Abstract
In the paper, we prove a new fixed point theorem of nonlinear quasi-contractions in non-normal cone metric spaces, which partially improve the recent results of Arandelović and Kečkić’s and of Li and Jiang since some of the essential conditions therein are removed. A suitable example is presented to show the usability of our theorem. It is worth mentioning that the results in this paper could not be derived from the corresponding results in the setting of metric spaces by using a scalarization function or a Minkowski functional. MSC:06A07, 47H10.
Highlights
In, Huang and Zhang [ ] introduced the concept of cone metric spaces, as a generalization of metric spaces, and gave the version of the Banach contraction principle and other basic theorems in the setting of cone metric spaces
There are some references concerned with the problem of whether cone metric spaces are equivalent to metric spaces in terms of the existence of the fixed points of the mappings in cone metric spaces; see [ – ]
It has been shown that each cone metric space (X, d) is equivalent to a usual metric space (X, de), where the real-valued metric function de is defined by a nonlinear scalarization function [ ] or by a Minkowski functional [ ]
Summary
In , Huang and Zhang [ ] introduced the concept of cone metric spaces, as a generalization of metric spaces, and gave the version of the Banach contraction principle and other basic theorems in the setting of cone metric spaces. By using Lemma , we prove a new fixed point theorems of nonlinear quasi-contractions in non-normal cone metric spaces, which improved the relevant results of [ , ] since the conditions (I – A)(intP) ⊂ intP and (H) are removed. Lemma Let P be a solid cone of a normed vector space (E, · ) and A : P → P a nondecreasing mapping. The cone metric space (X, d) is complete [ , ], if each Cauchy sequence {xn} of X converges to a point x ∈ X. where u ∈ {d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx)}. Theorem Let (X, d) be a complete cone metric space over a solid cone P of a normed vector space (E, · ) and T : X → X a quasi-contraction.
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