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TOPOLOGICAL METHODS
IN
NONLINEAR ANALYSIS
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TABLE OF CONTENTS
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Rate of convergence of global attractors of some perturbed reaction-diffusion problems
José M. Arrieta, Flank D. M. Bezerra and Alexandre N. Carvalho
| ABSTRACT.
In this paper we treat the problem of~the rate of~convergence of~ attractors of~dynamical systems for some autonomous semilinear parabolic problems. We consider a prototype problem, where the diffusion $a_0(\,\cdot\,)$ of~a reaction-diffusion equation in a bounded domain $\Omega$ is perturbed to~$a_\eps(\,\cdot\,)$. We show that the equilibria and the local unstable manifolds of~the perturbed problem are at a distance given by the order of~$\|a_\eps-a_0\|_\infty$. Moreover, the perturbed nonlinear semigroups are at a distance $\|a_\eps-a_0\|_\infty^\theta$ with $\theta<1$ but arbitrarily close to 1.
Nevertheless, we can only prove that the distance of~attractors is of~order $\|a_\eps-a_0\|_\infty^\beta$
for some $\beta<1$, which depends on some other parameters of~the problem and may be significantly smaller than $1$.
We also show how this technique can be applied to other more complicated problems.
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229
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Solutions to some singular nonlinear boundary value problems
Beata Medak, Alexey A. Tret'yakov and Henryk ¯o³¹dek
| ABSTRACT.
We apply the so-called $p$-regularity theory to prove the existence of
solutions to two nonlinear boundary value problems: an equation of rod
bending and some nonlinear Laplace equation.
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255
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Dimension of attractors and invariant sets of damped wave equations in unbounded domains
Martino Prizzi
ABSTRACT.
Under fairly general assumptions, we prove that every compact invariant set $\Cal I$ of the semiflow generated
by the semilinear damped wave equation
u_{tt}+\alpha u_t+\beta(x)u-\Delta u&\, =f(x,u), (t,x)\in[0,+\infty\mathclose[\times\Omega,
u&\,=0,&\quad &(t,x)\in[0,+\infty\mathclose [\times\partial\Omega,
in $H^1_0(\Omega)\times L^2(\Omega)$ has finite Hausdorff and
fractal dimension. Here $\Omega$ is a~regular, possibly unbounded,
domain in~$\R^3$ and $f(x,u)$ is a~nonlinearity of critical
growth. The nonlinearity $f(x,u)$ needs not to satisfy any
dissipativeness assumption and the invariant subset $\Cal I$
needs not to be an~attractor. If $f(x,u)$ is dissipative and
$\Cal I$ is the global attractor, we give an~explicit bound on
the Hausdorff and fractal dimension of $\Cal I$ in~terms of
the structure parameters of the equation.
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267
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A Hartman-Nagumo type condition for a class of contractible domains
Pablo Amster and Julian Haddad
| ABSTRACT.
We generalize an existence result on second order
systems with a nonlinear term satisfying
the so-called Hartman-Nagumo condition.
The generalization is based on the use of Gauss second fundamental form and continuation techniques.
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287
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Infinite many positive solutions for nonlinear first-order BVPs with integral boundary conditions on time scales
Yongkun Li and Lijuan Sun
ABSTRACT.
In this paper, we investigate the existence of infinite many positive solutions for the nonlinear
first-order BVP with integral boundary conditions
x^{\Delta}(t)+p(t)x^{\sigma}(t)=f(t,x^{\sigma}(t)), t\in (0,T)_{\mathbb{T}},
\dis x(0)-\beta x^{\sigma}(T)=\alpha\int_{0}^{\sigma(T)}x^{\sigma}(s)\Delta g(s),
where $x^{\sigma}=x\circ\sigma$, $f\colon [0,T]_{\mathbb{T}}\times\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$
is continuous, $p$ is regressive and rd-continuous, $\alpha,\beta\geq0$,
$g\colon [0,T]_{\mathbb{T}}\rightarrow \mathbb{R}$ is a nondecreasing function.
By using the fixed-point index theory and a new fixed point
theorem in a cone, we provide sufficient conditions for the existence of
infinite many positive solutions to the above boundary value problem on time scale $\mathbb{T}$.
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305
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Generic properties of critical points of the boundary mean curvature
Anna Maria Micheletti and Angela Pistoia
| ABSTRACT.
Given a bounded domain $\Omega\subset\rr^N$ of class $C^k$ with $k\ge3$, we prove that for a generic deformation $I+\psi$, with $\psi$ small enough, all the critical points of the mean curvature of the boundary of the domain $(I+\psi)\Omega$ are non degenerate.
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323
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Analytic robustness of parameter-dependent perturbations of difference equations
Luis Barreira and Claudia Valls
| ABSTRACT.
We establish the robustness of nonuniform exponential dichotomies under sufficiently small analytic parameter-dependent perturbations. We also show that the stable and unstable subspaces of the exponential dichotomies depend analytically on the parameter.
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335
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Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem
Isabella Ianni
| ABSTRACT.
We consider the Schr\"odinger-Poisson-Slater (SPS) system in~$\R^3$ and
a~nonlocal SPS type equation in balls of $\mathbb R^3$ with Dirichlet boundary conditions.
We show that for every $k\in\mathbb N$ each problem considered admits a nodal radially symmetric solution
which changes sign exactly $k$ times in~the~radial variable.
Moreover, when the domain is the ball of $\mathbb R^3$ we obtain the existence
of~radial global solutions for the associated nonlocal parabolic problem having $k+1$ nodal regions at every time.
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365
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Notes on circadian rhythm
Claudio Saccon and Robert E. L. Turner
| ABSTRACT.
We discuss a class of models arising in the study of circadian rhythm and the properties of the matrix equations providing the bifurcation points for a wide parameter class. In particular, we prove that the five
dimensional system studied in the cited work of Gonze, Halloy, and Goldbeter
can have only a simple Hopf bifurcation.
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387
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On vector fields having properties of Reeb fields
Bogus³aw Hajduk and Rafa³ Walczak
| ABSTRACT.
We study constructions of vector fields
with properties which are characteristic to Reeb vector fields
of contact forms. In particular, we prove that all closed oriented
odd-dimensional manifold have geodesible vector fields.
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401
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Weak Local Nash Equilibrium
Carlos Biasi and Thais Monis
| ABSTRACT.
In this paper, we consider a concept of local Nash equilibrium for
non-cooperative games - the so-called weak local Nash equilibrium.
We prove its existence for a significantly more general class of
sets of strategies than compact convex sets. The theorems on
existence of the weak local equilibrium presented here are
applications of Brouwer and Lefschetz fixed point theorems.
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409
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Syndetic proximality and scrambled sets
T. K. Subrahmonian Moothathu and Piotr Oprocha
| ABSTRACT.
This paper is a systematic study about the syndetically proximal relation and the possible existence of syndetically scrambled sets for the dynamics of continuous self-maps of compact metric spaces. Especially we consider various classes of transitive subshifts, interval maps, and topologically Anosov maps. We also present many constructions and examples.
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421
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