
TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS

Vol. 40, No. 1 September 2012 
TABLE OF CONTENTS
Title and Author(s) 
Page 

On convergence and compactness in parabolic problems with globally large diffusion and nonlinear boundary conditions
Maria C. Carbinatto and Krzysztof P. Rybakowski
ABSTRACT.
We establish some abstract convergence and compactness results for families of singularly perturbed semilinear parabolic equations and apply them to reactiondiffusion equations with nonlinear boundary conditions and large diffusion. This refines some previous results of \cite{\rfa{Wi}}.


1


Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the
PoincareBirkhoff theorem
Alessandro Fonda, Marco Sabatini and Fabio Zanolin
ABSTRACT.
We prove the existence of periodic solutions for a planar nonautonomous Hamiltonian system
which is a small perturbation of an autonomous system, in the presence of a nonisochronous
period annulus. To this aim we use the Poincar\'eBirkhoff fixed point theorem, even if
the boundaries of the annulus are neither assumed to be invariant for the Poincar\'e map,
nor to be starshaped. As a consequence, we show how to deal with the problem of bifurcation
of subharmonic solutions near a given nondegenerate periodic solution. In this framework,
we only need little regularity assumptions, and we do not need to introduce any Melnikov
type functions.


29


Nontrivial solutions of fourthorder singular boundary value problems with signchanging nonlinear terms
Kemei Zhang
ABSTRACT.
In this paper, the fourthorder singular boundary value problem (BVP)
\aligned &u^{(4)}(t)=h(t)f(u(t)),\ \ t\in(0,1),
&u(0)=u(1)=u'(0)=u'(1)=0\endaligned
is considered under some conditions concerning the first
characteristic value corresponding to the relevant
linear operator, where $h$ is allowed to be singular at both $t=0$
and $t=1$. In particular, $f\colon (\infty,\infty)\rightarrow
(\infty,\infty)$ may be a signchanging and unbounded function
from below, and it is not also necessary to exist a control
function for $f$ from below. The existence results of nontrivial
solutions and positivenegative solutions are given by
the topological degree theory and the fixed point index theory, respectively.


53


Planar nonautonomous polynomial equations III. Zeros of the vector field
Pawel Wilczynski
ABSTRACT.
We give a few sufficient conditions for the existence of periodic solutions
of the equation $\dot{z}=\sum_{j=0}^n a_j(t)z^j$. We prove the existence
of one up to $n$ periodic solutions and heteroclinic ones.


71


What an infranilmanifold endomorphism really should be . . .
Karel Dekimpe
ABSTRACT.
Infranilmanifold endomorphisms were introduced in the late sixties. They
play a very crucial role in dynamics, especially when studying expanding
maps and Anosov diffeomorphisms.
However, in this note we will explain that the two main results in this area
are based on a false result and that although we can repair one of these
two theorems, there remains doubt on the correctness of the other one.
Moreover, we will also show that the notion of an
infranilmanifold endomorphism itself has not always been interpreted
in the same way.
Finally, we define a slightly more general concept of the notion of an
infranilmanifold endomorphism and explain why this is really the right
concept to work with.


111


Z_2homology of weak (p2)faceless ppseudomanifolds may be computed in O(n) time
Mateusz Juda and Marian Mrozek
ABSTRACT.
We consider the class of weak $(p2)$faceless $p$pseudomanifolds
with bounded boundaries and coboundaries.
We show that in this class the Betti numbers with $\ZZ_2$ coefficients
may be computed in time $O(n)$ and the $\ZZ_2$ homology generators
in time $O(nm)$ where $n$ denotes
the cardinality of the $p$pseudomanifold on input and
$m$ is the number of homology generators.


137


Central points and measures and dense subsets of compact metric spaces
Piotr Niemiec
ABSTRACT.
For every nonempty compact convex subset $K$ of a normed linear space a (unique) point $c_K \in K$, called the generalized
Chebyshev center, is distinguished. It is shown that $c_K$ is a common fixed point for the isometry group of the metric
space $K$. With use of the generalized Chebyshev centers, the central measure $\mu_X$ of an arbitrary compact metric space
$X$ is defined. For a large class of compact metric spaces, including the interval $[0,1]$ and all compact metric groups,
another `central' measure is distinguished, which turns out to coincide with the Lebesgue measure and the Haar one
for the interval and a compact metric group, respectively. An idea of distinguishing infinitely many points forming
a dense subset of an arbitrary compact metric space is also presented.


161


On the Kuratowski measure of noncompactness for duality mappings
George Dinca
ABSTRACT.
Let $(X,\Vert \cdot\Vert ) $ be an infinite dimensional real Banach
space having a Fr\'{e}chet differentiable norm and
$\varphi\colon \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ be a gauge function.
Denote by $J_{\varphi}\colon X\rightarrow X^{\ast}$ the duality mapping on $X$
corresponding to $\varphi$.
Then, for the Kuratowski measure of noncompactness of $J_{\varphi}$, the
following estimate holds:
\alpha( J_{\varphi}) \geq
\sup\bigg\{ \frac{\varphi(r) }{r}\ \bigg\ r>0\bigg\} .
In particular, for $\!\Delta_{p}\colon W_{0}^{1,p}( \Omega)\!\rightarrow\!\!
W^{1,p^{\prime}}( \Omega) $, $1\!<\!p<\!\infty$, ${1}/{p}+\!{1}/{p^{\prime}}\nobreak=\nobreak1$,
viewed as duality mapping on $W_{0}^{1,p}(\Omega)$,
corresponding to the gauge function $\varphi(t)=t^{p1}$, one has
\alpha( \Delta_{p}) =\cases
1 & \text{for }p=2,
\infty & \text{for }p\in( 1,2) \cup( 2,\infty).
\endcases


181


Extensions of theorems of Rattray and Makeev
Pavle V. M. Blagojevic and Roman Karasev
ABSTRACT.
We consider extensions of the Rattray theorem and two Makeev's theorems, showing that they hold for several maps, measures, or functions simultaneously,
when we consider orthonormal $k$frames in $\R^n$ instead of orthonormal bases (full frames).
We also present new results on simultaneous partition of several measures into parts by $k$ mutually orthogonal hyperplanes.
In the case $k=2$ we relate the Rattray and Makeev type results with the well known embedding problem for projective spaces.


189


Around Ulam's question on retractions
Chaoha Phichet, Goebel Kazimierz and Termwuttipong Imchit
ABSTRACT.
It is known that the unit ball in infinitely dimensional Hilbert space can be retracted onto its boundary via a lipschitzian mapping. The magnitude of Lipschitz constant is only roughly estimated. The note contains a number of observations connected to this result and opens some new problems.


215

