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TOPOLOGICAL METHODS

IN

NONLINEAR ANALYSIS



Vol. 19, No. 2           June 2002



TABLE OF CONTENTS

Title and Author(s) Page
item Recent Results on Thin Domain Problems II
M. Prizzi and K. P. Rybakowski
ABSTRACT. In this paper we survey some recent results on parabolic equations on curved squeezed domains. More specifically, consider the family of semilinear Neumann boundary value problems
(\text{\rm E}_\eps) u_t=\Delta u + f(u), t>0, x\in \Omega_\eps, \partial_{\nu_\eps}u=0, t>0, x\in \partial\Omega_\eps
where, for \epsilon > 0 small, the set \Omega_\epsilon is a thin domain in Rl, possibly with holes, which collapses, as \eps\to0^+, onto a (curved) k-dimensional submanifold M of Rl. If f is dissipative, then equation (\text{\rm E}_\eps) has a global attractor {\Cal A}_\eps. We identify a ``limit'' equation for the family (\text{\rm E}_\eps), establish an upper semicontinuity result for the family {\Cal A}_\eps and prove an inertial manifold theorem in case M is a k-sphere.
199
item Some Results for Jumping Nonlinearities
E. N. Dancer
ABSTRACT. We discuss the calculation of critical groups for jumping nonlinearities as the resonance set is crossed. In addition, we produce a counter-example showing that even ``generically'' the resonance set is more complicated than previously thought.
221
item Note on the Deck Transformations Group and the Monodromy Group
H. Zoladek
ABSTRACT. For a ramified covering between Riemann surfaces the groups Deck of deck transformations and Mon of monodromy permutations are introduced. We associate with them groups of automorphisms of certain extensions of function fields. We study relations between these objects.
237
item A Remark about Homogeneous Polynomial Maps
A. Tret'yakov and H. Zoladek
ABSTRACT. We consider homogeneous polynomial maps F: Rn ->Rn of degree p. We classify the pairs (p,n) for which there exists a surjective and non-proper such map and when the right inverse to F exists but is unbounded.
257
item Periodic Solutions of Ordinary Differential Equations with Bounded Nonlinearities
J. R. Ward, Jr.
ABSTRACT. In this article we discuss the existence and non-existence of forced T-periodic solutions to ordinary differential equations of the form u'' + g(u) = e(t). The results concern equations with bounded nonlinear terms g satisfying g(s) > 0 (or g(s) < 0) for all real numbers s, and g(\pm \infty )=0. Variational and topological methods are employed.
275
item Structure of Steady States for Streater's Energy-Transport Models of Gravitating Particles
P. Biler and T. Nadzieja
ABSTRACT. Energy-transport models introduced by R. F. Streater describe the evolution of the density and temperature of a cloud of self-gravitating particles. We study the existence of steady states with prescribed mass and energy for these models.
283
item Infinitely Many Solutions of Superlinear Fourth Order Boundary Value Problems
B. P. Rynne
ABSTRACT. We consider the boundary value problem
u^{(4)}(x)=g(u(x)) + p(x,u^{(0)}(x),\dots,u^{(3)}(x)), x \in (0,1),\cr u(0) =u(1)=u^{(b)}(0)=u^{(b)}(1)=0
where:
  1. g: R -> R is continuous and satisfies
    \lim_{|\xi| \tends \infty} g(\xi)/\xi =\infty
    (g is superlinear as |\xi| \tends \infty,
  2. p :[0,1] x R4 -> R is continuous and satisfies
    |p(x,\xi_0,\xi_1,\xi_2,\xi_3)|\le C + \frac{1}{4} |\xi_0|, x \in [0,1],\ (\xi_0,\xi_1,\xi_2,\xi_3) \in \R^4,
    for some C > 0,
  3. either b = 1 or b = 2.
We obtain solutions having specified nodal properties. In particular, the problem has infinitely many solutions.
303
item Free Boundary Problem for a Viscous Heat-Conducting Flow with Surface Tension
E. Zadrzynska
ABSTRACT. In the paper the equations describing the motion of a drop of a~viscous heat-conducting capillary fluid bounded by a free surface are examined. Assuming that the viscosity coefficients, the coefficient of heat-conductivity, the pressue and the specific heat at constant volume of the fluid depend on its density and temperature we prove the existence of a~global in time solution which is close to a constant state for any moment of time.
313
item Lefschetz Fixed Point Theorem for Acyclic Maps with Multiplicity
F. v. Haeseler, H.-O. Peitgen and G. Skordev
ABSTRACT. The Lefschetz fixed point theorem for multivalued upper semi-continuous acyclic maps with multiplicity with respect to (w.r.t.) a given field F of F-simplicial spaces is proved.
339
item Upper and Lower Solutions for Problems with Singular Sign Changing Nonlinearities and with Nonlinear Boundary Data
D. O'Regan
ABSTRACT. An upper and lower solution approach is presented for singular boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign.
375
item Minimal Displacement of Random Variables under Lipschitz Random Maps
I. Beg
ABSTRACT. Let (\Omega,\Sigma) be a measurable space and X be a separable metric space. It is shown that for measurable maps \zeta,\eta \colon \Omega \rightarrow X, if a random map T\colon\Omega \times X\rightarrow X satisfies
d(T(\omega ,\zeta(\omega)),T(\omega,\eta (\omega)))\leq \alpha d(\zeta (\omega ),\eta (\omega ))+\gamma
then
\inf\{d(\xi (\omega ),T(\omega ,\xi (\omega )))\}\leq \gamma/ (1-\alpha),
where \gamma > 0, \alpha in (0,1) and inf is taken over all measurable maps \xi \colon \Omega \rightarrow X Several consequences of this result are also obtained.
391



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