that, for all x, y, z ∈ X, we have (p1) x = y ⇔ p(x, x) = p(x, y) = p(y, y),. A basic example of a partial metric space is the pair ([0, +∞), p), where p(x, y) = max{x, y} for all x, y ∈ [0, +∞). Other examples of partial metric spaces which are interesting from a computational point of view can be found in [1]. A partial metric space (X, p) is said to be complete if every Cauchy sequence {xn} in X converges, with respect to τp, to a point x ∈ X such thatp(x, x) = limn,m → +∞p(xn, xm).
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