Approximate controllability for second order nonlinear evolution hemivariational inequalities

The goal of this paper is to study approximate controllability for control systems driven by abstract second order nonlinear evolution hemivariational inequalities in Hilbert spaces. First, the concept of a mild solution of our problem is defined by using the cosine operator theory and the generalized Clarke subdifferential. Next, the existence and the approximate controllability of mild solutions are formulated and proved by means of the fixed points strategy. Finally, an example is provided to illustrate our main results

Existence of solutions for implicit obstacle problems of fractional laplacian type involving set-valued operators

The paper is devoted to a new kind of implicit obstacle problem given by a fractional Laplacian-type operator and a set-valued term, which is described by a generalized gradient. An existence theorem for the considered implicit obstacle problem is established, using a surjectivity theorem for set-valued mappings, Kluge’s fixed point principle and nonsmooth analysis

Set-Valued Analysis

This Special Issue contains eight original papers with a high impact in various domains of set-valued analysis. Set-valued analysis has made remarkable progress in the last 70 years, enriching itself continuously with new concepts, important results, and special applications. Different problems arising in the theory of control, economics, game theory, decision making, nonlinear programming, biomathematics, and statistics have strengthened the theoretical base and the specific techniques of set-valued analysis. The consistency of its theoretical approach and the multitude of its applications have transformed set-valued analysis into a reference field of modern mathematics, which attracts an impressive number of researchers

Approximations and generalized Newton methods

We present approaches to (generalized) Newton methods in the framework of generalized equations $0\in f(x)+M(x)$, where $f$ is a function and $M$ is a multifunction. The Newton steps are defined by approximations $\hat f$ of $f$ and the solutions of $0\in \hat{f}(x)+M(x)$. We give a unified view of the local convergence analysis of such methods by connecting a certain type of approximation with the desired kind of convergence and different regularity conditions for $f+M$. Our paper is, on the one hand, thought as a survey of crucial parts of the topic, where we mainly use concepts and results of the monograph (Klatte and Kummer, Nonsmooth equations in optimization: regularity, calculus, methods and applications, Kluwer Academic Publishers, Dordrecht, 2002). On the other hand, we present original results and new features. They concern the extension of convergence results via Newton maps (Klatte and Kummer, Nonsmooth equations in optimization: regularity, calculus, methods and applications, Kluwer Academic Publishers, Dordrecht, 2002; Kummer in: Oettli, Pallaschke (eds) Advances in optimization, Springer, Berlin, 1992) from equations to generalized equations both for linear and nonlinear approximations $\hat f$, and relations between semi-smoothness, Newton maps and directional differentiability of $f$. We give a Kantorovich-type statement, valid for all sequences of Newton iterates under metric regularity, and recall and extend results on multivalued approximations for general inclusions $0\in F(x)$. Equations with continuous, non-Lipschitzian $f$ are considered, too

Fractional Differential Equations, Inclusions and Inequalities with Applications

During the last decade, there has been an increased interest in fractional differential equations, inclusions, and inequalities, as they play a fundamental role in the modeling of numerous phenomena, in particular, in physics, biomathematics, blood flow phenomena, ecology, environmental issues, viscoelasticity, aerodynamics, electrodynamics of complex medium, electrical circuits, electron-analytical chemistry, control theory, etc. This book presents collective works published in the recent Special Issue (SI) entitled "Fractional Differential Equation, Inclusions and Inequalities with Applications" of the journal Mathematics. This Special Issue presents recent developments in the theory of fractional differential equations and inequalities. Topics include but are not limited to the existence and uniqueness results for boundary value problems for different types of fractional differential equations, a variety of fractional inequalities, impulsive fractional differential equations, and applications in sciences and engineering

Art of Modeling in Contact Mechanics

International audienceIn this chapter, we will first address general issues of the art and craft of modeling-contents, concepts, methodology. Then, we will focus on modeling in contact mechanics, which will give the opportunity to discuss these issues in connection with non-smooth problems. It will be shown that the non-smooth character of the contact laws raises difficulties and specificities at every step of the modeling process. A wide overview will be given on the art of mod-eling in contact mechanics under its various aspects: contact laws, their mechanical basics, various scales, underlying concepts, mathematical analysis, solvers, identification of the constitutive parameters and validation of the models. Every point will be illustrated by one or several examples. 1 Modeling: the bases It would be ambitious to try to give a general definition of the concept either of a model itself or of model processing. Modeling relates to the general process of production of scientific knowledge and also to the scientific method itself. It could be deductive (from the general to the particular, as privileged by Aristotle) or inductive (making sense of a corpus of raw data). Descartes (38) saw in the scientific method an approach to be followed step by step to get to a truth. Modeling can be effectively regarded as a scientific method that proceeds step by step, but its objective is more modest: to give sense of an observation or an experiment, and above all to predict behaviors within the context of specific assumptions. This concept of " proceeding step by step " is fundamental in modeling. In this first section, we will examine the notion of model in the general context of mechanical systems