Abstract
In this paper we present some (unidimensional and) multidimensional fixed point results under -contractivity conditions in the framework of -metric spaces, which are spaces that result from G-metric spaces (in the sense of Mustafa and Sims) omitting one of their axioms. We prove that these spaces let us consider easily the product of -metrics. Our result clarifies and improves some recent results on this topic because, among other different reasons, we will not need a partial order on the underlying space. Furthermore, the way in which several contractivity conditions are proposed imply that our theorems cannot be reduced to metric spaces. MSC: 46T99, 47H10, 47H09, 54H25.
Highlights
In the sixties, inspired by the mapping that associated the area of a triangle to its three vertices, Gähler [, ] introduced the concept of -metric spaces
The main aim of the present paper is to prove new unidimensional and multidimensional fixed point results in the framework of the G-metric spaces provided with a partial preorder
6 Multidimensional Υ-fixed point results in partially preordered G∗-metric spaces we extend Theorem to an arbitrary number of variables
Summary
In the sixties, inspired by the mapping that associated the area of a triangle to its three vertices, Gähler [ , ] introduced the concept of -metric spaces. Lemma Let {am}m∈N be a sequence of non-negative real numbers which has not any subsequence converging to zero. Lemma If (X, G) is a G∗-metric space and {xm} ⊆ X is a sequence, the following conditions are equivalent. Proposition The limit of a G-convergent sequence in a G∗-metric space is unique. Lemma Let {(Xi, Gi)}ni= be a family of G∗-metric spaces, consider the product space X = X × X × · · · × Xn and define Gnmax and Gnsum on X by n. Consider the sequence of non-negative real numbers {G(xn(k)– , xm(k)– , xn(k))} If this sequence has a partial subsequence converging to zero, we can take the limit in ( ) using this partial subsequence and we would deduce < ψ(ε ) ≤ , which is impossible.
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