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Using elements of the theory of linear operators in Hilbert spaces and monotonicity tools we obtain the existence and uniqueness results for a wide class of nonlinear problems driven by the equation Tx=N(x)Tx=N\left(x), where TTis a self-adjoint operator in a real Hilbert space ℋ{\mathcal{ {\mathcal H} }}and NNis a nonlinear perturbation. Both potential and nonpotential perturbations are considered. This approach is an extension of the results known for elliptic operators.

Using the Ekeland variational principle and the mountain pass lemma we prove some properties about potential homeomorphisms between a real Banach space and its dual. In particular, we show that a locally monotone homeomorphism is necessarily strictly monotone if it is potential.

Using monotonicity methods and some variational argument we consider nonlinear problems which involve monotone potential mappings satisfying condition (S) and their strongly continuous perturbations. We investigate when functional whose minimum is obtained by a direct method of the calculus of variations satisfies the Palais-Smale condition, relate minimizing sequence and Galerkin approximaitons when both exist, then provide structure conditions on the derivative of the action functional under which bounded Palais-Smale sequences are convergent. Finally, we make some comment concerning the convergence of Palais-Smale sequence obtained in the mountain pass theorem due to Rabier.

We provide sufficient conditions for the existence of saddle-point solutions to a system driven by the weak Laplacian on the Sierpiński gasket. We analyze also its stability by proving its continuous dependence on parameters. [ABSTRACT FROM AUTHOR]

By combining monotonicity theory related to the parametric version of the Browder-Minty theorem with fixed point arguments, hybrid existence results for a system of two operator equations are obtained. Applications are given to a system of boundary value problems with mixed nonlocal and Dirichlet conditions.

Modern industry expectations in terms of milling operations often demand the milling of the flexible details by using slender ball-end tools. This is a difficult task because of possible vibration occurrence. Due to existence of certain conditions (small depths of cutting, regeneration phenomena), cutting process may become unstable and self-excited chatter vibration may appear. Frequency of the chatter vibration is close to dominant natural frequency of the workpiece or the tool. One of the methods of chatter vibration avoidance is matching the spindle speed to the optimum phase shift between subsequent cutting edges passes (i.e. the Liao-Young condition). However, the set of optimum spindle speeds from the point of view of vibration reduction may be not optimum one from other points of view. For example milling efficiency or machine tool capabilities cannot be assured. This article presents the idea of a workpiece holder with adjustable stiffness and discusses a new variant of its realization. In the holder, milling process is performed at constant spindle speed and feed speed. In order to avoid vibration the holder stiffness is modified. Stiffness changes modify natural frequencies of the workpiece and thus it is possible to modify dynamic properties of the workpiece in such a way that arbitrary chosen, constant spindle speed will be optimum, due to the Liao-Young condition performance. Calculation of the optimum stiffness is performed before milling, based on the workpieces modal identification results and the finite element model simulations.

In this work we investigate the existence of solutions, their uniqueness and finally dependence on parameters for solutions of second order neutral nonlinear difference equations. The main tool which we apply is Darbo fixed point theorem.

The paper presents the method of the surveillance of the self-excited chatter vibration. At first, the workpiece modal parameters are estimated based on experimental data which leads to verification of computational model. Then, for selected surface points optimal spindle speeds are calculated. By considering sufficient amount of points it is possible to build a map of optimal spindle speeds. Experimental results show that this map may be used effectively for eliminating chatter in case of the process of ball end milling of a curved flexible detail.

The paper presents method of the modal parameters identification based on the Particle Swarm Optimization (PSO) algorithm. The basic PSO algorithm is modified in order to achieve fast convergence and low estimation error of identified parameters values. The procedure of identification as well as algorithm modifications are presented and some simple examples for the SISO systems are provided. Results are compared with the results obtained for Least Square Complex Exponential (LSCE) method.

The milling process considered in the paper is a finishing milling of ship bow thruster. It is really challenging task because this kind of object can be prone to vibration, and due to certain conditions, the self-excited chatter vibration may appear [. One of the methods of the chatter reduction depends upon the selection of the spindle speed by matching it to the optimal phase shift between subsequent passes of the tool cutting edges [. However, in case of complex-shaped objects there may be different values of optimal spindle speeds for various points of the workpiece surface. The paper presents an example of implementation of the procedure of building the optimal spindle speed map [. In order to create the map for specific surface points, optimal spindle speed is obtained with the use of various modal analysis techniques. The map seems to be effective way for eliminating chatter in case of the process of milling flexible details.

The main goal of modern machining operations is to achieve increasingly better performance. High Speed Machining and/or High Performance Cutting, despite a lot of advantages, have also some drawbacks, for example, a possibility of losing stability and development of self-excited chatter vibration. This paper presents an approach of vibration surveillance during high speed milling with a use of active optimal control. Non-stationary model of the milling process is described with the use of convention of the discrete hybrid system, and the problem of optimal control at energy performance index is defined and resolved. Numerous computer simulations disclosed efficiency of the surveillance by the active optimal control of flexible workpiece vibration. The performed real experiments of the milling process support the meaning of the method of vibration surveillance.

Using critical point theory and some monotonicity results we consider the existence and multiplicity of solutions to nonlinear discrete boundary value problems represented as a nonlinear system Au= λf(u) with parameter λ > 0 and with matrix Abeing not necessarily positive definite. We provide applications for discrete version of the Emden–Fowler equation.

We consider by a dual variational method the existence of solutions to certain fourth order Dirichlet problems with nonlinearities corresponding to the derivatives of a sum of a convex and a concave function. The growth conditions are imposed only on the convex part.

The paper is devoted to vibration monitoring of rotating tools in modern milling machines. Dynamic analysis of slender ball-end milling process was performed and dynamics of the controlled structure was described. Instantaneous change in the spindle speed is applied in order to reduce vibration level. The method of vibration monitoring by means of spindle speed optimal-linear control was developed and implemented with success. Vibration monitoring during high speed milling was performed on the basis of results of computer simulation. These results were verified during experimental investigation on the Alcera Gambin 120CR milling machine.

Abstract: We prove an existence principle that would apply for elliptic problems with nonlinearity separating into a difference of derivatives of two convex functions in the case when the growth conditions are imposed only on the minuend term. We present abstract result and its application. We modify the so called dual variational method.

We consider the Dirichlet boundary value problem for higher order O.D.E. with nonlinearity being the sum of a derivative of a convex and of a concave function in case when no growth condition is imposed on the concave part.

AbstractWe provide the existence results for a nonlinear operator equationΛ*Φ′ (Λ𝑥) = 𝐹′(𝑥),in case 𝐹 – Φ is not necessarily convex. We introduce the dual variational method which is based on finding global minima of primal and dual action functionals on certain nonlinear subsets of their domains and on investigating relations between the minima obtained. The solution is a limit of a minimizng sequence whose existence and convergence are proved. The application for the non-convex Dirichlet problem with P.D.E. is given.