Abstract
AbstractAfter the appearance of Nieto and Rodríguez-López’s theorem, the branch of fixed point theory devoted to the setting of partially ordered metric spaces have attracted much attention in the last years, especially when coupled, tripled, quadrupled and, in general, multidimensional fixed points are studied. Almost all papers in this direction have been forced to present two results assuming two different hypotheses: the involved mapping should be continuous or the metric framework should be regular. Both conditions seem to be different in nature because one of them refers to the mapping and the other one is assumed on the ambient space. In this paper, we unify such different conditions in a unique one. By introducing the notion of continuity of a mapping from a metric space into itself depending on a function α, which is the case that covers the partially ordered setting, we extend some very recent theorems involving control functions that only must be lower/upper semi-continuous from the right. Finally, we use metric spaces endowed with transitive binary relations rather than partial orders.
Highlights
In recent times, some extensions of the Banach contractive mapping principle have been introduced using a contractivity condition that involves two different functions
Functions like ψ verifying the previous properties are known in the literature as altering distance functions
We present the kind of control functions we will involve in the contractivity condition
Summary
Some extensions of the Banach contractive mapping principle have been introduced using a contractivity condition that involves two different functions. Given a metric space (X, d), a point z ∈ X, a function α : X × X → [ , ∞) and two mappings T, g : X → X, we say that T is (d, g, α)-right-continuous at z if we have that {Txn} converges to Tz for all sequence {xn} ⊆ X such that {gxn} is convergent to gz and verifying that α(gxn, gxn+ ) ≥ for all n ∈ N.
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