Contractive Condition Research Articles

Common Attractors of Generalized Hutchinson–Wardowski Contractive Operators

The aim of this paper is to obtain a fractal set of ℑ-iterated function systems comprising generalized ℑ-contractions. For a variety of Hutchinson–Wardowski contractive operators, we prove that this kind of system admits a unique common attractor. Consequently, diverse outcomes are obtained for generalized iterated function systems satisfying various generalized contractive conditions. An illustrative example is also provided. Finally, the existence results of common solutions to fractional boundary value problems are obtained.

The Invariant Measure for a Countable Generalized Iterated Function System

The aim of this paper is to answer one of the open questions raised in Strobin [Qual. Theory Dyn. Syst. 19, 85 (2020)] of whether there exists an invariant (Hutchinson) measure for generalized iterated function systems of any order, consisting of a countably infinite number of maps. Our results likewise strengthen those obtained in Secelean [Mediterr. J. Math. 11, 361–372 (2014)], where the existence of the invariant measure is ascertained only for the case of generalized iterated function systems of order 2, consisting of functions which satisfy a particular contractive condition.

Fixed Point Results for Fuzzy Enriched Contraction in Fuzzy Banach Spaces with Applications to Fractals and Dynamic Market Equillibrium

We introduce fuzzy enriched contraction, which extends the classical notion of fuzzy Banach contraction and encompasses specific fuzzy non-expansive mappings. Our investigation establishes both the presence and uniqueness of fixed points considering this broad category of operators using a Krasnoselskij iterative scheme for their approximation. We also show the graphical representation of fuzzy enriched contraction and analyze its graph for different values of beta. The implications of these findings extend to significant results within fuzzy fixed-point theory, enriching the understanding of iterative processes in fuzzy metric spaces. To demonstrate the versatility of our innovative concepts and the associated fixed-point theorems, we provide illustrative examples that showcase their applicability across diverse domains, including the generation of fractals. This demonstrates the relevance of fuzzy enriched contraction to iterated function systems, enabling the study of fractal structures under various contractive conditions. Additionally, we explore practical applications of fuzzy enriched contraction in dynamic market equilibrium, offering new insights into stability and convergence in economic models. Through this unified framework, we open new avenues for both theoretical advancements and real world applications in fuzzy systems.

Abdominal wall muscle ultrasound assessment and level of experience in Pilates - an exploratory study

The core refers to a group of muscles that allow optimal force transfer along the body’s kinetic chain and is critical for movement. The goal of this quasi-experimental study was to investigate the relationship between the experience level of Pilates participants and abdominal wall thickness. For this, we compared the thickness of four abdominal muscles—transversus abdominis (TrA), internal oblique (IO), external oblique (EO), and rectus abdominis (RA)—as measured before and after 8 weeks of Pilates training in three conditions, namely, relaxation, abdominal hollowing, and plank exercise. Eighteen participants were distributed into three groups with different levels of Pilates practice—experienced (EG; n=6), inexperienced (IG; n=7), and control (CG; n=5) groups. The RA showed a tendency to increase post-intervention in the EG (all conditions: p=0,002, p=0,006, and p=0,002, respectively for relaxation condition, abdominal hollowing and standing plank). Additionally, significant differences were found in relaxation (p=0.003, d=−0.744) and plank (p=0.009, d=−0.630) conditions in the IG. Significant differences were also registered in the EO muscle in the IG (all conditions: p=0.046, p=0.013, and p=0.008, respectively; d=0.464, d=−0.596, and d=−0.637, respectively). IO muscle thickness tended to increase in the CG in all conditions (p=0.044, p=0.006, and p<0.001, respectively) and the EG in relaxation and plank conditions (p=0.009 and p=0.007, respectively). Within groups, the effects of Pilates practice were more significant post-intervention, with the exceptions being under the contraction condition in the deepest muscles (IO: p=0.109, d=0.083; TrA: p=0.194, d=0.062). Our hypothesis was partially confirmed because 8 weeks of Pilates practice have improved significantly the thickness of the RA and EO muscles in the IG.

A Theorem on k-Expansive Mapping and its Fixed Point in G𝐽 𝑆 – Metric Space

Objectives: To find a fixed point of k-expansive mapping on GJS-metric space by establishing a theorem on it and to prove the uniqueness of such a fixed point. Methods: A theorem for the k-expansive map on GJS-metric space is established by generalizing an existing contractive condition theorem. Generalization is done by replacing the contractive condition with an expansive type of condition. Findings: Certain conditions that have to be satisfied by a k-expansive mapping for the existence and uniqueness of fixed points in GJS-metric space were found in this article. An example is also given to illustrate the existence of a fixed point. Novelty: GJS-metric space is a new metric space that has been established recently. Furthermore, the fixed point theorem of k-expansive mapping on GJS - metric space has not yet been proved by anyone, and hence it is proved in this article. 2020 Mathematics Subject Classification: 47H10, 54H25. Keywords: GJS-Metric space; Symmetric GJS-Metric space; GJS-Continuous function; GJS-Cauchy sequence; Expansive mapping; k-Expansive mapping

Boundary rigidity of 3D CAT(0) cube complexes

The boundary rigidity problem is a classical question from Riemannian geometry: if (M,g) is a Riemannian manifold with smooth boundary, is the geometry of M determined up to isometry by the metric dg induced on the boundary ∂M? In this paper, we consider a discrete version of this problem: can we determine the combinatorial type of a finite cube complex from its boundary distances? As in the continuous case, reconstruction is not possible in general, but one expects a positive answer under suitable contractibility and non-positive curvature conditions. Indeed, in two dimensions Haslegrave gave a positive answer to this question when the complex is a finite quadrangulation of the disc with no internal vertices of degree less than 4. We prove a 3-dimensional generalisation of this result: the combinatorial type of a finite CAT(0) cube complex with an embedding in R3 can be reconstructed from its boundary distances. Additionally, we prove a direct strengthening of Haslegrave’s result: the combinatorial type of any finite 2-dimensional CAT(0) cube complex can be reconstructed from its boundary distances.

Fixed Point Theorems in G-Metric Vector Spaces and their applications in Machine Learning and Optimization Algorithms

This research aims to develop a novel mathematical formulation of fixed point theorems in G-metric vector spaces and explore its applications in machine learning and optimization algorithms. The study begins by defining a complete G-metric space and establishing a generalized contraction condition for an operator representing iterative updates in optimization processes. Utilizing gradient descent with regularization as a case study, a numerical example is presented to validate the proposed formulation. The methodology involves iterative calculations demonstrating the convergence of a sequence of parameters toward a fixed point, highlighting the effectiveness of the auxiliary function in promoting sparsity while accommodating the non-Euclidean characteristics of high-dimensional data. The main results reveal that the proposed framework not only enhances the convergence properties of iterative algorithms but also aligns well with contemporary regularization techniques in machine learning. The conclusions drawn emphasize the significance of integrating generalized G-metric spaces and auxiliary functions in fixed point theories, suggesting that this approach provides a robust foundation for developing adaptive optimization algorithms. The findings indicate substantial implications for future research, highlighting the need for broader exploration of G-metric spaces and their applicability in various machine learning paradigms to address complex data challenges.

A Revised Fuzzy Differential Equations Using Weakly Compatible Self-Mappings in Revised Fuzzy Metric Spaces

In this paper, we demonstrated certain coincidence point and Common Fixed Point [briefly, CFP] findings under the rational type weakly compatible revised fuzzy-contraction conditions in full Revised Fuzzy Metric spaces [briefly, RFMSs] utilizing the "triangular property of revised fuzzy metric" with examples in this work. We also provided Revised fuzzy differential equations [briefly, RFDEs] as an application and demonstrated that the solution of the RFDEs has a unique CFP of the integral operators B,C and g. This new path of weakly-compatible Revised fuzzy-contraction with the use of RFDEs in RFM space will be crucial. With different sorts of weakly-compatible Revised fuzzy contraction conditions for self-mappings and different forms of differential equations in the context of RFMSs, this approach may be expanded and developed in diverse directions.

A Revised Fuzzy Differential Equations Using Weakly Compatible Self-Mappings in Revised Fuzzy Metric Spaces

In this paper, we demonstrated certain coincidence point and Common Fixed Point [briefly, CFP] findings under the rational type weakly compatible revised fuzzy-contraction conditions in full Revised Fuzzy Metric spaces [briefly, RFMSs] utilizing the "triangular property of revised fuzzy metric" with examples in this work. We also provided Revised fuzzy differential equations [briefly, RFDEs] as an application and demonstrated that the solution of the RFDEs has a unique CFP of the integral operators and . This new path of weakly-compatible Revised fuzzy-contraction with the use of RFDEs in RFM space will be crucial. With different sorts of weakly-compatible Revised fuzzy contraction conditions for self-mappings and different forms of differential equations in the context of RFMSs, this approach may be expanded and developed in diverse directions.

Enriched Z-Contractions and Fixed-Point Results with Applications to IFS

In this manuscript, we initiate a large class of enriched (d,Z)-Z-contractions defined on Banach spaces and prove the existence and uniqueness of the fixed point of these contractions. We also provide an example to support our results and give an existence condition for the uniqueness of the solution to the integral equation. The results provided in the manuscript extend, generalize, and modify the existence results. Our research introduces novel fixed-point results under various contractive conditions. Furthermore, we discuss the iterated function system associated with enriched (d,Z)-Z-contractions in Banach spaces and define the enriched Z-Hutchinson operator. A result regarding the convergence of Krasnoselskii’s iteration method and the uniqueness of the attractor via enriched (d,Z)-Z-contractions is also established. Our discoveries not only confirm but also significantly build upon and broaden several established findings in the current body of literature.

Immediate effect of local vibration on motor unit firing behavior and muscle strength in healthy young adult males.

The aim of this study was to examine the effect of vibration on motor unit (MU) firing behavior and physical performance of antagonist muscles in healthy young adult males. Fourteen males (age = 24.3 ± 3.6years) were included in this study. There were two conditions, one in which participants received 80Hz vibration in the distal tendon of the hamstring for 30s and the control condition (no vibration). High-density surface electromyography (HD-SEMG) signals and maximal voluntary contraction (MVC) of knee extensor muscles were evaluated before and after the respective conditions and recorded from the vastus lateralis muscle during submaximal ramp-up and sustained contractions at 30% MVC. Convolution blind source separation was used to decompose the HD-SEMG signals into individual MU firing behaviors. In total, 739 MUs were detected (control; 360 MUs and vibration; 379 MUs), and a total of 312 matched MUs were identified across both submaximal contraction conditions (control: 150 MUs; vibration: 162 MUs). Vibration significantly increased the discharge rate (p = 0.047) and decreased the recruitment threshold before and after intervention (p = 0.001) but not in the control condition. Furthermore, the recruitment threshold is a factor that influences discharge rate. Significant correlations were observed between the recruitment threshold and both the ∆ discharge rate and the ∆ recruitment threshold under the vibration condition (p < 0.001). Vibration increased in the discharge rate and decreased the recruitment threshold of the antagonist muscle. These findings suggested that vibration contributes to immediate changes in the neural control of antagonist muscles.

Common Fixed Point Theorems in Extended Rectangular b-Metric Spaces

In this paper, we establish common fixed point theorems for quadruple weakly compatible mappings satisfying a new generalized contraction condition. Our results generalize the corresponding result of Budi Nurwahyu et al. [6]. Non-trivial examples are further provided to support the hypotheses of our results.

Multi-period quality and pricing decision in production-focused and market-focused supply chains: How to manage a long-term contractual relationship

This study investigates the effect of the long-term contract on multi-period decisions regarding product quality and sales price in a production-focused supply chain (PSC) and a market-focused supply chain (MSC). By introducing cumulative quality investment over time as in practice and comparing long-term supply chain performance, we reveal that the relationship between supply chain players is the key factor impacting long-term performance. To enhance the supply chain’s long-term profit performance, we find the contract condition should be more advantageous to the supplier responsible for product quality than to the buyer deciding on the sales price, regardless of supply chain types and environmental changes. We also reveal that the MSC can outperform the PSC in the long run. However, it should be noted that the contract condition needs to be more advantageous to the supplier in the MSC than in the PSC. We provide new and significant guidelines for practicing managers by revealing that long-term decisions should differ from single-period decisions.

The Banach Fixed Point Theorem: selected topics from its hundred-year history

On June 24, 1920 Stefan Banach presented his doctoral dissertation titled O operacjach na zbiorach abstrakcyjnych i ich zastosowaniach do równañ całkowych (On operations on abstract sets and their applications to integral equations) to the Philosophy Faculty of Jan Kazimierz University in Lvov. He passed his PhD examinations in mathematics, physics and philosophy, and in January 1921 he became a doctor. A year later, he published the results of his doctorate in Fundamenta Mathematicae. Among them there was the theorem known today as the Banach Fixed Point Theorem or the Banach Contraction Principle. It is one of the most famous theorems in mathematics, one of many under the name of Banach. It concerns certain mappings (called contractions) of a complete metric space into itself and it gives the conditions sufficient for the existence and uniqueness of a fixed point of such mapping. In 2022 we had a centenary of publishing this theorem. In the paper, we want to present its most important modifications and generalizations, several contractive conditions, the converse theorems and some applications. It is not possible to provide complete information about what has been written during the last hundred years about the Banach Fixed Point Theorem and we are just trying to touch on some breakthrough moments in the development of the metric fixed point theory. The main purpose of this article is to organize the knowledge on this subject and to elaborate a broad bibliography which all interested persons can refer to.

A Bakry–Émery criterion for weighted contractivity and L2-Hardy inequalities

We show a symmetric Markov diffusion semigroup satisfies a weighted contractivity condition if and only if a [Formula: see text]-Hardy inequality holds, and we give a Bakry–Émery-type criterion for the former. We then give some applications.

Peri-ankle muscles architecture and performance changes inpatients with chronic ankle instability: A retrospective cross-sectional study.

This study aimed to identify changes in the architecture and performance of the peri-ankle muscles in patients with chronic ankle instability (CAI) and investigate the relationship between them. In total, 17 subjects were evaluated retrospectively. Each subject underwent anthropometric and isokinetic test, and peroneus longus (PL) and brevis (PB), medial gastrocnemius (MGCM), and tibialis anterior (TA) ultrasound imaging were performed at rest and maximum voluntary contraction (MVC) conditions. Regarding muscle architectural variables, the pennation angle (PA) of the MGCM at rest and the PA of the TA, MGCM, and PL in MVC were significantly reduced on the injured side compared to the intact side. There were no significant differences in muscle thickness of PL, PB, MGCM, and TA observed between intact and injured side during both rest and MVC. Regarding muscle performance parameters, significant decreased were observed in the muscle strength for both limbs in all four directions under the two different conditions. A secondary finding was that the relative PA ratio of the TA showed moderate correlation with the relative dorsiflexion ratio at 30°/s. These findings can provide opportunities to better understand how injuries in patients with CAI may be related to changes in ankle and foot function.

Results on fixed points in b-metric space by altering distance functions

We discuss the existence of a fixed point for a self mapping and its uniqueness satisfying (ϕ˙,η˙)-generalized contractive condition including altering distance functions of rational terms in an ordered b-metric space. It is also discussed whether the two self-maps under the same contraction condition can be coincident and coupled coincident. The results are backed up by a dearth of numerical examples and application to nonlinear quadratic integral equation.

An Implicit Function Implies Several Contraction Conditions

An Implicit Function Implies Several Contraction Conditions

Common Fixed Point Theorems for Weakly Compatible Mapping by (CLR) Property in Dislocated Metric Spaces

In this paper, we employ the concepts of commutativity, weak compatibility and common limit in the range (CLR) property to derive some common fixed point theorems for six mappings satisfying integral type contractive conditions in complete dislocated metric spaces. Our findings improve and extends some previous results found in the literature. 2010 Mathematical Sciences Classification: 54H25, 47H10

Type (K) Compatible Mappings and Common Fixed Points in Complete Cone S-metric Spaces

Objective: This research work is an attempt to explore common fixed points for four self-maps that are pairwise type ( ) compatible in a complete cone -metric space. Method: We have applied a contractive condition together with a weaker condition, compatible of type (K) to establish fixed points that are common for four self-maps. Findings: Some already existing results in the literature have been generalized, expanded upon, and new fixed point results have been obtained in cone S-metric spaces. To back up our conclusions, a suitable example is provided. Novelty: By adopting an appropriate contractive condition, we have proven the existence of a unique common fixed point for four maps in a cone -metric space, despite the weak compatible condition. MSC 2020: 47H10, 54H25. Keywords: Normal Cone, Partial Ordering, Cone S-metric Space, Compatible Mappings of Type (K), Common Fixed Point