In 2007 H. Long-Guang and Z. Xian, [H. Long-Guang and Z. Xian, Cone Metric Spaces and Fixed Point Theorems of Contractive Mapping, J. Math. Anal. Appl., 322(2007), 1468-1476], generalized the concept of a metric space, by introducing cone metric spaces, and obtained some fixed point theorems for mappings satisfying certain contractive conditions. The main question was "Are cone metric spaces a real generalization of metric spaces?" Throughout this paper we answer the question in the negative, proving that every cone metric space is metrizable and the equivalent metric satisfies the same contractive conditions as the cone metric. So most of the fixed point theorems which have been proved are straightforward results from the metric case.
In this paper we show that by renorming an ordered Banach space, every cone P can be converted to a normal cone with constant K = 1 and consequently due to this approach every cone metric space is really a metric one and every theorem in metric space valid for cone metric space automatically.
This note tries to give an answer to the following question: Is there a sufficiently rich class of metric vector spaces such that sufficiently large spaces of continuous linear maps between them are metrizable?
In this manuscript, we claim that the newly introduced $\mathcal{F}$-metric space \cite[\, M.~Jleli and B.~Samet, On a new generalization of metric spaces, J. Fixed Point Theory Appl, 20(3) 2018]{JS1} is metrizable. Also, we deduce that the notions of convergence, Cauchy sequence, completeness due to Jleli and Samet for $\mathcal{F}$-metric spaces are equivalent with that of usual metric spaces. Moreover, we assert that the Banach contraction principle in the context of $\mathcal{F}$-metric spaces is a direct consequence of its standard metric counterpart.
In this manuscript, we claim that the newly introduced $\mathcal{F}$-metric spaces are Hausdorff and also first countable. Moreover, we assert that every separable $\mathcal{F}$-metric space is second countable. Additionally, we acquire some interesting fixed point results concerning altering distance functions for contractive-type mappings and Kannan-type contractive mappings in this exciting context. However, most of the findings are well-furnished by several non-trivial numerical examples. Finally, we raise an open problem regarding the metrizability of such kind of spaces.
In this paper, we prove that the topology induced by algebraic cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e. any algebraic cone metric space is metrizable. Furthermore, the notion of algebraic cone normed space is introduced and also normability of the topology of this space is proved.
The main aim of this paper is to study of fixed point theory in partial cone metric spaces. Infact, some common fixed point theorems for two mappings in partial cone metric spaces are obtained.
Here we have introduced the idea of rough Cauchyness of sequences in a cone metric space. Also here we have discussed several basic properties of rough Cauchy sequences in a cone metric space using the idea of Phu.
In this survey, at first we review to many examples which have been made on cone metric spaces to verify some properties of cones on real Banach spaces and cone metrics and second, in continue like as examples that sandwich theorem doesn't hold and we shall present an other example that comparison test doesn't hold with an example for normal cones.
There have been a number of generalizations of fixed point results to the so called TVS-cone metric spaces, based on a distance function that takes values in some cone with nonempty interior (solid cone) in some topological vector space. In this paper we prove that the TVS-cone metric space can be equipped with a family of mutually equivalent (usual) metrics such that the convergence (resp. property of being Cauchy sequence, contractivity condition) in TVS sense is equivalent to convergence (resp. property of being Cauchy sequence, contractivity condition) in all of these metrics. As a consequ…
In the present work we shall show that nonlinear contractive conditions on TVS-cone metric spaces can be reduced to nonlinear contractive conditions on usual metric spaces. Our results extend the results of Du (2010) [W.S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259-2261] and results of Abdeljawad and Rezapour [T. Abdeljawad, Sh. Rezapour, Some fixed point results in TVS-cone metric spaces, Fixed Point Theory (to appear 2012)].
In this paper we shall show that the metrics are equivalent which obtained by Feng and Mao in [[1], Y. Feng and W. Mao, Equivalence of Cone Metric spaces and Metric Spaces, Fixed Point Theory, 11(2)(2010), 259-264.] and Du in [[2], Wei-Shih Du, A Note on Cone Metric Fixed Point Theory and its equivalence, Nonlinear Analysis, 72 2259-2261 (2010)]. Also in the sequel we calculate some examples by one of the metrics.
In this paper, we study the fixed point theory for multi-valued mappings on partial cone metric spaces. We prove an analogous to the well-known Kannan$'s$ fixed point theorem and Chatterjea$'s$ fixed point theorem for multi-valued mappings on partial cone metric spaces.
The aim of this paper is to present some fixed point theorems for generalized contractions by altering distance functions in a complete cone metric spaces endowed with a partial order. We also generalize fixed point theorems of J. Harjani, K. Sadarangani [J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Analysis 72 (2010) 1188-1197] from metric spaces to cone metric spaces.
In this paper we have introduced arithmetic ff-continuity and arithmetic fb-continuity utilizing the concept of forward and backward arithmetic convergence in quasi cone metric spaces. These concepts are used to prove some fascinating results. The notions of forward and backward arithmetic compactness in quasi cone metric spaces are also established which are further utilized to prove some related results.
In this paper the notion of modular cone metric space is introduced and some properties of such spaces are investigated. Also we define convex modular cone metric which takes values in CR(Y) where Y is a compact Hausdorff space. Then a fixed point theorem is proved for contractions in these spaces. Furthermore, we make a remark on paper [9] and it will be proved that their fixed point result in modular metric spaces is not true.
In this paper we study the existence and uniqueness of fixed points of a class of mappings defined on complete, (sequentially compact) cone metric spaces, without continuity conditions and depending on another function.
The purpose of this paper is to obtain sufficient conditions for the existence of a unique fixed point of T-Zamfirescu and T-weak contraction mappings in the framework of complete cone metric spaces.
The purpose of this paper is to obtain sufficient conditions for the existence of a unique fixed point of T-Kannan type mappings on complete cone metric spaces depended on another function.
In this paper, we study the existence of fixed points for mappings defined on complete, (sequentially compact) cone metric spaces, satisfying a general contractive inequality depending of two additional mappings.
Let Y be a locally convex Hausdorff space, K \subset E a cone and \leq_K the partial order defined by K. Let (X, p) be a TV S- cone metric space, ϕ : K \rightarrow K a vectorial comparison function and f : X \rightarrow X such that p(f(x), f(y)) \leq_K ϕ(p(x, y)), for all x, y \in X. We shall show that there exists a scalar comparison function ψ : R+ \rightarrow R+ and a metric d_p(in usual sense) on X such that d_p(f(x), f(y)) \leq ψ(d_p(x, y)), for all x, y \in X. Our results extend the results of Du (2010) [Wei-Shih Du, A note on cone metric fixed point theory and its equivalence, Nonlinear…
The purpose of this new survey paper is, among other things, to collect in one place most of the articles on cone (abstract, K-metric) spaces, published after 2007. This list can be useful to young researchers trying to work in this part of functional and nonlinear analysis. On the other hand, the existing review papers on cone metric spaces are updated. The main contribution is the observation that it is usually redundant to treat the case when the underlying cone is solid and non-normal. Namely, using simple properties of cones and Minkowski functionals, it is shown that the problems can be …
This paper is devoted to prove the S. L. Singh's common fixed point Theorem for commuting mappings in cone metric spaces. In this framework, we introduce the notions of Generalized Kannan Con- traction, Generalized Zamfirescu Contraction and Generalized Weak Contraction for a pair of mappings, proving afterward their respective fixed point results.
We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr{H}$ generated by a sequence $\mathcal{V} = \{v_n\}_{n=0}^\infty$. The first main result of this paper provides a sufficient condition under which we can identify the closed convex cone generated by $\mathcal{V}$ with the following set: \[ \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in $\mathscr{H}$}\bigg\}. \] Then, by adapting classical results on general convex cones, we give a useful description of the metr…
We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces. We define two classes of monotone mappings between an ordered linear space and a metric space (resp. ordered linear space): $K$-monotone dominated and cone-to-cone monotone mappings. First we show some relationships between these classes. Then, we study continuity and differentiability (also in the metric and $w^*$ senses) of mappings in these classes.
If $K$ and $L$ are mutually dual closed convex cones in a Hilbert space with the metric projections onto them denoted by $P_K$ and $P_L$ respectively, then the following two assertions are equivalent: (i) $P_K$ is isotone with respect to the order induced by $K$ (i. e. $v-u\in K$ implies $P_Kv-P_Ku\in K$); (ii) $P_L$ is subadditive with respect to the order induced by $L$ (i. e. $P_Lu+P_Lv-P_L(u+v)\in L$ for any $u, v \in \R^n$). This extends the similar result of A. B. Németh and the author for Euclidean spaces. The extension is essential because the proof of the result for Euclidean spaces i…
In this paper, we introduce double controlled cone metric spaces via two control functions. An example of a double controlled cone metric space by two incomparable functions, which is not a controlled metric space, is given. We also provide some fixed point results involving Banach type and Kannan type contractions in the setting of double controlled cone metric spaces.
In this paper the concept of a partial cone metric space is investigated, some continuity type theorems, and fixed point theorems of contractive mappings in this generalized setting are proved as well as some theorems related to topological properties.
It is well known that fixed point problems of contractive-type mappings defined on cone metric spaces over Banach algebras are not equivalent to those in usual metric spaces (see [3] and [10]). In this framework, the novelty of the present paper represents the development of some fixed point results regarding sequences of contractions in the setting of cone metric spaces over Banach algebras. Furthermore, some examples are given in order to strengthen our new concepts. Also, based on the powerful notion of a cone metric space over a Banach algebra, we present important applications to systems …
