Uniqueness Of Best Proximity Points Research Articles

Integral Equations: New Solutions via Generalized Best Proximity Methods

This paper introduces the concept of proximal (α,F)-contractions in F-metric spaces. We establish novel results concerning the existence and uniqueness of best proximity points for such mappings. The validity of our findings is corroborated through a non-trivial example. Furthermore, we demonstrate the applicability of these results by proving the existence of solutions for Volterra integral equations related to population growth models. This approach not only extends best proximity theory, but also paves the way for further research in applied mathematics and beyond.

Best Proximity Points for p–Cyclic Infimum Summing Contractions

We investigate fixed points for p cyclic maps by introducing a new notion of p–cyclic infimum summing maps and a generalized best proximity point for p–cyclic maps. The idea generalizes some results about best proximity points in order to widen the class of sets and maps for which we can ensure the existence and uniqueness of best proximity points. The replacement of the classical notions of best proximity points and distance between the consecutive set arises from the well-known group traveling salesman problem and presents a different approach to solving it. We illustrate the new result with a map that does not satisfy the known results about best proximity maps for p–cyclic maps.

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 $A$.

Best Proximity Points of Multivalued Hardy-Roger’s Type (Cyclic) Contractive Mappings of b-Metric Spaces

In this article, we introduce a new type of generalized multivalued Hardy and Roger’s type proximal contractive and proximal cyclic contractive mappings of b -metric spaces and develop some results for the existence of best proximity point(s). Moreover, we obtain some results for the existence and uniqueness of best proximity points for single-valued mappings. Examples are given to explain the main results.

Some results on Best proximity in geodesic spaces

Eldred and Veeramani have proved the existence and uniqueness of best proximity for contraction cyclic contraction mapping in uniformly convex banach spaces. In this paper, Eldred and Veeramani study is extended to geodesic spaces particularly in CAT(0) spaces, prove the sequences (x_2n) and (x_2n+1) are Cauchy sequences and prove the existenace and uniqueness of best proximity point for contraction cyclic mapping in CAT(0) spaces.

Best proximity point of modified Suzuki-Edelstein-Geraghty type proximal contractions

In this paper, we introduce the notion of modified Suzuki-Edelstein-Geraghty proximal contraction and prove the existence and uniqueness of best proximity point for such mappings. Our results extend and unify many existing results in the literature. We draw corollaries and give illustrative example to demonstrate the validity of our result.

Integral Type Contractions in Partially Ordered Metric Spaces and Best Proximity Point

In this paper, we introduce Banach and Kannan integral type contractions in partially ordered metric spaces and investigate the existence and uniqueness of best proximity points for them. Additionally, we pose two non-trivial examples for our main results.

Generalization of best proximity points theorem for non-self proximal contractions of first kind

The primary objective of this paper is the study of the generalization of some results given by Basha (Numer. Funct. Anal. Optim. 31:569–576, 2010). We present a new theorem on the existence and uniqueness of best proximity points for proximal β-quasi-contractive mappings for non-self-mappings S:Mrightarrow N and T:Nrightarrow M. Furthermore, as a consequence, we give a new result on the existence and uniqueness of a common fixed point of two self mappings.

Best proximity points for p-cyclic summing iterated contractions

We generalize the p - summing contractions maps. We found sufficient conditions for these new type of maps, that ensure the existence and uniqueness of best proximity points in uniformly convex Banach spaces. We apply the result for Kannan and Chatterjea type cyclic contractions and we obtain sufficient conditions for these maps, that ensure the existence and uniqueness of best proximity points in uniformly convex Banach spaces.

Best proximity points in partially ordered metric spaces

The existence of best proximity point is an important aspect of optimization theory. We define the concept of proximally monotone Lipschitzian mappings on a partially ordered metric space. Then we obtain sufficient conditions for the existence and uniqueness of best proximity points for these mappings in partially ordered CAT(0) spaces. This work is a continuation of the work of Ran and Reurings [Proc. Amer. Math. Soc. 132 (2004), 1435–1443] and Nieto and Rodr´ iguez-Lopez [Order, 22 (2005), 223–239] for the new class of mappings introduced herein.

Existence and convergence theorems for best proximity points

In this paper, we give new conditions for existence and uniqueness of best proximity point. Also, we introduce the concept of cyclic contraction and nonexpansive for multivalued map and we give existence and convergence theorems for best proximity point in the complete metric space.

Best proximity point results in dislocated metric spaces via R-functions

In this paper, we investigate the existence of best proximity of R-contractions in the frame of dislocated metric spaces. We also propose some conditions to guarantee the uniqueness of best proximity point for such contractions. We consider an illustrative example to support the given results. This result generalizes a number of recent results on the topic in the literature.

Fixed and best proximity points for Kannan cyclic contractions in modular function spaces

We generalize the notion of best proximity points for cyclic contraction maps in modular function spaces about Kannan maps. We have found sufficient conditions for the existence and uniqueness of best proximity points and fixed points for cyclic Kannan maps in modular function spaces. As corollaries we get sufficient conditions for the existence and uniqueness of best proximity points and fixed points for cyclic maps in Orlicz spaces, endowed with an Orlicz function modular. We present an application of the results for solving integral equations.

Best Proximity Point Results for Some Contractive Mappings in Uniform Spaces

We introduce the concept of Jav-distance (an analogue of b-metric), ϕp-proximal contraction, and ϕp-proximal cyclic contraction for non-self-mappings in Hausdorff uniform spaces. We investigate the existence and uniqueness of best proximity points for these modified contractive mappings. The results obtained extended and generalised some fixed and best proximity points results in literature. Examples are given to validate the main results.

Existence and uniqueness of best proximity points under rational contractivity conditions

Abstract The main aim of this paper is to present some theorems in order to guarantee existence and uniqueness of best proximity points under rational contractivity conditions using very general test functions. To illustrate the variety of possible test functions, we include some examples of pairs of functions which are included in innovative papers published in the last years. As a consequence, we prove that our results unify and extend some recent results in this field.

Best Proximity point for Z-contraction and Suzuki type Z-contraction mappings with an application to fractional calculus

In this article, we introduced the best proximity point theorems for $\mathcal{Z}$-contraction and Suzuki type $\mathcal{Z}$-contraction in the setting of complete metric spaces. Also by the help of weak $P$-property and $P$-property, we proved existence and uniqueness of best proximity point. There is a simple example to show the validity of our results. Our results extended and unify many existing results in the literature. Moreover, an application to fractional order functional differential equation is discussed.<br /><br />

Best proximity point results in partially ordered metric spaces via simulation functions

AbstractWe obtain sufficient conditions for the existence and uniqueness of best proximity points for a new class of non-self mappings involving simulation functions in a metric space endowed with a partial order. Some interesting consequences including fixed point results via simulation functions are presented.

Best proximity points for p-summing cyclic orbital Meir–Keeler contractions

We generalize the cyclic orbital Meir–Keeler contractions, which were introduced by S. Karpagam and S. Agrawal in the context of p-summing maps. We found sufficient conditions for these new type of maps, that ensure the existence and uniqueness of fixed points in complete metric spaces, when the distances between the sets are zero, and the existence and uniqueness of best proximity points in uniformly convex Banach spaces.

Best proximity points in modular function spaces

We generalize the notion of best proximity points in the context of modular function spaces. We have found sufficient conditions for the existence and uniqueness of best proximity points for cyclic maps in modular function spaces. We present an application of the main result for cyclic integral operators in Orlicz function spaces, endowed with an Orlicz function modular.

A variational principle and best proximity points

We generalize Ekeland’s Variational Principle for cyclic maps. We present applications of this version of the variational principle for proving of existence and uniqueness of best proximity points for different classes of cyclic maps.