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In this study, we establish the convergence of Ishikawa iterates defined by nonlinear quasicontractive mappings on TVS-cone metric space. Further, our results generalize many existing results in the literature.

We discuss and give some insights on the Collatz conjecture, known as 3N + 1, and Kurepa’s hypothesis on the left factorial. First, the Collatz conjecture is considered and the density of values is compared to Planck’s black body radiation in physics, showing a remarkable agreement between the two. We also briefly discuss a generalization of Collatz conjecture for a generic sequence qN +1 by means of numerical analysis. Then, we give a brief historical excursus and prove in a simple way some properties of Kurepa’s function, also called the left factorial. We introduce Kurepa’s hypothesis, propose a new description, and the relation to Bezout’s parameters and the Diophantine equation. A numerical analysis supports Kurepa’s hypothesis and the conjecture about distribution for Kurepa’s function.

Purpose This study aims to obtain and present the closed-form solution of coupled axial-bending vibration problem for the general case of axially functionally graded (AFG) beams, to create the frequency equation and to propose numerical method for computing natural frequencies. Methods The general model of a beam is introduced using two cylindrical springs, one rotational spring, and a rigid body at each beam end. Mass centers of bodies are placed eccentrically in axial and transverse direction with respect to end points of the beam. The general boundary conditions are modelled by linearized Newton-Euler differential equations and the general case of the in-plane axial-bending vibrations of AFG beams is covered. It is assumed that the beam is made of a functionally graded material whose material and cross-sectional characteristics change along its longitudinal axis without any restrictions. The Euler-Bernoulli constitutive theory is applied for modelling. Partial differential equations of motion are transformed into a system of ordinary differential equations with variable coefficients, the form suitable for the implementation of the symbolic-numeric methods of initial parameters. Natural frequencies of the beam are computed as numerical solutions of the exact frequency equation. Results and conclusions The closed-form solution of coupled axial-bending vibrations is derived for general case of AFG beams. Orthogonality conditions of mode shapes are derived, and constants in time function. Also, the frequency equation is derived and solved numerically in order to obtain natural frequencies. Obtained results of natural frequencies and mode shapes are compared to those available in open literature.

This paper presents the continuation of the research by the authors on the topics of vibration analysis. Namely, a tapered cantilever beam has been studied in theoretical and experimental terms. The beam is 3D printed using the FDM technique of PLA material with two characteristics patterns, linear layered and the two-dimensional periodic structure. The former of the two is conducted to tensile test in order to determine elasticity modulus, so the computations of natural frequencies could be conducted. The experimental part of the research, in terms of natural frequency, is conducted on six specimens with various eccentricities. The angular frequencies of one representative sample, obtained by analytical computations and experimental measurement are compared.