Fixed Points of Closed and Compact Composite Sequences of Operators and Projectors in a Class of Banach Spaces

Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces which can be also considered as Banach's spaces are discussed through the paper. The class of composite operators under study can include, in particular, sequences of projection operators under, in general, oblique projective operators. In this paper we are concerned with composite operators which include sequences of pairs of contractive operators involving, in general, oblique projection operators. The results are generalized to sequences of, in general, nonconstant bounded closed operators which can have bounded, closed, and compact limit operators, such that the relevant composite sequences are also compact operators. It is proven that in both cases, Banach contraction principle guarantees the existence of unique fixed points under contractive conditions.

Analysis of the Response Speed of Musculature of the Knee in Professional Male and Female Volleyball Players

The aim of this study was to evaluate the normalized response speed (Vrn) of the knee musculature (flexor and extensor) in high competitive level volleyball players using tensiomyography (TMG) and to analyze the muscular response of the vastus medialis (VM), rectus femoris (RF), vastus lateralis (VL), and biceps femoris (BF) in accordance with the specific position they play in their teams. One hundred and sixty-six players (83 women and 83 men) were evaluated. They belonged to eight teams in the Spanish women's superleague and eight in the Spanish men's superleague. The use of Vrn allows avoiding possible sample imbalances due to anatomical and functional differences and demands. We found differences between Vrn in each of the muscles responsible for extension (VM, RF, and VL) and flexion (BF) regardless of the sex. Normalized response speed differences seem to be larger in setters, liberos and outside players compared to middle blockers and larger in males when compared to females. These results of Vrn might respond to the differences in the physical and technical demands of each specific position, showing an improved balance response of the knee extensor and flexor musculature in male professional volleyball players.