Ulam-Hyers stability and exponentially dichotomic evolution equations in Banach spaces

For finite-dimensional linear differential systems with bounded coefficients we prove that their exponential dichotomy on R is equivalent to their Ulam–Hyers stability on R with uniqueness. We also consider abstract non-autonomous evolution equations which are exponentially bounded and exponentially dichotomic and prove that Ulam– Hyers stability with uniqueness is maintained when perturbing them with a nonlinear term having a sufficiently small Lipschitz constant

Addendum to "Ulam-Hyers stability and exponentially dichotomic equations in Banach spaces" [Electron. J. Qual. Theory Differ. Equ. 2023, No. 8, 1-10]

We add relevant references about which we learned after the completion of the initial work. We mainly show how the concept of exponential trichotomy can successfully replace the one of exponential dichotomy in some results from the paper in the title

Many periodic solutions for a second order cubic periodic differential equation

The aim of this work is to provide results that assure the existence of many isolated T-periodic solutions for T-periodic second-order differential equations of the form x'' = a(t)x+b(t)x2+c(t)x3. We use bifurcation methods, including Malkin functions and results of Fonda, Sabatini and Zanolin. In addition, we give a general result that assures the existence of a T-periodic perturbation of a non-isochronous center with an arbitrary number of T-periodic solutions

Some remarks on the monotone iterative technique

We consider an abstract operator equation in coincidence form $Lu=N(u)$ and establish some comparison results and existence results via the monotone iterative technique. We use a generalized iteration method developed by Carl-Heikkila (1999). An application to a boundary value problem for a second-order functional differential equation is considered

Note on the abstract generalized quasilinearization method

The abstract generalized quasilinearization method established in [2] for ordered Banach spaces with regular or normal cone and continuous mappings is revisited for strongly minihedral cones and (o)-continuous operators

Zero-Hopf bifurcation in the FitzHugh-Nagumo system

We characterize the values of the parameters for which a zero--Hopf equilibrium point takes place at the singular points, namely, $O$ (the origin), $P_+$ and $P_-$ in the FitzHugh-Nagumo system. Thus we find two $2$--parameter families of the FitzHugh-Nagumo system for which the equilibrium point at the origin is a zero-Hopf equilibrium. For these two families we prove the existence of a periodic orbit bifurcating from the zero--Hopf equilibrium point $O$. We prove that exist three $2$--parameter families of the FitzHugh-Nagumo system for which the equilibrium point at $P_+$ and $P_-$ is a zero-Hopf equilibrium point. For one of these families we prove the existence of $1$, or $2$, or $3$ periodic orbits borning at $P_+$ and $P_-$

Management of occupational safety and health in the commissioning of defective work equipment in the manufacturing industry

Work equipment must provide compliance with the essential safety and health requirements applicable from the design and manufacturing stage to ensure a high level of protection of social interests, safety and health at work, and environmental protection. Non-conformities are discovered during the initial commissioning of work equipment in some circumstances due to incorrect design and/or manufacture, resulting in unsafe and non-compliant equipment that is defective when placed on the market and marketed. The purpose of this article is to present the research findings. The goal was to assess the risks associated with the security aspects of digital paper and cardboard printing machinery, to ensure compliance with current safety and health regulations, and to limit the risk of operating defective products. The study offered research results on the establishment of safety and compliance conditions, in order to design a methodology for assessment and technical inspection of work equipment, based on risk assessment and taking into account the usual and anticipated use of the work equipment. Introduction on the market, to improve the efficiency of technical inspection and market surveillance operations by detecting and generating risk profiles for work equipment manufacturer non-conformity

Aspects of the earthing and short-circuit device’s safety quality

Earthing and short-circuit devices are part of the category of equipment and means of electric shock protection, and their purpose is to protect workers in the event of an accidental voltage in the work area while doing electrical installation work. The purpose of this study is to convey the findings of research into the level of safety that these devices must provide, not only in terms of electrodynamic and electro thermal consequences that occur during a short-circuit, but also in terms of mechanical, chemical, and environmental aspects. The study's risk analysis of safety performance provides critical information for earthing and shortcircuiting device manufacturers to ensure the safety function throughout their use, as well as for workers to pick, use, and maintain