A comparative study in cone metric spaces and cone normed spaces. Some fixed point theorems in cone rectangular metric spaces. In this paper, we prove a unique common fixed point theorem for four selfmappings in cone metric spaces by using the continuity and commuting mappings. When x is compact and hausdorff essentially, when x can be embedded in euclidean space, then the cone can be visualized as the collection of lines joining every point of x to a single point. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. In any dcone metric space, limits of dconvergent sequences are unique. Chapter 5 functions on metric spaces and continuity when we studied realvalued functions of a real variable in calculus, the techniques and theory built on properties of continuity, differentiability, and integrability. As an important result for tvscone metric spaces, it is proved that each tvscone metric space is metrizable, 14, e. So, proper generalizations when passing from normvalued cone metric spaces of 7 to tvsvalued cone metric spaces can be obtained only in the case of nonnormal cones. Also, we consider hardyrogers type mappings on partial cone metric spaces and prove some fixed point results without. The concept of metric space can be expanded, including bmetric space, cone metric space, cone bmetric space and typemetric space. X, iterative sequence f n x converges to the fixed point.
Finally it is proven that every separable fuzzy cone metric space is second countable and a subspace of a separable fuzzy cone metric space is separable. In this thesis we made a comparison between cone metric spaces and cone normed spaces and ordinary metric spaces and normed spaces as a way to find an answer for our main contribution. A similar notion was also considered by rzepecki in. Then d is called a cone metric on x, and x, d is called a cone metric space. Cone metric spaces and fixed point theorems of contractive. Metric spaces a metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. Some notes on the paper the equivalence of cone metric. We do not develop their theory in detail, and we leave the veri. By using of needing same definition and as same results in it, we consider fixed point theorem in cone metric space to complete and extending the theorem 1 in it.
It is obvious that cone metric spaces generalize metric space. A comparative study in cone metric spaces and cone normed spaces by duaa abdullah mohammad alafghani supervisor dr. Also every cone metric space is partial cone metric space with zero. By using of variation iteration method and an effective modification of hes variation iteration method discusses some integral and differential equations, we give out some new conclusion and more new examples. Jungck, common fixed point results for non commuting mappings without continuity in cone metric spaces, j. Some fixed point theorems of integral type contraction in. Some fixed point theorems for contractive maps in n cone metric spaces some fixed point theorems for contractive maps in n cone metric spaces. Dcone metric space and fixed point theorem 211 theorem 2. Cone metric spaces, cone rectangular metric spaces and. A cone metric space is an ordered pair x, d, where x is any set and d. In this paper we establish some topological properties of the cone b metric spaces and then improve some recent results about kkm mappings in the setting of a cone b metric space. Then, it is shown that each solid cone in a topological vector space can be essentially replaced by a solid and normal cone in a normed space even with normal constant equal to 1. Pdf fixed point theorem in cone metric space semantic scholar.
This paper investigates superspaces and of a tvscone metric space, where and are the space consisting of nonempty subsets of and the space consisting of nonempty compact subsets of, respectively. Throughout, we shall say that t he cone metric space x is isometric with the cone. Common fixed point theorem for thardyrogers contraction mapping in a cone metric space r. Topological vector spacevalued cone metric spaces and. Remarks on cone metric spaces and fixed point theorems of. A survey article pdf available in middle east journal of scientific research 1112 february 2011 with 2,068 reads how we measure reads. A comparative study in cone metric spaces and cone. In many occasions the answer was proved, not to be affirmative. Let be the set of lebesgue measurable functions on such that. May 21, 2012 the pair x, d is then called a tvscone metric space.
After carefully defining convergence and completeness in cone metric spaces, the authors proved some fixed point theorems of contractive mappings. The concept of a cone metric space is more general than that of a metric space, because each metric space is a cone metric. However, we notice that some results related to nontopological properties, for example, metric properties including hemimetric. Paper open access sequences and its properties on type. Cone metric space and some fixed point results for pair. Note that iff if then so thus on the other hand, let. At the end of this paper, we construct a typemetric function on the cone metric space. If for any sequence x n in x, there is a subsequence x n i of x n such that x n i is convergent in x. Some unique fixed point theorem in partial cone metric spaces 2 a partial cone metric space is a pair x,p. Many studies appears during 2008 and 2009 about xed point in cone metrics and coupled xed point theorems. This does happen in the covariant method, but in the light cone method the space direction x1 is split o.
In this paper, which is split into two parts, our aim is to develop the theory of cone metric spaces called modular cone metric spaces. If every cauchy sequence is convergent in x, then x is called a complete cone metric space. The concept of metric space can be expanded, including b metric space, cone metric space, cone b metric space and type metric space. Then x is called a sequentially compact cone metric space. Some topological properties of fuzzy cone metric spaces. Fixed point theorems for multivalued mappings in g cone metric spaces fixed point theorems for multivalued mappings in g cone metric spaces. Let, xd be a cone metric space and p be a normal cone with normal constant k. Hakawati abstract cone metric spaces are, not yet proven to be generalization of metric spaces. Chapter 5 functions on metric spaces and continuity when we studied realvalued functions of a real variable in calculus, the techniques and theory built on properties of. The pair x, d is then called a tvscone metric space. Then is a cone metric space with coefficient, but it is not a cone metric space. Nov 25, 20 in this section we shall prove some fixed point theorems of generalized lipschitz mappings in the setting of cone metric spaces with banach algebras. Clearly, a cone metric space, in the sense of huang and zhang, is a special case of a tvscone metric space. It is proved in 1 that every cone metric space is metraizable.
There is a natural definition of cone in the context of pointed metric spaces. In this paper, we give some new results of common fixed point theorems and coincidence point case for some iterative method. Moreover, cone metric spaces generalize cone metric spaces, metric spaces, and metric spaces. Let x,d be a cone metric space, if every cauchy sequence is convergent in x, then xis called a complete cone metric space. Fixed point theorems in this section we shall prove some fixed point theorems of contractive mappings. Indeed, the authors there replace the real numbers. E with 0 c there is a natural number n such that dxn,x c for all n. Let x,d be a complete cone metric space, p be a normal cone with normal constant k.
The purpose of this paper is to establish some relationships between the lower topology and the lower tvscone hemimetric topology resp. At the end of this paper, we construct a type metric function on the cone metric space. In any d cone metric space, limits of dconvergent sequences are unique. Introduction and preliminaries in recent years, several authors see 1,3,4,5,6 have studied the strong convergence to a. F cone metric spaces over banach algebra springerlink. If every cauchy sequence is convergent in, then x is called a complete cone rectangular metric space. Moreover the underlying banach space and the associated cone subset are not necessary. The concept of a cone metric space is more general than that of a metric space.
An example is presented which shows that the generalizations of this paper are proper. Some equivalences between cone metric spaces and metric spaces. Paper open access sequences and its properties on typemetric. Partial cone metric space, self mappings, normal cone, regular cone, fixed point. Fcone metric spaces over banach algebra pdf paperity. Let x, d be a complete cone metric space over a normal. Fixed point theorems for multivalued maps in cone metric. Fixed point theorems for multivalued maps in cone metric spaces. In this paper a completion theorem for cone metric spaces and a completion theorem for cone normed spaces are proved. For a metric space let us consider the space of all nonempty closed bounded subset of with the following metric. Cone metric spaces, cone rectangular metric spaces and common. X x satisfies the contractive condition dtx,tylessorequalslantkdx,y, for all x,y. Some unique fixed point theorem in partial cone metric spaces.
One of the important questions that will appear is. In this section we shall prove some fixed point theorems of generalized lipschitz mappings in the setting of cone metric spaces with banach algebras. I thought a cone would be an easy place to start with calculating a metric, shape operator, what have you. We shall give some results about characterization of best approximations in the cone metric spaces. D be a d cone metric space and suppose that the sequence fx. As an important result for tvs cone metric spaces, it is proved that each tvs cone metric space is metrizable, 14, e. 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. Some equivalences between cone metric spaces and metric. The cone is used in algebraic topology precisely because it embeds a space as a subspace of a contractible space. Let x, d be a complete cone metric space over a normal solid cone. A cone metric space is a partial cone metric space. Then, it is shown that each solid cone in a topologicalvectorspace can be essentially replaced by a solid and normal cone in a normed space even with normal constant equal to 1. Common fixed point theorems for cyclic contractive.
If is a compact subset of, then, for any, there is a finite subset of such that. Also every cone metric space is partial cone metric space with zero self distance, but there are partial cone metric spaces which are not cone metric space. Then f has a unique fixed point in x and for any x. D cone metric space and fixed point theorem 211 theorem 2. For more on metric fixed point theory, the reader may. Cone metric spaces are, not yet proven to be generalization of metric spaces. He and the russian school which he founded have made an extensive study of the local properties of such spaces. Notes on metric spaces these notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Theorem a see theorem 1 3 assume that x, d be a complete cone metric space.
Topological questions were answered in cone metric spaces, where it was proved that cone metric spaces are. Such generalized spaces called cone metric spaces, were introduced by rzepecki 10. Chapter 5 functions on metric spaces and continuity. Introduction one of the topics dicussed in functional analysis is metric spaces. Fixed point theorems in this section we shall prove some fixed. Huang and zhang discuss the case in which y is a real banach space and call a vectorvalued function d. In the present paper we develop further the theory of topological vector space valued cone metric spaces with nonnormal cones. Pdf fixed point theorem in cone metric space semantic. Cone metric spaces and fixed point theorems of contractive mappings.
Let mappings satisfying following lipchitz conditions for any. The completion spaces are defined by means of an equivalence relation. Some approximation in cone metric space and variational. E with 0 c there is a natural number n such that dxn.