Fixed Point Theorems on Some New Types of Cyclic Contractions and Their Generalization

There are different types of contraction in the existing literature for the generalization of Banach’s contraction principle. Our aim in this paper is to generalize cyclic contraction so that it can explain all types of cyclic contraction as a particular case. Besides all contractions in the existing literature we introduce some new types of cyclic contraction before defining the generalized cyclic contraction.

Symmetric Spaces Approach to Various Cyclic Contractions and Application to Probabilistic Spaces

This paper aims to prove fixed point results for cyclic compatible contraction and Hardy–Rogers cyclic contraction in symmetric spaces. Our results generalize the results of Kumari and Panthi (2016) proved for cyclic compatible contraction and modified Hardy–Rogers cyclic contraction in the generating space of a b-quasi metric family and b-dislocated metric family. After that, as an application, we prove a fixed point result in symmetric pre-probabilistic metric spaces (PPM-spaces).

Coincidence best proximity point theorems for proximal Berinde g-cyclic contractions in metric spaces

In this paper, we introduce the notions of proximal Berinde g-cyclic contractions of two non-self-mappings and proximal Berinde g-contractions, called proximal Berinde g-cyclic contraction of the first and second kind. Coincidence best proximity point theorems for these types of mappings in a metric space are presented. Some examples illustrating our main results are also given. Our main results extend and generalize many existing results in the literature.

Global optimal approximate solutions of best proximity points

In this paper, we introduce the concept of contractive pair maps and give some necessary and sufficient conditions for existence and uniqueness of best proximity points for such pairs. In our approach, some conditions have been weakened. An application has been presented to demonstrate the usability of our results. Also, we introduce the concept of cyclic ?-contraction and cyclic asymptotic ?-contraction and give some existence and convergence theorems on best proximity point for cyclic ?-contraction and cyclic asymptotic ?-contraction mappings. The presented results extend, generalize and improve some known results from best proximity point theory and fixed-point theory.

Multivalued weak cyclic δ-contraction mappings

In this paper, we propose some new type of weak cyclic multivalued contraction mappings by generalizing the cyclic contraction using the δ-distance function. Several novel fixed point results are deduced for such class of weak cyclic multivalued mappings in the framework of metric spaces. Also, we construct some examples to validate the usability of the results. Various existing results of the literature are generalized.

Characterization of AZ91 magnesium alloy processed by cyclic contraction/expansion extrusion using the experimental and micromechanical cellular automaton finite element approach

The feasibility to fabricate ultra-fine-grained AZ91 Mg alloy is investigated with a newly presented severe plastic deformation method entitled cyclic contraction/expansion extrusion. In this method, an initial cylindrical AZ91 Mg billet is placed into a die and the moving punch causes the large deformations by extruding the material in two different regions entitled contraction and extrusion. The evolution of AZ91 microstructure and mechanical properties during cyclic contraction/expansion extrusion was studied through different experimental observations. The microstructure observations showed the ultra-fine-grained structure of AZ91 at the end of the third pass where the average grain size of 600 nm obtained from the initial value of 130 µm. The tensile tests showed that the ultimate tensile strength, yield strength, hardness, and elongation of AZ91 cyclic contraction/expansion extrusion-processed samples are increased significantly. Discontinuous dynamic recrystallization has a main role in the grain refinement of Mg alloys during hot deformations. The evolution of grains in microlevel is analyzed by the cellular automaton finite element method in the DEFORM software environment. The macroscopic flow parameters including effective plastic strain, stain rate, and temperature were calculated in finite element. By tracing these data in defined domain of cellular automaton, the discontinuous dynamic recrystallization of material is analyzed through a devised cellular automaton finite element post-processing step. The imposed plastic strain and variation of dislocation density are the two main driven forces in discontinuous dynamic recrystallization of AZ91 samples during cyclic contraction/expansion extrusion processing. The experimentally observed grains and the cellular automaton finite element predicted microstructure were reasonably in good agreement.

Best Proximity Point for the Sum of Two Non-Self-Operators

In the present paper, we focus our attention on the existence of the fixed point for the sum of the cyclic contraction and the noncyclic accretive operator. Also, we study the best proximity point for the sum of two non-self-mappings. Moreover, we provide the existence of the best proximity point for the cyclic contraction through the notion of the nonlinear D-set contraction. Finally, we give the existence of the best proximity point for the sum of the nonlinear D-set contraction mapping and partially completely continuous mapping in the setting of the partially ordered complete normed linear space.

Development of a trunk motor paradigm for use in neuroimaging

The purpose of this study was to quantify head motion between isometric erector spinae (ES) contraction strategies, paradigms, and intensities in the development of a neuroimaging protocol for the study of neural activity associated with trunk motor control in individuals with low back pain. Ten healthy participants completed two contraction strategies; (1) a supine upper spine (US) press and (2) a supine lower extremity (LE) press. Each contraction strategy was performed at electromyographic (EMG) contraction intensities of 30, 40, 50, and 60% of an individually determined maximum voluntary contraction (MVC) (±10% range for each respective intensity) with real-time, EMG biofeedback. A cyclic contraction paradigm was performed at 30% of MVC with US and LE contraction strategies. Inertial measurement units (IMUs) quantified head motion to determine the viability of each paradigm for neuroimaging. US vs LE hold contractions induced no differences in head motion. Hold contractions elicited significantly less head motion relative to cyclic contractions. Contraction intensity increased head motion in a linear fashion with 30% MVC having the least head motion and 60% the highest. The LE hold contraction strategy, below 50% MVC, was found to be the most viable trunk motor control neuroimaging paradigm.