Evolutionary Quasi-Variational-Hemivariational Inequalities I: Existence and Optimal Control

We study a nonlinear evolutionary quasi–variational–hemivariational inequality (in short, (QVHVI)) involving a set-valued pseudo-monotone map. The central idea of our approach consists of introducing a parametric variational problem that defines a variational selection associated with (QVHVI). We prove the solvability of the parametric variational problem by employing a surjectivity theorem for the sum of operators, combined with Minty’s formulation and techniques from the nonsmooth analysis. Then, an existence theorem for (QVHVI) is established by using Kluge’s fixed point theorem for set-valued operators. As an application, an abstract optimal control problem for the (QVHVI) is investigated. We prove the existence of solutions for the optimal control problem and the weak sequential compactness of the solution set via the Weierstrass minimization theorem and the Kuratowski-type continuity properties.

On Limit Sets of Monotone Maps on Dendroids

Let X be a dendrite, f : X → X be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that ω-limit set ω(x, f ) of any point x ∈ X has the next properties: (1)\omega (x,f) \subseteq \overline {Per(f)} , where Per( f ) is the set of periodic points of f ;(2)ω(x, f ) is either a periodic orbit or a minimal Cantor set.In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that (3)\Omega (f) = \overline {Per(f)} , where Ω( f ) is the set of non-wandering points of f.The aim of this note is to show that the above results (1) – (3) do not hold for monotone maps on dendroids.

Relaxation methods for optimal control problems

We consider a nonlinear optimal control problem with dynamics described by a differential inclusion involving a maximal monotone map [Formula: see text]. We do not assume that [Formula: see text], incorporating in this way systems with unilateral constraints in our framework. We present two relaxation methods. The first one is an outgrowth of the reduction method from the existence theory, while the second method uses Young measures. We show that the two relaxation methods are equivalent and admissible.

A strong convergence theorem for maximal monotone operators in Banach spaces with applications

An algorithm is constructed to approximate a zero of a maximal monotone operator in a uniformly convex anduniformly smooth real Banach space. The sequence of the algorithm is proved to converge strongly to a zeroof the maximal monotone map. In the case where the Banach space is a real Hilbert space, our theorem com-plements the celebrated proximal point algorithm of Martinet and Rockafellar. Furthermore, our convergencetheorem is applied to approximate a solution of a Hammerstein integral equation in our general setting. Finally,numerical experiments are presented to illustrate the convergence of our algorithm.

Invariant Sets for Monotone Local Dendrite Maps

Let [Formula: see text] be a local dendrite and let [Formula: see text] be a monotone map. Denote by [Formula: see text], RR[Formula: see text], UR[Formula: see text], [Formula: see text] the set of periodic (resp., regularly recurrent, uniformly recurrent, recurrent) points and [Formula: see text] the union of all [Formula: see text]-limit sets of [Formula: see text]. We show that if [Formula: see text] is nonempty, then (i) [Formula: see text]. (ii) R[Formula: see text] if and only if every cut point is a periodic point. If [Formula: see text] is empty, then (iii) [Formula: see text]. (iv) R[Formula: see text] if and only if [Formula: see text] is a circle and [Formula: see text] is topologically conjugate to an irrational rotation of the unit circle [Formula: see text]. On the other hand, we prove that [Formula: see text] has no Li–Yorke pair. Moreover, we show that the family of all [Formula: see text]-limit sets of [Formula: see text] is closed with respect to the Hausdorff metric.

Well-founded coalgebras, revisited

Theoretical models of recursion schemes have been well studied under the names well-founded coalgebras, recursive coalgebras, corecursive algebras and Elgot algebras. Much of this work focuses on conditions ensuring unique or canonical solutions, e.g. when the coalgebra is well founded.If the coalgebra is not well founded, then there can be multiple solutions. The standard semantics of recursive programs gives a particular solution, typically the least fixpoint of a certain monotone map on a domain whose least element is the totally undefined function; but this solution may not be the desired one. We have recently proposed programming language constructs to allow the specification of alternative solutions and methods to compute them. We have implemented these new constructs as an extension of OCaml.In this paper, we prove some theoretical results characterizing well-founded coalgebras, along with several examples for which this extension is useful. We also give several examples that are not well founded but still have a desired solution. In each case, the function would diverge under the standard semantics of recursion, but can be specified and computed with the programming language constructs we have proposed.

A Fixed Point Theorem for Monotone Maps and Its Applications to Nonlinear Matrix Equations

By using the fixed point theorem for monotone maps in a normal cone, we prove a uniqueness theorem for the positive definite solution of the matrix equationX=Q+A⁎f(X)A, wherefis a monotone map on the set of positive definite matrices. Then we apply the uniqueness theorem to a special equationX=kQ+A⁎(X^-C)qAand prove that the equation has a unique positive definite solution whenQ^≥Candk>1and0<q<1. For this equation the basic fixed point iteration is discussed. Numerical examples show that the iterative method is feasible and effective.

Fuzzy -Continuous Posets

The aim of this paper is to generalize fuzzy continuous posets. The concept of fuzzy subset system on fuzzy posets is introduced; some elementary definitions such as fuzzy -continuous posets and fuzzy -algebraic posets are given. Furthermore, we try to find some natural classes of fuzzy -continuous maps under which the images of such fuzzy algebraic structures can be preserved; we also think about fuzzy -continuous closure operators in alternative ways. An extension theorem is presented for extending a fuzzy monotone map defined on the -compact elements to a fuzzy -continuous map defined on the whole set.