Title and Author(s) 
Page 

Conley index and homology index braids in singular perturbation problems without uniqueness of solutions
Maria C. Carbinatto and Krzysztof P. Rybakowski
ABSTRACT.
We define the concept of a Conley index and a homology index braid class for
ordinary differential equations of the form
\dot x= F_1(x),
\leqno(\EE)
where $\M$ is a $C^2$manifold and $F_1$ is the principal part of
a {\it continuous vector field} on $\M$.
This allows us to extend our previously obtained results
from \cite{\rfa{CR9}} on singularly perturbed systems
of ordinary differential equations
\aligned
\eps\dot y&=f(y,x,\eps),\\
\dot x&=h(y,x,\eps)
\endaligned
\leqno(\EE_\eps)
on $Y\times \M$, where $Y$ is a finite dimensional Banach space
and $\M$ is a $C^2$manifold, to the case where the vector field
in $(E_\eps)$ is continuous, but not necessarily locally Lipschitzian.


1


Ground state solutions for the nonlinear KleinGordonMaxwell equations
Antonio Azzollini and Alessio Pomponio
ABSTRACT.
In this paper we prove the existence of a ground state solution
for the nonlinear KleinGordonMaxwell equations in the electrostatic case.


33


Traveling Front Solutions in Nonlinear Diffusion Degenerate FisherKPP and Nagumo Equations via Conley index
Fatiha El Adnani and Hamad Talibi Alaoui
ABSTRACT.
Existence of one dimensional traveling wave solutions
$u( x,t)$ $:=\phi ( xct) $
at the stationary equilibria, for the nonlinear
degenerate reactiondiffusion equation
$u_{t}=[K( u)u_{x}]_{x}+F( u) $ is studied, where $K$ is the density
coefficient and $F$ is the reactive part. We use the Conley index theory to
show that there is a traveling front solutions connecting the critical
points of the reactiondiffusion equations. We consider the nonlinear
degenerate generalized FisherKPP and Nagumo equations.


43


Multiple solutions for the mean curvature equation
Sebastiaan Lorca and Marcelo Montenegro
ABSTRACT.
We perturb the mean curvature operator and
find multiple critical points of functionals that are not even.
As a consequence we find infinitely many solutions for a quasilinear
elliptic equation. The generality of our results are also reflected
in the relaxed hypotheses related to the behavior of the functions
around zero and at infinity.


61


Incompressibility and global inversion
Eduardo Cabral Balreira
ABSTRACT.
Given a local diffeomorphism $f\colon \Rn\to\Rn$, we consider
certain incompressibility conditions on the parallelepiped
$Df(x)([0,1]^n)$ which imply that the preimage of an affine
subspace is nonempty and has trivial homotopy groups. These conditions
are then used to establish criteria for $f$ to be globally invertible,
generalizing in all dimensions the previous results of M. Sabatini.


69


Bifurcations of random differential equations with bounded noise on surfaces
Ale Jan Homburg and Todd R. Young
ABSTRACT.
In random differential equations with bounded noise minimal forward
invariant (MFI) sets play a central role since they support stationary
measures. We study the stability and possible bifurcations of MFI sets.
In dimensions 1 and 2 we classify all minimal forward
invariant sets and their codimension one bifurcations in bounded noise
random differential equations.


77


Index at infinity and bifurcations of twice degenerate vector fields
Alexander Krasnosel'skii
ABSTRACT.
We present a method to study twice degenerate at infinity
asymptotically linear vector fields, i.e. the fields with
degenerate principal linear parts and next order bounded terms.
The main features of the method
are sharp asymptotic expansions
for projections of nonlinearities onto the kernel of the linear part.
The method includes theorems in abstract Banach spaces,
the expansions which are the main assumptions of these abstract theorems,
and lemmas on the exact form of the expansions for generic
functional nonlinearities with saturation.
The method leads to several new results on solvability and bifurcations
for various classic BVPs.
If the leading terms in the expansions
are of order $0$, then solvability conditions (and conditions for
the index at infinity to be nonzero) coincide with LandesmanLazer
conditions, traditional for the BVP theory.
If the terms of
order $0$ vanish (the LandesmanLazer conditions
fail), then it is necessary to determine and to take into account
nonlinearities that are smaller at infinity.
The presented method uses such nonlinearities and makes it possible
to obtain the expansions with the leading terms of arbitrary
possible orders.
The method is applicable if the linear part has simple degeneration,
if the corresponding eigenfunction vanishes,
and if the small nonlinearities decrease at infinity sufficiently fast.
The Dirichlet BVP for a second order ODE is the main model example,
scalar and vector cases being considered separately. Other applications
(the Dirichlet problem for the Laplace PDE and the Neumann
problem for the second order ODE) are given rather schematically.


99


Existence of noncollision periodic solutions for second order singular dynamical systems
Shuqing Liang
ABSTRACT.
In this paper, we study the existence of noncollision
periodic solutions for the second order singular dynamical systems.
We consider the systems where the potential have a repulsive or
attractive type behavior near the singularity. The proof is based on
Schauder's fixed point theorem involving a new type of cone. The
socalled strong force condition is not needed and the nonlinearity
could have sign changing behavior. We allow that the Green function
is nonnegative, so the critical case for the repulsive case is
covered. Recent results in the literature are generalized and
improved.


127


On the existence of periodic solutions for a class of nonautonomous differential delay equations
Rong Cheng, Junxiang Xu and Dongfebg Zhang
ABSTRACT.
This paper considers the existence of periodic solutions for a
class of nonautonomous differential delay equations
x'(t)=\sum_{i=1}^{n1}f(t,x(ti\tau)), \leqno{(*)}
where $\tau>0$ is a given
constant. It is shown that under some conditions on $f$ and by
using symplectic transformations, Floquet theory and some results
in critical point theory, the existence of single periodic
solution of the differential delay equation $(*)$ is obtained.
These results generalize previous results on the cases that the
equations are autonomous.


139


Inertial manifolds for a singular perturbation of the viscous CahnHilliardGurtin equation
Ahmed Bonfoh, Maurizio Grasselli and Alain Miranville
ABSTRACT.
We consider a singular
perturbation of the generalized viscous CahnHilliard equation
based on constitutive equations introduced by
M. E. Gurtin and we establish the existence of a family
of inertial manifolds which is continuous with respect
to the perturbation parameter $\varepsilon>0$ as $\varepsilon$
goes to 0. In a recent paper, we proved a similar result
for the singular perturbation of the standard viscous CahnHilliard equation,
applying a construction due to X. Mora and J. Sol\`aMorales
for equations involving linear selfadjoint operators only.
Here we extend the result to the singularly perturbed CahnHilliardGurtin
equation which contains a nonselfadjoint operator. Our method can be
applied to a larger class of nonlinear dynamical systems.


155


Twin Positive Solutions for Singular Nonlinear Elliptic Equations
Jianqing Chen, Nikolaos S. Papageorgiou and Eugenio M. Rocha
ABSTRACT.
For a bounded domain $\vZ\subseteq\bkR^N$ with a $C^2$boundary, we prove the existence of an ordered pair of smooth positive strong solutions for the nonlinear Dirichlet problem
\Delta_p\, \vx(\vz) = \beta(\vz)\vx(\vz)^{\eta}+f(\vz,\vx(\vz))
\quad \text{a.e. on } \vZ
\text{ with } \vx\in\Wz,
which exhibits the combined effects of a singular term ($\eta\geq 0$) and a $(p1)$linear term $f(\vz,\vx)$ near
$+\infty$, by using a combination of variational methods, with upperlower solutions and with suitable truncation techniques.


187
