TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS

TABLE OF CONTENTS

	Title and Author(s)	Page
	Classification of diffeomorphisms of S$^4$ induced by queternionic Riccati equations with periodic coefficients Henryk Żołądek ABSTRACT. The monodromy maps for the quaternionic Riccati equations with periodic coefficients $\dot{z}=zp(t)z+q(t)z+zr(t)+s(t)$ in $\HP^{1}$ are quternionic M\"{o}bius transformations. We prove that, like in the case of automorphisms of $\CP^{1}$, the quaternionic homografies are divided into three classes: hyperbolic, elliptic and parabolic.	205
	Constants of Motion for Non-Differentiable Quantum Variational Problems Jacky Cresson, Gastao S. F. Frederico and Delfim F. M. Torres ABSTRACT. We extend the DuBois-Reymond necessary optimality condition and Noether's symmetry theorem to the scale relativity theory setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus of variations with functionals defined on sets of non-differentiable functions, as well as more general non-differentiable problems of optimal control. As an application we obtain constants of motion for some linear and nonlinear variants of the Schr\"{o}dinger equation.	217
	Global regular solutions to the Navier-Stokes equations in an axially symmetric domain Wojciech M. Zajączkowski ABSTRACT. We prove the existence of global regular solutions to the Navier-Stokes equations in an axially symmetric domain in $\R^3$ and with boundary slip conditions. We assume that initial angular component of velocity and angular component of the external force and angular derivatives of the cylindrical components of initial velocity and of the external force are sufficiently small in corresponding norms. Then there exists a solution such that velocity belongs to $W_{5/2}^{2,1}(\Omega^T)$ and gradient of pressure to $L_{5/2}(\Omega^T)$, and we do not have restrictions on $T$.	233
	Periodic solutions for a neutral differential equation with variable parameter Bo Du, Jianxin Zhao and Weigao Ge ABSTRACT. By means of Mawhin's continuation theorem, we present some sufficient conditions which guarantee the existence of at least one $T$-periodic solution for a first-order neutral equation with variable parameter. The interest is that the coefficient $c$ is not a constant, which is different from the corresponding ones of past work.	275
	A decomposition formula for equivariant stable homotopy classes Wacław Marzantowicz and Carlos Prieto ABSTRACT. For any compact Lie group $G$, we give a decomposition of the group $\{X,Y\}_G^k$ of (unpointed) stable $G$-homotopy classes as a direct sum of subgroups of fixed orbit types. This is done by interpreting the $G$-homotopy classes in terms of the generalized fixed-point transfer and making use of conormal maps.	285
	Abelianized Obstruction for fixed points of fiber-preserving maps of Surface bundles Daciberg Lima Goncalves Dirceu Penteado and João Peres Vieira ABSTRACT. Let $f\colon M \to M$ be a fiber-preserving map where $S\to M \to B$ is a bundle and $S$ is a closed surface. We study the abelianized obstruction, which is a cohomology class in dimension 2, to deform $f$ to a fixed point free map by a fiber-preserving homotopy. The vanishing of this obstruction is only a necessary condition in order to have such deformation, but in some cases it is sufficient. We describe this obstruction and we prove that the vanishing of this class is equivalent to the existence of solution of a system of equations over a certain group ring with coefficients given by Fox derivatives.	293
	Retracting ball onto sphere in $BC_0(R)$ Łukasz Piasecki ABSTRACT. In infinite dimensional Banach spaces the unit sphere is a lipschitzian retract of the unit ball. We use the space of continuous functions vanishing at a point to provide an example of such retraction having relatively small Lipschitz constant.	307
	Fixed point results for generalized $\varphi$-contraction on a set with two metrics Tunde Petra Petru and Monica Boriceanu ABSTRACT. The aim of this paper is to present fixed point theorems for multivalued operators $ T\colon X \to P(X)$, on a nonempty set $X$ with two metrics $d$ and $\varrho$, satisfying the following generalized $\varphi$-contraction condition: $$ H_{\varrho}(T(x),T(y))\leq \varphi(M^T(x,y)),\quad \text{for every } x,y \in X, $$ where $$ \multline M^T(x,y):=\max \{ \varrho(x,y),D_{\varrho}(x,T(x)),D_{\varrho}(y,T(y)),\\ 2^{-1} [ D_{\varrho}(x,T(y))+D_{\varrho}(y,T(x)) ]\}. \endmultline $$	315
	The admissibility of topological degree in Herz-type Besov and Triebel-Lizorkin spaces Jingshi Xu ABSTRACT. In this paper, an admissibility for topological degree of Herz spaces, Herz-type Besov and Triebel-Lizorkin spaces is given.	327
	Function Bases For Topological Vector Spaces Yilmaz Yilmaz ABSTRACT. Our main interest in this work is to characterize certain operator spaces acting on some important vector-valued function spaces such as $(V_{a}) _{c_{0}}^{a\in{\Bbb A}}$, by introducing a new kind basis notion for general Topological vector spaces. Where ${\Bbb A}$ is an infinite set, each $V_{a}$ is a Banach space and $( V_{a}) _{c_{0}}^{a\in{\Bbb A}}$ is the linear space of all functions $x\colon{\Bbb A} \rightarrow\bigcup V_{a}$ such that, for each $\varepsilon>0$, the set $\{ a\in{\Bbb A}:\Vert x_{a}\Vert >\varepsilon\} $ is finite or empty. This is especially important for the vector-valued sequence spaces $( V_{i}) _{c_{0}}^{i\in{\Bbb N}}$ because of its fundamental place in the theory of the operator spaces (see, for example, [12]).	335
	Not finitely but countably Hopf-equivalent clopen sets in a Cantor minimal system Hisatoshi Yuasa ABSTRACT. We investigate topological Hopf-equivalences in the Morse substitution system. In particular, we give an example of not finitely but countably Hopf-equivalent clopen sets in a Cantor minimal system.	355
	Computer-assisted proof of a periodic solution in a nonlinear feedback DDE Mikołaj Zalewski ABSTRACT. In this paper, we rigorously prove the existence of a non-trivial periodic orbit for the nonlinear DDE: $x'(t) = - K \sin(x(t-1))$ for $K=1.6$. We show that the equations for the Fourier coefficients have a solution by computing the local Brouwer degree. This degree can be computed by using a homotopy, and its validity can be proved by checking a finite number of inequalities. Checking these inequalities is done by a computer program.	373