Suzuki ψF-contractions and some fixed point results

The purpose of this paper is to combine and extend some recent fixed point results of Suzuki, T., [A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313–5317] and Secelean, N. A. & Wardowski, D., [ψF-contractions: not necessarily nonexpansive Picard operators, Results Math., 70 (2016), 415–431]. The continuity and the completeness conditions are replaced by orbitally continuity and orbitally completeness respectively. It is given an illustrative example of a Picard operator on a non complete metric space which is neither nonexpansive nor expansive and has a unique continuity point.

Development of complicated oscillation algorithm for continuity assurance at arc welding deposition

A theoretical development of formulae for the definition of a speed of welding deposition on a flat is presented where in the first case cross oscillations are applied on it and in the second one – longitudinal oscillations. The law of oscillation motion for both variants and also the law of sinusoidal oscillations of a welding tool are specified. In such a way, the computation of the rate of welding deposition in both cases is carried out under conditions of an additive oscillation effect. The realization of continuity conditions with the application of a heating source motion according to a sinusoidal law at a motionless surface to be under welding deposition – a uniform formation of a beaded weld on a width which is characterized by the position of a continuity point is assumed as a basis of the computation. In case of a shift of the surface under welding deposition this condition will change considerably as a point of continuity will be shifted in which connection in the first case – crosswise, and in the second – lengthwise. The module of the shift of this point is specified by a differential equation which describes motion character equally both at longitudinal and transversal shift of the surface under welding deposition. Final expressions for a welding deposition rate will differ that is caused by a difference in a mathematical description of a continuity condition.

On approximation by Function having a strong Entropy Point

The paper deals with approximation of functions from the unit interval into itself by means of functions having strong entropy point. For this purpose we define a family of functions having the fixed point property: ConnC (which is a subfamily of the class Conn introduced in [Korczak-Kubiak. E.. Paw- lak. R.J.: Trajectories, first return limiting notions and rings of H-connected and iteratively H-connected functions. Czechoslovak Math. J. 63 (2013). 679-700]). The main result of the paper Is a theorem saying that for any function ƒ ∈ ConnC and any point x0 ∈ Fix(ƒ) there exists a ring R ⊂ ConnC containing function ƒ and in the intersection of any “graph neighbourhood of ƒ” and “neighbourhood of ƒ in topology of uniform convergence”, one can find functions ξ,Ψ ∈ R having a strong entropy point y0 located close to the point x0 and being a discontinuity point of the function ξ and a continuity point of the function Ψ.

Shadowing, expansiveness and hyperbolic homeomorphisms

The purpose of this paper is to complete results concerning the class ℋ of expansive homeomorphisms having the pseudo orbits tracing property on a compact metric space. We show that hyperbolic homeomorphisms introduced by Mañé in [8] are exactly those in the class ℋ then by the result of [12, 20] they form a class equal to the Smale space introduced by Ruelle in [18]. Next, assuming that the phase space is a smooth manifold, we show that a diffeomorphism is Anosov if and only if it is in the class ℋ and is a lower semi-continuity point of the map which assigns to any diffeomorphism the supremum of its expansive constants (possibly zero). Then we discuss the behavior of the dynamical systems generated by homeomorphisms from ℋ near their basic sets.