Weak Contraction Research Articles

Reich type weak contractions on metric spaces endowed with a graph

In this paper, we define a new class of Reich type multi-valued contractions on a complete metric space satisfying the g-graph preserving condition and we study the fixed point theorem for such mappings. In addition, we present the existence and uniqueness of the fixed point for at least one of two multi-valued mappings. The results of this paper extend and generalize several well-known results. Some examples illustrate the usability of our results.

Phenotypic expansion of visceral myopathy associated with ACTG2 tandem base substitution.

Familial visceral myopathy (FVM) is a rare heritable and heterogeneous condition due to impaired smooth muscle function. We identified a family segregating 11 individuals with a spectrum of visceral symptoms involving the small intestine, colon, biliary tract, urinary tract and uterus. Whole-exome sequencing revealed a novel heterozygous tandem base substitution c.806_807delinsAA (p.(Gly269Glu)) in ACTG2, encoding smooth muscle actin γ-2, in affected family members. Variants in ACTG2 were recently identified in FVM with intestinal pseudo-obstruction as well as with the congenital megacystics-microcolon-intestinal hypoperistalsis syndrome. In our family, eight affected members presented with severe complications from the biliary and/or the urinary tracts in addition to gastrointestinal pseudo-obstructions. Furthermore, all affected mothers had a history of assisted deliveries owing to poor progress during labor and weak uterine contractions. The variable involvement of multiple smooth muscle-dependent organs in our family, including the biliary tract and the uterus, add to the phenotypic spectrum associated with ACTG2 missense variants.

Discrete Optimization for Ordered Weak Proximal Kannan Contractions

Abstract Let us consider a non-self mapping T : A → B, where A and B are two nonempty subsets of a partially ordered set that is equipped a metric. A best proximity point x⋆ for such a mapping T is a point such that d(x⋆, T x⋆) = dist(A,B). In this work, we provide different existence results of best proximity points and so, we establish some new fixed point theorems in the setting of partially ordered set.

Tendon palpation during agonist contraction and antagonist co-contraction to assess wrist flexor and extensor muscle function

The aim of this study was to validate direct tendon palpation during agonist contraction and antagonist co-contraction as a method to assess wrist flexor and extensor muscle function in cases of upper limb paralysis. On one occasion, five doctors examined 17 patients with partial paralysis of the upper limb resulting from brachial plexus or cervical spinal cord injury. We asked examiners to determine if the extensor carpi radialis longus (ECRL), extensor carpi radialis brevis (ECRB), extensor carpi ulnaris (ECU), flexor carpi radialis (FCR), flexor carpi ulnaris (FCU) and palmaris longus (PL) were paralyzed, weak or strong in each patient. Examiners tested flexion – extension and radial – ulnar deviation against resistance and palpated wrist motor tendons. While palpating tendons, co-contractions were encouraged by soliciting finger extension to evaluate the FCU, thumb extension to evaluate the ECU, and finger flexion to evaluate the ECRB. Kappa values were 0.8 for the ECRL, 0.7 for the ECRB, 0.5 for the ECU, 0.8 for the FCR, 0.6 for the PL, and 0.8 for the FCU, indicating moderate to almost perfect agreement between examiners. Tendon palpation during muscle examination was adequate to identify complete paralysis, as well as weak and strong muscle contractions. This assessment helps to identify muscles that could be used during nerve or tendon transfer for reconstruction of extensive upper limb paralysis.

Fixed Point Theorems for Set-Valued Mappings under Contractive Condition in Quasi-Metric Spaces

Abstract In this paper, we introduce the concept of a generalized weak contraction for set-valued mappings defined on quasi-metric spaces. We show the existence of fixed points for generalized weakly contractive set-valued mappings. Indeed, we have a generalization of Nadler’s fixed point theorem and Banach’s fixed point theorem in quasi-metric spaces and, further, investigate the convergence of iterate scheme of the form xn+1 ∈ Fxn with error estimates.

Agonist contraction during intermittent theta burst stimulation enhances motor cortical plasticity of the wrist flexors

Differences in cortical control across the different muscles of the upper limb may mitigate the efficacy of TMS interventions targeting a specific muscle. The current study sought to determine whether weak concurrent contraction during TMS could enhance the efficacy of intermittent theta burst stimulation (iTBS) in the forearm flexors. Motor evoked potentials (MEP) were elicited from the flexor (FCR) and extensor carpi radialis (ECR) motor cortical hotspots before and after iTBS over the FCR cortical hotspot. During iTBS the FCR was either relaxed (iTBS-Relax) or tonically contracted to 10% of maximum voluntary force (iTBS-Contract). iTBS-Relax failed to produce consistent potentiation of MEPFCR amplitude. Individuals with a relatively lower RMTFCR compared RMTECR demonstrated MEPFCR facilitation post-iTBS-Relax. Individuals with relatively higher RMTFCR demonstrated less facilitation and even suppression of MEPFCR amplitude. iTBS-Contract facilitated MEPFCR amplitude but only for MEPFCR evoked from the ECR hotspot. Interactions between overlapping cortical representations determine the efficacy of iTBS. Tonic contraction increases the efficacy of iTBS by enhancing the volume of the cortical representation. However, metaplastic effects may attenuate the enhancement of MEP gain at the motor cortical hotspot. The use of TMS as an adjunct to physical therapy should account for inter-muscle interactions when targeting muscles of the forearm.

Unitary Equivalence and Similarity to Jordan Models for Weak Contractions of Class C0

AbstractWe obtain results on the unitary equivalence of weak contractions of class C0 to their Jordan models under an assumption on their commutants. In particular, our work addresses the case of arbitrary finite multiplicity. The main tool in this paper is the theory of boundary representations due to Arveson. We also generalize and improve previously known results concerning unitary equivalence and similarity to Jordan models when the minimal function is a Blaschke product.

Effects of muscle contraction on cervical vestibular evoked myogenic potentials in normal subjects

ObjectiveCervical vestibular evoked myogenic potentials (cVEMPs) are vestibular-dependent muscle reflexes recorded from the sternocleidomastoid (SCM) muscles in humans. cVEMP amplitude is modulated by stimulus intensity and SCM muscle contraction strength, but the effect of muscle contraction is less well-documented. The effects of intensity and contraction were therefore compared in 25 normal subjects over a wide range of contractions. MethodscVEMPs were recorded at different contraction levels while holding stimulus intensity constant and at different intensities while holding SCM contraction constant. ResultsThe effect of muscle contraction on cVEMP amplitude was linear for most of the range of muscle contractions in the majority of subjects (mean R2=0.93), although there were some nonlinearities when the contraction was either very weak or very strong. Very weak contractions were associated with absent responses, incomplete morphology and prolonged p13 latencies. Normalization of amplitudes, by dividing the p13–n23 amplitude by the muscle contraction estimate, reduced the effect of muscle contraction, but tended to underestimate the amplitude with weak contractions. ConclusionsMinimum contraction levels are required for accurate interpretation of cVEMPs. SignificanceThese data highlight the importance of measuring SCM contraction strength when recording cVEMPs.

A new type of contraction in a complete $G$-metric space

In this paper we extend and generalize the concept of $F$-contraction to $F$-weak contraction and prove a fixed point theorem for $F$-weak contraction in a complete $G$-metric space. The article includes a nontrivial example which verify the effectiveness and applicability of our main result.

Common fixed points for an uncountable family of weakly contractive operators

In this paper, we consider some behavior concerning common fixed points of an uncountable family of operators. We apply here the concept of circular metric spaces, and the operators are assumed to satisfy different rates of weak contractivity. We show under certain assumptions that weakly contractive family have some strong relationships to its common selector in terms of their fixed points.

Weak Contractions and Fixed Points

SummaryThe notation of a contraction is an important concept in analysis. We explore here the weaker notion of a weak contraction in a variety of contexts, involving the reader throughout with questions to provide additional details or proofs, and suggest possible further areas of exploration.

Common fixed point theorem for reciprocally continuous maps with weak contraction in IFMS

Common fixed point theorem for reciprocally continuous maps with weak contraction in IFMS

A fixed point theorem for generalized weak contractions

A fixed point theorem for a generalized weak contractive map in a metric space is proven by generalizing some recent results of Djoric [4, Theorem 2.2], Zhang and Song [30, Corollary 2.2] and others. The result is illustrated by examples.

Best Proximity Points for Generalized Proximal Weak Contractions Satisfying Rational Expression on Ordered Metric Spaces

We introduce a generalized proximal weak contraction of rational type for the non-self-map and proved results to ensure the existence and uniqueness of best proximity point for such mappings in the setting of partially ordered metric spaces. Further, our results provides an extension of a result due to Luong and Thuan (2011) and also it provides an extension of Harjani (2010) to the case of self-mappings.

Fixed Points of Weakly Compatible Mappings Satisfying Generalized $$\varphi $$ φ -Weak Contractions

In this paper, utilizing the notion of the common limit range property, we prove some new integral type common fixed point theorems for weakly compatible mappings satisfying a \(\varphi \)-weak contractive condition in metric spaces. Moreover, we extend our results to four finite families of self mappings, and furnish an illustrative example and an application to support our main theorem. Our results improve, extend, and generalize well-known results on the topic in the literature.

Asynchronous recruitment of low-threshold motor units during repetitive, low-current stimulation of the human tibial nerve.

Motoneurons receive a barrage of inputs from descending and reflex pathways. Much of our understanding about how these inputs are transformed into motor output in humans has come from recordings of single motor units during voluntary contractions. This approach, however, is limited because the input is ill-defined. Herein, we quantify the discharge of soleus motor units in response to well-defined trains of afferent input delivered at physiologically-relevant frequencies. Constant frequency stimulation of the tibial nerve (10–100 Hz for 30 s), below threshold for eliciting M-waves or H-reflexes with a single pulse, recruited motor units in 7/9 subjects. All 25 motor units recruited during stimulation were also recruited during weak (<10% MVC) voluntary contractions. Higher frequencies recruited more units (n = 3/25 at 10 Hz; n = 25/25 at 100 Hz) at shorter latencies (19.4 ± 9.4 s at 10 Hz; 4.1 ± 4.0 s at 100 Hz) than lower frequencies. When a second unit was recruited, the discharge of the already active unit did not change, suggesting that recruitment was not due to increased synaptic drive. After recruitment, mean discharge rate during stimulation at 20 Hz (7.8 Hz) was lower than during 30 Hz (8.6 Hz) and 40 Hz (8.4 Hz) stimulation. Discharge was largely asynchronous from the stimulus pulses with “time-locked” discharge occurring at an H-reflex latency with only a 24% probability. Motor units continued to discharge after cessation of the stimulation in 89% of trials, although at a lower rate (5.8 Hz) than during the stimulation (7.9 Hz). This work supports the idea that the afferent volley evoked by repetitive stimulation recruits motor units through the integration of synaptic drive and intrinsic properties of motoneurons, resulting in “physiological” recruitment which adheres to Henneman’s size principle and results in relatively low discharge rates and asynchronous firing.

Shorter proofs of some recent even-tupled coincidence theorems for weak contractions in ordered metric spaces

In this paper, we prove some recent even coincidence theorems due to Imdad et al. (Bull Math Anal Appl 5(4): 19-39, 2013) using a method of reduction from the respective coincidence theorems for mappings with one variable in ordered complete metric spaces. Our technique of proof is different, slightly simpler, shorter and more effective than the ones used in Imdad et al.

Fixed point theorems on b-metric spaces for weak contractions with auxiliary functions

Abstract In this paper, we obtain some fixed point results for generalized weakly contractive mappings with some auxiliary functions in the framework of b-metric spaces. The proved results generalize and extend the corresponding well-known results of the literature. Some examples are also provided in order to show that these results are more general than the well-known results existing in literature. MSC:47H10, 54H25.

Fixed point results for generalized mappings

In this paper, first we establish fixed point results for weak asymptotic pointwise contraction type mappings in metric spaces. Then we study the existence of fixed points for weak asymptotic pointwise nonexpansive type mappings in spaces. Our results improve and extend some corresponding known results in the literature. MSC:47H09, 47H10, 54H25.

Best Proximity Points for Weak Proximal Contractions

In this article, we introduce a new class of non-self mappings, called weak proximal contractions, which contains the proximal contractions as a subclass. Existence and uniqueness results of a best proximity point for weak proximal contractions are obtained. Also, we provide sufficient conditions for the existence of common best proximity points for two non-self mappings in metric spaces having appropriate geometric property. Examples are given to support our main results.