Proximity Point Results Research Articles

Best Proximity Point Results for Fuzzy Proximal Quasi Contractions with Applications

In this work, we introduce a new type of multivalued fuzzy proximal quasi-contraction. These are generalized contractions which are a hybrid of H-contractive mappings and quasi-contractions. Furthermore, we establish the best proximity point results for newly introduced fuzzy contractions in the context of fuzzy b-metric spaces. Fuzzy b-metric spaces are more general than fuzzy metric spaces and are linked with the cosine distance, which is used in various contexts of artificial intelligence to measure the similarity between elements of a vector space.

An Application of φ-Metric and Related Best Proximity Point Results Generalizing Wardowski’s Fixed Point Theorem

Abstract In this article, an application of φ-metric is given via a geometrical example to show how it can help to measure distance for non-planar surfaces where the classical metric becomes incapable. We also introduce the concept of best proximity point and proximal contraction for a class of mappings in a φ-metric space and prove a best proximity point theorem for such class of contraction mappings from which the famous “Wardowski’s fixed point theorem” can be deduced as a particular case. To support our theorem, we provide an example in which Wardowski’s metric fixed point theorem cannot be applied.

Fixed Point and Best Proximity Point Results in PIV-S-Metric Spaces

Fixed Point and Best Proximity Point Results in PIV-<i>S</i>-Metric Spaces

Best Proximity Point Results for Multi-Valued Mappings in Generalized Metric Structure

In this paper, we introduce the novel concept of generalized distance denoted as Jθ and call it an extended b-generalized pseudo-distance. With the help of this generalized distance, we define a generalized point to set distance Jθ(u,H★), a generalized Hausdorff type distance and a PJθ-property of a pair (H★,K★) of nonempty subsets of extended b-metric space (U★,ρθ). Additionally, we establish several best proximity point theorems for multi-valued contraction mappings of Nadler type defined on b-metric spaces and extended b-metric spaces. Our findings generalize numerous existing results found in the literature. To substantiate the introduced notion and validate our main results, we provide some concrete examples.

Darbo Type Best Proximity Point Results via R-function using Measure of Noncompactness with an Application

Darbo Type Best Proximity Point Results via R-function using Measure of Noncompactness with an Application

Existence of the solution of nonlinear fractional differential equations via new best proximity point results

AbstractIn this paper, we obtain some best proximity point results by introducing the concepts of proximal p-contractions of the first type and proximal p -contractions of the second type on partial metric spaces. Thus, some famous results in the literature such as the main result of Altun et al. (Acta Math Hung 162:393–402, 2020) and Basha (J Approx Theory 163(11):1772–1781, 2011) have been extended. Also, we provide some examples where our results are applicable and the results in Haghi et al. (Topol Appl 160:450–454, 2013) are not. Hence, our results are a real generalization of some results in metric spaces and partial metric spaces. Finally, we obtain sufficient conditions for the existence of the solution of nonlinear fractional differential equations via our results.

Finding a solution to an optimization problem for the homotopic mappings via some best proximity point results

Finding a solution to an optimization problem for the homotopic mappings via some best proximity point results

A new type of $ \mathcal{R} $-contraction and its best proximity points

&lt;abstract&gt;&lt;p&gt;In this paper, we aim to overcome the problem given by Abkar et al. [&lt;italic&gt;Abstr. Appl. Anal.&lt;/italic&gt;, &lt;bold&gt;2013&lt;/bold&gt; (2013), 189567], and so to obtain real generalizations of fixed point results in the literature. In this direction, we introduce a new class of functions, which include $ \mathcal{R} $-functions. Thus, we present a new type of $ \mathcal{R} $ -contraction and weaken $ \mathcal{R} $-contractions that have often been studied recently. We also give a new definition of the $ P $-property. Hence, we obtain some best proximity point results, including fixed point results for the new kind of $ \mathcal{R} $-contractions. Then, we provide an example to show the effectiveness of our results. Finally, inspired by a nice and interesting technique, we investigate the existence of a best proximity point of the homotopic mappings with the help of our main result.&lt;/p&gt;&lt;/abstract&gt;

Proximity Point Results for Generalized p-Cyclic Reich Contractions: An Application to Solving Integral Equations

This article studies new classes of contractions called the p-cyclic Reich contraction and p-cyclic Reich contraction pair and develops certain best proximity point results for such contractions in the setting of partial metric spaces. Furthermore, the best proximity point results for p-proximal cyclic Reich contractions of the first and second types are also discussed.

Certain Interpolative Proximal Contractions, Best Proximity Point Theorems in Bipolar Metric Spaces with Applications

In this manuscript, we present several types of interpolative proximal contraction mappings including Reich–Rus–Ciric-type interpolative-type contractions and Kannan-type interpolative-type contractions in the setting of bipolar metric spaces. Further, taking into account the aforementioned mappings, we prove best proximity point results. These results are an extension and generalization of existing ones in the literature. Furthermore, we provide several nontrivial examples, an application to find the solution of an integral equation, and a nonlinear fractional differential equation to show the validity of the main results.

Geraghty–Pata–Suzuki-Type Proximal Contractions and Related Coincidence Best Proximity Point Results

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context of a complete metric space: an α-θ-Geraghty–Pata–Suzuki-type proximal contraction; an α-θ-generalized Geraghty–Pata–Suzuki-type proximal contraction; and an α-θ-modified Geraghty–Pata–Suzuki-type proximal contraction. These new results generalize, extend, and unify various results from the existing literature. Symmetry plays a crucial role in solving nonlinear problems in operator theory, and the variables involved in the metric space are symmetric. Several illustrative examples are provided to showcase the superiority of our results over existing approaches.

Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach

Abstract In this article, a class of cyclic (noncyclic) operators are defined on Banach spaces via concept of measure of noncompactness using some abstract functions. The best proximity point (pair) results are manifested for the said operators. The obtained main results are applied to demonstrate the existence of optimum solutions of a system of fractional differential equations involving ( k , ψ ) \left(k,\psi ) -Hilfer fractional derivatives.

Existence of Best Proximity Point in O-CompleteMetric Spaces

In this work, we prove the existence of the best proximity point results for ⊥-contraction (orthogonal-contraction) mappings on an O-complete metric space (orthogonal-complete metric space). Subsequently, these existence results are employed to establish the common best proximity point result. Finally, we provide suitable examples to demonstrate the validity of our results.

Generalized Common Best Proximity Point Results in Fuzzy Metric Spaces with Application

The symmetry of fuzzy metric spaces has benefits for flexibility, ambiguity tolerance, resilience, compatibility, and applicability. They provide a more comprehensive description of similarity and offer a solid framework for working with ambiguous and imprecise data. We give fuzzy versions of some celebrated iterative mappings. Further, we provide different concrete conditions on the real valued functions J,S:(0,1]→R for the existence of the best proximity point of generalized fuzzy (J,S)-iterative mappings in the setting of fuzzy metric space. Furthermore, we utilize fuzzy versions of J,S-proximal contraction, J,S-interpolative Reich–Rus–Ciric-type proximal contractions, J,S-Kannan type proximal contraction and J,S-interpolative Hardy Roger’s type proximal contraction to examine the common best proximity points in fuzzy metric space. Also, we establish several non-trivial examples and an application to support our results.

Best Proximity Point Results for n-Cyclic and Regular-n-Noncyclic Fisher Quasi-Contractions in Metric Spaces

In this work, we introduce some new concepts such as n-cyclic Fisher quasi-contraction mappings, full-n-noncyclic and regular-n-noncyclic Fisher quasi-contraction mappings in metric spaces. We then generalize the results by Safari-Hafshejani, Amini-Harandi and Fakhar. Meanwhile, we answer the question “under what conditions does a full-n-noncyclic Fisher quasi-contraction mapping have n(n−1)/2 unique optimal pairs of fixed points?”. Further, to support the main results, we highlight all of the new concepts via non-trivial examples.

Best Proximity Point Results in Fuzzy Normed Spaces

Fixed point (briefly FP ) theory is a potent tool for resolving several actual problems since many problems may be simplified to the FP problem. The idea of Banach contraction mapping is a foundational theorem in FP theory. This idea has wide applications in several fields; hence, it has been developed in numerous ways. Nevertheless, all of these results are reliant on the existence and uniqueness of a FP on some suitable space. Because the FP problem could not have a solution in the case of nonself-mappings, the idea of the best proximity point (briefly Bpp) is offered to approach the best solution. This paper investigates the existence and uniqueness of the Bpp of nonself-mappings in fuzzy normed space(briefly FN space) to arrive at the best solution. Following the introduction of the definition of the Bpp, the existence, and uniqueness of the Bpp are shown in a FN space for diverse fuzzy proximal contractions such as ?????? fuzzy proximal contractive mapping and ????h ????h - fuzzy proximal contractive mapping.

Best Proximity Points for Generalized Geraghty Quasi-Contraction Type Mappings in Metric Spaces

In this paper, we introduce a new concept of α-φ-Geraghty proximal quasi-contraction type mappings and establish best proximity point theorems for those mappings in proximal T-orbitally complete metric spaces. This generalizes and complements the proofs of some known fixed and best proximity point results.

Some Best Proximity Point Results in the Orthogonal $$0$$-Complete $$b$$-Metric-Like Spaces

Some Best Proximity Point Results in the Orthogonal $$0$$-Complete $$b$$-Metric-Like Spaces

A New Best Proximity Point Results in Partial Metric Spaces Endowed with a Graph

For a given mapping f in the framework of different spaces, the fixed-point equations of the form fx=x can model several problems in different areas, such as differential equations, optimization, and computer science. In this work, the aim is to find the best proximity point and to prove its uniqueness on partial metric spaces where the symmetry condition is preserved for several types of contractive non-self mapping endowed with a graph. Our theorems generalize different results in the literature. In addition, we will illustrate the usability of our outcomes with some examples. The proposed model can be considered as a theoretical foundation for applications to real cases.

Best Proximity Point Results for Generalization of α ̌–η ̌ Proximal Contractive Mapping in Fuzzy Banach Spaces

&lt;p&gt;The best proximity point is a generalization of a fixed point that is beneficial when the contraction map is not a self-map. On other hand, best approximation theorems provide an approximate solution to the fixed-point equation Tҳ = ҳ. It is used to solve the problem to determine an approximate solution that is optimum. The main goal of this paper is to present new types of proximal contraction for nonself mappings in a fuzzy Banach space. At first, the notion of the best proximity point is presented. We introduce the notion of α ̌–η ̌-β ̌ proximal contractive. After that, the best proximity point theorem for such type of mappings in a fuzzy Banach space is proved. In addition, the concept of α ̌–η ̌-φ ̌ proximal contractive mapping is presented in a fuzzy Banach space and under specific conditions, the best proximity point theorem for such type of mapping is proved. Additionally, some examples are supplied to show the results' applicability.&lt;/p&gt;