Proximal Point Research Articles

Best proximity points for alternative p-contractions

Cyclic mappings describe fixed paths for which each point is sequentially transmitted from one set to another. Cyclic mappings satisfying certain cyclic contraction conditions have been used to obtain the best proximity points, which constitute a suitable framework for the mirror reflection model. Alternative contraction mappings introduced by Chen (Symmetry 11:750, 2019) built a new model containing several mirrors in which the light reflected from a mirror does not go to the next mirror sequentially, and its path may diverge to any other mirror. The aim of this paper is to present a new variant of alternative contraction called alternative p-contraction and study its properties. The best proximity point result for such contractions under the alternative UC property is proved. An example to support the result proved herein is provided.

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.

Height Assessment Using the Length of Humerus of Selected Group of Hausa Ethnic Students in Yusuf Maitama Sule University, Kano State

Height assessment from bones plays a vital role in detecting unidentified bodies and parts of bodies or skeletal remnants, the aim of this study was to estimate the height of individual using the length of humerus of selected group of Hausa ethnic Students in Yusuf Maitama Sule University, Kano State, Nigeria. Two hundred and two (202) subjects were selected from the various faculties of the University, using random sampling method. The age range of the participants was between 18-50 years. Ethical approval was required from the Ethical Committee of the College of Health Science of the institution before the beginning of the research, Informed consents were sought from the participants. Using a stadiometer, standing height of individual was measured to the nearest centimeter (cm) with subject standing barefoot in an anatomical position. The humeral length was measured in between two points:- Distal point-Midpoint of elbow joint and Proximal point- Acromian landmarks, line was measured and documented. Length of humerus obtained was put in regression equation for humerus estimation. From the result obtained indicated that height can be estimated from the humeral length with Coefficient of significance (R=0.628) Coefficient of determinant (R2=0.395). The present study found significant differences between the sex in both humeral length and height of the study population and resolved that the maximum height can be estimated from the length of humerus with effective precision so that may help the archaeological, anthropometric and also forensic investigations.

Modified proximal point algorithms for variational inclusion problems on Hadamard manifolds

Modified proximal point algorithms for variational inclusion problems on Hadamard manifolds

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

Global optimization problem and probabilistic distance

In this article, the probabilistic metric distance between two disjoint sets is utilised to define the essential criteria for the existence and uniqueness of the best proximity point, which takes into account the global optimization problem. In order to solve this problem, we pretend that we are trying to obtain the optimal approximation to the solution of a fixed point equation. Here, we introduce two types of probabilistic proximal contraction mappings and use a geometric property called Ω-property in the context of probabilistic metric spaces. We also obtain some consequences for self-mappings, which give the fixed point results. Some examples are provided to validate the findings. As an application, we obtain the solution to a second-order boundary value problem using a minimum t-norm in the context of probabilistic metric spaces.

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

<abstract><p>In this paper, we aim to overcome the problem given by Abkar et al. [<italic>Abstr. Appl. Anal.</italic>, <bold>2013</bold> (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.</p></abstract>

A New Approach to the Existence of Best Proximity Points With R‐Functions in the Framework of w‐Distances

In this paper, we present a new approach to check the existence of best proximity points for R‐proximal contractions in the framework of metric spaces equipped with a w0‐distance and present elementary proofs of recent results by using the corresponding fixed point problems without the continuity assumption of the considered self‐mapping. We also discuss on cyclic Geraghty contractions and resolve the problem of the existence and convergence of the best proximity point for such a class of mappings.

On the Proximal Point Algorithm of Hybrid-Type in Flat Hadamard Spaces with Applications

In this paper, we introduce a hybrid-type proximal point algorithm for approximating zero of monotone operator in Hadamard-type spaces. We then prove that a sequence generated by the algorithm involving Mann-type iteration converges strongly to a zero of the said operator in the setting of flat Hadamard spaces. To the best of our knowledge, this result presents the first hybrid-type proximal point algorithm in the space. The result is applied to convex minimization and fixed point problems.

Best proximity points for proximal Górnicki mappings and applications to variational inequality problems

<abstract><p>We introduce a large class of mappings called proximal Górnicki mappings in metric spaces, which includes Górnicki mappings, enriched Kannan mappings, enriched Chatterjea mappings, and enriched mappings. We prove the existence of the best proximity points in metric spaces and partial metric spaces. Moreover, we utilize appropriate examples to illustrate our results, and we verify the convergence behavior. As an application of our result, we prove the existence and uniqueness of a solution for the variational inequality problems. The obtained results generalize the existing results in the literature.</p></abstract>

Best proximity point theorems for tricyclic diametrically contractive mappings

Best proximity point theorems for tricyclic diametrically contractive mappings

Best proximity point of $ \alpha $-$ \eta $-type generalized $ F $-proximal contractions in modular metric spaces

<abstract><p>The purpose of this paper is to present a study of $ \alpha $-$ \eta $-type generalized $ F $-proximal contraction mappings in the framework of modular metric spaces and to prove some best proximity point theorems for these types of mappings. Some examples are given to show the validity of our results. We also apply our results to establish the existence of solutions for a certain type of non-linear integral equation.</p></abstract>

Generalized proximal point algorithms with correction terms and extrapolation

Generalized proximal point algorithms with correction terms and extrapolation

Best Proximity Point Theorems in Non-Archimedean Menger Probabilistic Spaces

In this work, we prove best proximity point theorems for γ-contractions with conditions the weak P-property in non-Archimedean Menger probabilistic metric spaces. We give the notion of γ- proximal contractions of first and second type in non-Archimedean Menger probabilistic metric spaces and also we establish best proximity point theorems for these proximal contractions. Lastly, we complete our study by giving examples that support our results.

On a Recent Question from the Orthogonal b-metric-like Spaces

In this paper, we announce a beautiful and important application of the results of the paper of two Romanian mathematicians: R. Miculescu, A. Mihail, New fixed point theorems for set-valued contractions in b-metric spaces, J. Fixed Point Theory Appl., 19, 2153-2163 (2017). Namely, the method they gave in the mentioned paper for b-metric spaces can be applied without difficulty to b-metric-like spaces as well. Thus, the results on b-metric spaces established in the papers: Hassen Aydi, Monica-Felicia Bota, Erdal Karapinar and Slobodanka Mitrović, A fixed point theorem for set-valued quasi-contraction in b-metric spaces, Fixed Point Theory and Applications 2012, 2012:88, and Nicola Fabiano, Zoran Kadelburg, Stojan Radenović, and Milica Savatović, Remarks on ”On new orthogonal contractions in b-metric spaces” and ”On orthogonal partial b-metric spaces with an application”, can be extended to b-metric-like spaces as well. Also, using the significant paper at the beginning of the mentioned Romanian professors, a positive answer can be directly given to the question given at the end of the recent paper: M. Gardašević-Filipović, K. Kukić, D. Gardašević, Z. D. Mitrović, Some best proximity point results in the orthogonal 0-complete b-metric-like spaces, J. Contemp. Math. Anal. Armen. Acad. Sci., 58 (2), 105–115 (2023). In the same way, the interval [0,1s ) belonging to the coefficient of some type of contraction within the b-metric-like space can be extended to the entire interval [0,1).

An inexact algorithm for stochastic variational inequalities

We present a new Progressive Hedging Algorithm to solve Stochastic Variational Inequalities in the formulation introduced by Rockafellar and Wets in 2017, allowing the generated subproblems to be approximately solved with an implementable tolerance condition. Our scheme is based on Hybrid Inexact Proximal Point methods and generalizes the exact algorithm developed by Rockafellar and Sun in 2019, providing stronger convergence results. We also show some numerical experiments in two-stage Nash games.

Strongly Convergent Inertial Proximal Point Algorithm Without On-line Rule

Strongly Convergent Inertial Proximal Point Algorithm Without On-line Rule

Robust Posegraph Optimization Using Proximity Points

This paper proposes a robust posegraph optimization (PGO) method for posegraphs with keypoints. In the conventional PGO formulation, a loop constraint is defined between a pair of nodes, whereas in the proposed method, it is defined between a pair of keypoints. In this manner, robust PGO based on switch variables can be realized in a more fine-grained manner. Loop constraint is defined based on the unique geometric property of proximity point, and implemented as a new edge type of the g2o solver. The proposed method is compared with other robust PGO methods using real world data recorded in Nakanoshima Robot Challenge 2021.

Existence and convergence of best proximity points for generalized pseudo-contractive and Lipschitzian mappings via an Ishikawa-type iterative scheme

In this article, we prove the existence of the best proximity point for the class of nonself generalized pseudo-contractive and Lipschitzian mappings. Also, we approximate the best proximity point through the proposed Ishikawa’s iteration process for the case of nonself-mappings. Finally, we provide an example to illustrate our main result.

Inertial Proximal Point Algorithms for Solving a Class of Split Feasibility Problems

Inertial Proximal Point Algorithms for Solving a Class of Split Feasibility Problems