Continuous dependence and optimal control of a dynamic elastic-viscoplastic contact problem with non-monotone boundary conditions

<p style='text-indent:20px;'>In this paper, we consider continuous dependence and optimal control of a dynamic elastic-viscoplastic contact model with Clarke subdifferential boundary conditions. Since the constitutive law of elastic-viscoplastic materials has an implicit expression of the stress field, the weak form of the model is an evolutionary hemivariational inequality coupled with an integral equation. By providing some equivalent weak formulations, we prove the continuous dependence of the solution on external forces and initial conditions in the weak topologies. Finally, the existence of optimal solutions to a boundary optimal control problem is established.</p>

Topology

This chapter discusses topological spaces and associated concepts, including first‐ and second‐countability, compactness, and separation properties. Weak topologies are defined. Product spaces, the product topology, and the Tychonoff theorem are treated and also ideas of embedding, compactification, and metrization.

On the lattice of weak topologies on the bicyclic monoid with adjoined zero

A Hausdorff topology τ on the bicyclic monoid with adjoined zero C0 is called weak if it is contained in the coarsest inverse semigroup topology on C0. We show that the lattice W of all weak shift-continuous topologies on C0 is isomorphic to the lattice SIF1×SIF1 where SIF1 is the set of all shift-invariant filters on ω with an attached element 1 endowed with the following partial order: F≤G if and only if G=1 or F⊂G. Also, we investigate cardinal characteristics of the lattice W. In particular, we prove that W contains an antichain of cardinality 2c and a well-ordered chain of cardinality c. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type t.

Elements of Geometry of Balls in Banach Spaces

One of the subjects of functional analysis is classification of Banach spaces depending on various properties of the unit ball. The need of such considerations comes from a number of applications to problems of mathematical analysis. The list of subjects contains: differential calculus in normed spaces, approximation theory, weak topologies and reflexivity, general theory of convexity and convex functions, metric fixed point theory, and others. The aim of this book is to present basic facts from this field. It is addressed to advanced undergraduate and graduate students interested in the subject. For some it may result in further interest, a continuation and deepening of their study of the subject. It may be also useful for instructors running courses on functional analysis, supervising diploma theses or essays on various levels.

Fractional differential inclusions of Hilfer type under weak topologies in Banach spaces

In this paper, we present some results concerning the existence of weak solutions for some functional Hilfer and Hadamard fractional differential inclusions. The Mönch’s fixed point theorem and the concept of measure of weak noncompactness are the main tools used to carry out our results.