Weak Contraction Research Articles

On Fuzzy Soft Set-Valued Maps with Application

Soft set and fuzzy soft set theories are proposed as mathematical tools for dealing with uncertainties. There has been tremendous progress concerning the extensions of these theories from different point of views of researchers so as to accommodate more robust and expressive applications in everyday life. In line with this development, in this paper, we combine the two aforementioned notions to initiate a novel concept of set-valued maps whose range set is a family of fuzzy soft sets. The later idea is employed to define Suzuki-type fuzzy soft $(e,K)$-weak contractions, thereby establishing some related fuzzy soft fixed point theorems. As a consequence, several well-known Suzuki-type fixed point theorems are derived as corollaries. Examples are also provided to validate the new concepts and to support the authenticity of the obtained results. Moreover, an application in homotopy is considered to show the usability of the obtained results.

A Method for Identification of Mechanical Response of Motor Units in Skeletal Muscle Voluntary Contractions Using Ultrafast Ultrasound Imaging—Simulations and Experimental Tests

The central nervous system coordinates movement through forces generated by motor units (MUs) in skeletal muscles. To analyze MUs function is essential in sports, rehabilitation medicine applications, and neuromuscular diagnostics. The MUs and their function are studied using electromyography. Typically, these methods study only a small muscle volume (1 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) or only a superficial (<1 cm) volume of the muscle. Here we introduce a method to identify so-called mechanical units, i.e., the mechanical response of electrically active MUs, in the whole muscle ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4\times 4$ </tex-math></inline-formula> cm, cross-sectional) under voluntary contractions by ultrafast ultrasound imaging and spatiotemporal decomposition. We evaluate the performance of the method by simulation of active MUs’ mechanical response under weak contractions. We further test the experimental feasibility on eight healthy subjects. We show the existence of mechanical units that contribute to the tissue dynamics in the biceps brachii at low force levels and that these units are similar to MUs described by electromyography with respect to the number of units, territory sizes, and firing rates. This study introduces a new potential neuromuscular functional imaging method, which could be used to study a variety of questions on muscle physiology that previously were difficult or not possible to address.

Existence of best proximity points satisfying two constraint inequalities

&lt;p style='text-indent:20px;'&gt;In this paper, we prove the existence of best proximity point and coupled best proximity point on metric spaces with partial order for weak proximal contraction mappings such that these critical points satisfy some constraint inequalities.&lt;/p&gt;

Physiotherapeutic Treatment for Levator Ani Avulsion after Delivery: A Transperineal Three-dimensional Ultrasound Assessment.

Background:Levator ani avulsion is defined as the interruption of the insertion of this muscle on the pubic bone. It is currently recognized as an important triggering factor for genital prolapse. Although surgical interventions are available, there are no conservative strategies for this muscular injury.Description:A 40-year-old female presented with urinary incontinence and levator ani avulsion, which was confirmed on transperineal three-dimensional ultrasound (3DUS). Upon referral for physiotherapy, she presented with incorrect and weak contractions of the pelvic floor. Her treatment comprised 13 sessions of intravaginal electrotherapy followed by pelvic floor muscle (PFM) exercises in different positions. At the end of the sessions, another transperineal 3DUS was performed, and it revealed rapprochement of the levator ani muscle. The avulsed levator ani muscle can be reinserted using physiotherapeutic interventions, especially a combination of electrotherapy and PFM exercises.Conclusion:Transperineal 3DUS is an important approach for the follow-up of conservative treatment until full recovery.

Some fixed point theorems for generalized (psi - phi)-weak contraction mappings in partial metric spaces

The aim of this paper is to introduce the concepts of generalized (psi - phi)-weak contraction mappings of type (A) and (B) and establish some fixed point theorems for said contraction mappings in complete partial metric spaces. Our results extend and generalize several results from the current existing literature.

Four-Step Iteration Scheme to Approximate Fixed Point for Weak Contractions

Fixed point theory is one of the most important subjects in the setting of metric spaces since fixed point theorems can be used to determine the existence and the uniqueness of solutions of such mathematical problems. It is known that many problems in applied sciences and engineering can be formulated as functional equations. Such equations can be transferred to fixed point theorems in an easy manner. Moreover, we use the fixed point theory to prove the existence and uniqueness of solutions of such integral and differential equations. Let <i>X</i> be a non-empty set. A fixed point for a self-mapping <i>T</i> on <i>X</i> is a point

Contract Enforcement and Productive Efficiency: Evidence From the Bidding and Renegotiation of Power Contracts in India

Weak contract enforcement may reduce the efficiency of production in developing countries. I study how contract enforcement affects efficiency in procurement auctions for the largest power projects in India. I gather data on bidding and ex post contract renegotiation and find that the renegotiation of contracts in response to cost shocks is widespread, despite that bidders are allowed to index their bids to future costs like the price of coal. To study heterogeneity in bidding strategies, I construct a new measure of firm connectedness, based on whether a firm has been awarded coal concessions by the Government. Connected firms choose to index less of the value of their bids to coal prices and, through this strategy, expose themselves to cost shocks to induce renegotiation. I use a structural model of bidding in a scoring auction to characterize equilibrium bidding when bidders are heterogeneous both in cost and in the payments they expect after renegotiation. The model estimates show that bidders offer powerbelowcost due to the expected value of later renegotiation. The model is used to simulate bidding and efficiency with strict contract enforcement. Contract enforcement is found to be pro‐competitive. With no renegotiation, equilibrium bids would rise to cover cost, but markups relative to total contract value fall sharply. Production costs decline, due to projects being allocated to lower‐cost bidders over those who expect larger payments in renegotiation.

Compatible mappings and its variants for generalized ψ-∅-weak contraction condition

In the present paper, first we introduce generalized ψ-∅-weak contraction condition that involves cubic and quadratic terms of distance function d(x,y) and then proved common fixed point theorems for compatible mappings. Secondly, we deal with variants of compatible mappings type (K), type (R) and type (E). At the end, we provide applications of our results.

Tactile Signatures and Hand Motion Intent Recognition for Wearable Assistive Devices.

Within the field of robotics and autonomous systems where there is a human in the loop, intent recognition plays an important role. This is especially true for wearable assistive devices used for rehabilitation, particularly post-stroke recovery. This paper reports results on the use of tactile patterns to detect weak muscle contractions in the forearm while at the same time associating these patterns with the muscle synergies during different grips. To investigate this concept, a series of experiments with healthy participants were carried out using a tactile arm brace (TAB) on the forearm while performing four different types of grip. The expected force patterns were established by analysing the muscle synergies of the four grip types and the forearm physiology. The results showed that the tactile signatures of the forearm recorded on the TAB align with the anticipated force patterns. Furthermore, a linear separability of the data across all four grip types was identified. Using the TAB data, machine learning algorithms achieved a 99% classification accuracy. The TAB results were highly comparable to a similar commercial intent recognition system based on a surface electromyography (sEMG) sensing.

Fixed point theorems for weakly beta-admissible pair of F-contractions with application

In this paper, we introduce a new set Fsb of nonlinear functions. We obtain unique common fixed point theorems for (β; F)-weak contractions under the effect of functions from Fsb. Moreover, we deduce new common fixed point results in ordered and graphic b-metric spaces. Our work generalizes several recent results existing in the literature. We set up an example to elucidate main result. We apply the main theorem to show the existence of common solution of the system of elliptic boundary value problems.

Existence results of delay and fractional differential equations via fuzzy weakly contraction mapping principle

&lt;div&gt;The purpose of this article is to extend the results derived through former articles with respect to the notion of weak contraction into intuitionistic fuzzy weak contraction in the context of (T,N,∝) -cut set of an intuitionistic fuzzy set. We intend to prove common fixed point theorem for a pair of intuitionistic fuzzy mappings satisfying weakly contractive condition in a complete metric space which generalizes many results existing in the literature. Moreover, concrete results on existence of the solution of a delay differential equation and a system of Riemann-Liouville Cauchy type problems have been derived. In addition, we also present illustrative examples to substantiate the usability of our main result.&lt;/div&gt;

Coupling between human brain activity and body movements: Insights from non-invasive electromagnetic recordings

Electroencephalographic and magnetoencephalographic data have characterized two types of brain–body interactions observed during various types of motor actions, “corticokinematic” and “corticomuscular” coupling. Here, we review the literature on these interactions in healthy individuals, discuss several open debates, and outline current limitations and directions for future research.Corticokinematic coupling (commonly referred to as corticokinematic coherence) probes the relationship between activity of sensorimotor network nodes and various movement-related signals (e.g., speed, velocity, acceleration). It is mainly driven by movement rhythmicity during active, passive, and observed dynamic motor actions. It typically predominates at the primary sensorimotor cortex contralateral to the moving limb, occurs at movement frequency and its harmonics, and predominantly reflects the cortical processing of proprioceptive feedback driven by movement rhythmicity in a broad range of dynamic motor actions.Corticomuscular coupling (commonly referred to as corticomuscular coherence) probes the interaction between sensorimotor cortical rhythms and electromyographic (EMG) activity that mainly occurs during steady isometric muscle contraction. We will here focus on the ~20-Hz coupling that is observed during weak isometric contraction and is linked to the modulation of the descending motor command by the ~20-Hz sensorimotor rhythm.This review argues that corticokinematic and corticomuscular couplings have different neural bases. Corticokinematic coupling is mainly driven by afferent signals, while corticomuscular coupling is mainly (but not solely) driven by efferent signals. This distinction should be considered when investigating interactions between brain and body movements.

Common fixed points for R-weakly commuting mappings satisfying a weak contraction

Common fixed points for R-weakly commuting mappings satisfying a weak contraction

Common Fixed Point Results of Set Valued Maps for Aφ-Contraction and Generalized ϕ-Type Weak Contraction

The solutions for many real life problems may be obtained by interpreting the given problem mathematically in the form f ( x ) = x . One such example is that of the famous Borsuk–Ulam theorem in which, using some fixed point argument, it can be guaranteed that at any given time we can find two diametrically opposite places in a planet with same temperature. Thus, the correlation of symmetry is inherent in the study of fixed point theory. In this article, some new results concerning coincidence and a common fixed point for an A φ -contraction and a generalized ϕ -type weak contraction are established. We prove our results for set valued maps without using continuity of the corresponding maps and completeness of the relevant space. Our results generalize and extend several existing results. Some new examples are given to demonstrate the generality and non-triviality of our results.

Iterated Function Systems with the Weak Average Contraction Conditions

This paper concerns the chaos game of random iterated function systems. We consider a random iterated function system IFS() generated by a finite family of Lipschitz maps on a compact ball of ℝn with the weak average contraction condition and show that it admits a quasi-attractor satisfying the deterministic chaos game. In particular, these properties are preserved under small perturbations of the iterated function system IFS() with respect to the Lipschitz topology.

SOME FIXED POINT THEOREMS FOR SET-VALUED (a, p)-WEAK CONTRACTIONS IN CONE METRIC SPACES OVER BANACH ALGEBRA

The purpose of this paper is to introduce the notion of set-valued (𝛼, 𝑝)- weak contractions in cone metric spaces over Banach algebra and to prove some fixed point theorems for such mappings. The fixed point results of this paper generalize and extend several known fixed point results on cone metric spaces. An example in support of our results is given.

Force-Temporal Characteristics of EEG-EMG Coherence during Isometric Contraction of Lateral Head of Gastrocnemius Muscle.

Coherence between an electroencephalogram (EEG) and an electromyogram (EMG) of the soleus (SOL) muscle during an isometric contraction is observed in the beta-band (15 to 35 Hz) regardless of the contraction force. However, the dynamics on how a variation in coherence occurs over time in the head of the gastrocnemius (GLH) muscle, which is also known to have the same role as the soleus muscle, have yet to be considered. In this study, we focused on GLH and measured an EEG and EMG taken of the GLH muscle when executing an isometric contraction through the dorsiflexion of the right ankle joint for a 1-min period. Moreover, we investigated changes in the EEG-EMG coherence based on the contraction force and elapsed time. As a result, in most subjects, the peak coherence during a weak contraction force was continuously observed in the β-band, whereas the peak coherence during a strong contraction force was observed in the γ-band (35 to 60 Hz) for only the first 12 s. In addition, no significant coherence was observed. Therefore, it was suggested that muscle fatigue induced by a strong contraction force affects the peak coherence. Meanwhile, the inconsistencies observed between the properties of the peak coherence and SOL might be due to the differences in muscle composition.

Some ϕ-fixed point results in b-metric spaces with applications

The fixed point theory is the most important topic in mathematics anaysis. This topic has many applications in the different fields. It demonstrates the uniqueness of the existence of a mapping and how it can be found. The Banach contraction mapping principle is most important theorem in the fixed point theory. Most authors have introduced new contractions and they proved some fixed point theorems in metric spaces. On the other hand, some authors introduced new metric spaces as a generalization of metric spaces. Later, they proved fixed point theorems which generalize the Banach contraction principle for various mappings in several metric spaces. The purpose of this study is to introduce the existence and uniqueness of ϕ-fixed point for some new contractions in complete b-metric spaces. Firstly, we presented new definitions called (F,ϕ,α,θ)_{s} and (F,ϕ,α,θ)_{s}-weak contractions in complete b-metric spaces as a generalization of metric spaces. Later, we proved ϕ-fixed point theorems for (F,ϕ,α,θ)_{s} and (F,ϕ,α,θ)_{s}-weak contractions in complete b-metric spaces. The presented theorems extend and generalize some ϕ-fixed point results which are known in the literature. As applications, we derived some fixed point results in complete partial b-metric spaces as a generalization of partial metric spaces. The results in this paper generalizes may existing results in the literature.

Cyclic $(\alpha ,\beta )$-Admissible Mappings in Modular Spaces and Applications to Integral Equations

The main concern of this study is to present a generalization of Banach's fixed point theorem in some classes of modular spaces, where the modular is convex and satisfying the $\Delta _{2}$-condition. In this work, the existence and uniqueness of fixed point for $(\alpha ,\beta )-(\psi ,\varphi )-$ contractive mapping and $\alpha -\beta -\psi -$weak rational contraction in modular spaces are proved. Some examples are supplied to support the usability of our results. As an application, the existence of a solution for an integral equation of Lipschitz type in a Musielak-Orlicz space is presented.

Determining Fuzzy Distance between Sets by Application of Fixed Point Technique Using Weak Contractions and Fuzzy Geometric Notions

In the present paper, we solve the problem of determining the fuzzy distance between two subsets of a fuzzy metric space. We address the problem by reducing it to the problem of finding an optimal approximate solution of a fixed point equation. This approach is well studied for the corresponding problem in metric spaces and is known as proximity point problem. We employ fuzzy weak contractions for that purpose. Fuzzy weak contraction is a recently introduced concept intermediate to a fuzzy contraction and a fuzzy non-expansive mapping. Fuzzy versions of some geometric properties essentially belonging to Hilbert spaces are considered in the main theorem. We include an illustrative example and two corollaries, one of which comes from a well-known fixed point theorem. The illustrative example shows that the main theorem properly includes its corollaries. The work is in the domain of fuzzy global optimization by use of fixed point methods.