Points In Metric Space Research Articles

Uzaklık Metriklerinin Performansı Üzerine Ampirik Bir Çalışma

Metrics are used to measure the distance, similarity, or dissimilarity between two points in a metric space. Metric learning algorithms perform the finding task of data points that are closest or furthest to a query point in m-dimensional metric space. Some metrics take into account the assumption that the whole dimensions are of equal importance, and vice versa. However, this assumption does not incorporate a number of real-world problems that classification algorithms tackle. In this research, the existing information gain, the information gain ratio, and some well-known conventional metrics have been compared by each other. The 1-Nearest Neighbor algorithm taking these metrics as its meta-parameter has been applied to forty-nine benchmark datasets. Only the accuracy rate criterion has been employed in order to quantify the performance of the metrics. The experimental results show that each metric is successful on datasets corresponding to its own domain. In other words, each metric is favorable on datasets overlapping its own assumption. In addition, there also exists incompleteness in classification tasks for metrics just like there is for learning algorithms.

Convergence results based on graph-Reich contraction in fuzzy metric spaces with application

This article introduces a novel class of Reich-type contractions that meet the graph preservation criteria in the context of complete fuzzy metric spaces. The provided contraction condition is satisfied through various forms of contractive mappings via directed graphs in the literature. Our key result is the natural extension of fuzzy metric spaces to fuzzy metrics enriched with a graph, which adds the understanding of fixed points in metric spaces within the realm of graph structure. The findings are further supported by examples and applications.

Paired contractive mappings and fixed point results

<abstract><p>In this study, we explored a novel type of contraction, known as paired contraction (PC), to establish fixed points in metric spaces. It has been demonstrated that mappings possessing the PC property are continuous. We have also provided proofs for the existence of fixed points for such mappings with the classical Banach fixed point theorem emerging as a corollary. Furthermore, we presented examples of mappings that do not satisfy the standard contraction condition, but do exhibit the PC property.</p></abstract>

On the UC and UC$^*$ properties and the existence of best proximity points in metric spaces

We investigate the connections between UC and UC$^*$ properties for ordered pairs of subsets $(A,B)$ in metric spaces, which are involved in the study of existence and uniqueness of best proximity points. We show that the UC property and the UC$^*$ property lead to one and the same corollaries, when iterated sequences, generated by cyclic maps, are investigated. We introduce some new notions: bounded UC (BUC) property and uniformly convex set about a function $\phi$. We prove that these new notions are generalizations of the UC property and that both of them are sufficient to ensure existence and uniqueness of best proximity points. We show that these two new notions are different from a uniform convexity and even from a strict convexity. If we consider the underlying space to be a Banach space, we find a sufficient condition which ensures that from the UC property follows the uniform convexity of the underlying Banach space. We illustrate the new notions with examples. We present an example of a cyclic contraction $T$ in a space, which is not even strictly convex and the ordered pair $(A,B)$ does not have the UC property, but has the BUC property and thus there is a unique best proximity point of $T$ in $A$.

On Unique and Nonunique Fixed Points in Metric Spaces and Application to Chemical Sciences

We introduce the notions of a generalized Θ -contraction, a generalized Θ E -weak contraction, a Ψ E -weak JS-contraction, an integral-type Θ E -weak contraction, and an integral-type Ψ E -weak JS-contraction to establish the fixed point, fixed ellipse, and fixed elliptic disc theorems. Further, we verify these by illustrative examples with geometric interpretations to demonstrate the authenticity of the postulates. The motivation of this work is the fact that the set of nonunique fixed points may include a geometric figure like a circle, an ellipse, a disc, or an elliptic disc. Towards the end, we provide an application of Θ -contraction to chemical sciences.

COMMON BEST PROXIMITY POINTS THEOREMS FOR H-CONTRACTIVE NON-SELF MAPPINGS

Fixed point theory and contractive mappings are popular tools in solving a variety of problems such as control theory, economic theory, nonlinear analysis and global analysis.There are many works on different types of contractions to find a fixed point in metric spaces. Improving and extending some kind of those, in this paper, we introduce a new version of H-contradiction for four mappings in a metric space (X,d). Then, we prove the existence and uniqueness of a common best proximity point for four non-self mappings. An example is also given to support our main result. The related fixed point theorem are also proved.

Fixed Point Theorems in Complex Valued Quasi b-Metric Spaces for Satisfying Rational Type Contraction

The notion of complex valued metric spaces proved the common fixed point theorem that satisfies rational mapping of contraction. In the contraction mapping theory, several researchers demonstrated many fixed-point theorems, common fixed-point theorems and coupled fixed-point theorems by using complex valued metric spaces. The idea of b-metric spaces proved the fixed point theorem by the principle of contraction mapping. The notion of complex valued b-metric spaces, and this metric space was the generalization of complex valued metric spaces. They explained the fixed point theorem by using the rational contraction. In the metric spaces, we refer to this metric space as a quasi-metric space, the symmetric condition d(x, y) = d(y, x) is ignored. Metric space is a special kind of space that is quasi-metric. The Quasi metric spaces were discussed by many researchers. Banach introduced the theory of contraction mapping and proved the theorem of fixed points in metric spaces. We are now introducing the new notion of complex quasi b-metric spaces involving rational type contraction which proved the unique fixed point theorems with continuous as well as non-continuous functions. Illustrate this with example.

Fixed point results for materials research

In this paper, we establish a new Class category of non expansive mappings in metric space which is wider than the class category of mappings satisfying contractive condition (CC). Special properties and various fixed point results for these mappings in metric space are obtained here. Also we discussed the results and examples of the convergence of some iteration schemes to the fixed point in metric space. Obtained results are useful in material research as well as in image processing.

Discussion on the existence of best proximity points in metric spaces

Discussion on the existence of best proximity points in metric spaces

Remotest Points and Best Proximity Points in Metric Spaces

In this paper, we give sufficient conditions for the existence of remotest points with recurrence relations. Then, we apply these relations to generalize recent best proximity point theorems. The findings of the present research contribute and enrich previous results reviewed in the literature.

Graphical interface as a teaching aid for nonlinear dynamical systems

Chaos theory is an important breakthrough in understanding nonlinear dynamical systems. It is generally accepted that chaotic systems involve transitivity, a high density of periodic points in metric space, and extreme sensitivity to changes in initial conditions. Many systems taught on undergraduate courses satisfy these conditions, such as electronic circuits or computer simulations of nonlinear dynamical systems. However, a good understanding of these concepts is broadly difficult to achieve on such courses, and especially so at the undergraduate level. In view of this, we have developed an interactive standalone graphical user interface, which is platform-independent. The interface didactically simulates six chaotic systems. It allows the user to study a particular system without prior knowledge of computer programming, demonstrating rich dynamical behaviour such as fixed points, periodic cycles and deterministic chaos. At the same time, the interface provides details of the simulation and the parameters used, as well as some of the leading references. We provide the complete code and a detailed installation guide, which should make it easy to use the interface reliably and with little overhead work. We also provide a comparison with other available software for reference, and as a guide to readers who would like to explore this fascinating subject in more depth or with alternative approaches. Lastly, we present the results of a survey of 30 students who used our graphical user interface, revealing that it is an easy-to-use tool in the process of learning about chaotic dynamical systems.

WEAKLY CONTRACTIVE MAPPING IN PARTIAL METRIC SPACE

Contractive mapping is one kind of mapping that guarantees a fixed point in a metric space Many experts has developed this kind of mapping to show the existence of a fixed point such as Kannan mapping and Chatterjea Contractive mapping. In this study, we will show the weakly contractive mapping to show the existence of fixed point in the partial metric space

Remotest Points and Approximate Remotest Points in Metric Spaces

In this paper the concept of remotest points and approximate remotest points are introduced and then we study the existence and convergence of these points for cyclic maps in metric spaces. We also introduce the notion of T-maximizing sequence and give existence result of approximate remotest points using this sequence.

A Survey on Kannan Mappings in Universalized Metric Spaces

In this manuscript, firstly, Kannan contraction type mapping is universalised. Secondly, the unique fixed point of the universalized Kannan contraction type mappings in universalized metric space is verified. Furthermore, we clarify that it can not have a fixed point in metric space.

Approximation of point of coincidence and common fixed points of quasi-contraction mappings using the Jungck iteration scheme

Let (X, d) be a cone metric space over a solid vector space (Y, ⪯). In this paper, we prove a convergence theorem with error estimates and localization formula for Jungck iteration process for approximating points of coincidence and common fixed points of two selfmappings T and f of X satisfying a quasi-contraction condition of the typed(Tx,Ty)⪯λco{d(fx,fy),d(fx,Tx),d(fy,Ty),d(fx,Ty),d(fy,Tx)}for all x, y ∈ X, where λ ∈ (0, 1) is a constant. Our result complements the recent result of Ding et al. [9].The main result is new even in the classical metric space setting. Moreover, this result answers an open question posed by Olaleru [17] concerning common fixed points in metric spaces.

Results on n-tupled fixed points in metric spaces with uniform normal structure

In this paper, we introduce the concept of new notions related to n-tupled fixed point and prove some related results for an asymptotically regular one-parameter semigroup ℑ={F(t):t∈G,where G is an unbounded subset of [0,∞)} of Lipschitzian self-mappings on ∏ i = 1 n X in the case when (X,d) is a complete bounded metric space with uniform normal structure. Our results extend the results due to Yao and Zeng (J. Nonlinear Convex Anal. 8(1):153-163, 2007) and Soliman (Fixed Point Theory Appl. 2013:346, 2013; J. Adv. Math. Stud. 7(2):2-14, 2014).

Quadrupled fixed point results in abstract metric spaces

In this paper we consider some new definitions about quadrupled fixed point in abstract metric spaces and obtain some new fixed point results in this field. These results unify, extend and generalize well-known comparable results in the existing literature. We also provide some examples and applications to support our results.

Employing Common Limit Range Property to Prove Unified Metrical Common Fixed Point Theorems

The purpose of this paper is to emphasize the role of “common limit range property” to ascertain the existence of common fixed point in metric spaces satisfying an implicit function essentially due to the paper of Ali and Imdad (2008). As an application to our main result, we derive a fixed point theorem for four finite families of self-mappings which can be utilized to derive common fixed point theorems involving any finite number of mappings. Our results improve and extend a host of previously known results including the ones contained in the paper of Ali and Imdad (2008). We also furnish some illustrative examples to support our main results.

Coupled best proximity point theorem in metric Spaces

In this article the concept of coupled best proximity point and cyclic contraction pair are introduced and then we study the existence and convergence of these points in metric spaces. We also establish new results on the existence and convergence in a uniformly convex Banach spaces. Furthermore, we give new results of coupled fixed points in metric spaces and give some illustrative examples. An open problems are also given at the end for further investigation.

Bounds on sets with few distances

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: • we improve the Ray-Chaudhuri–Wilson bound of the size of uniform intersecting families of subsets; • we refine the bound of Delsarte–Goethals–Seidel on the maximum size of spherical sets with few distances; • we prove a new bound on codes with few distances in the Hamming space, improving an earlier result of Delsarte. We also find the size of maximal binary codes and maximal constant-weight codes of small length with 2 and 3 distances.