Recursos de colección
Project Euclid (Hosted at Cornell University Library) (192.977 recursos)
Topological Methods in Nonlinear Analysis
Topological Methods in Nonlinear Analysis
Czapla, Małgorzata; Pawłucki, Wiesław
Let $R$ be a real closed field and let some o-minimal
structure extending $R$ be given. Let $F\colon X \rightrightarrows R^m$ be a definable
multivalued lower semicontinuous mapping with nonempty definably connected values defined on a
definable subset $X$ of $R^n$ of dimension $1$ ($X$ can be identified with a finite graph immersed
in $R^n$). Then $F$ admits a definable continuous selection.
López-Gómez, Julian; Rabinowitz, Paul H.
In [Nodal solutions for a class of degenerate
boundary value problems, Adv. Nonl. Studies 15 (2015), 253-288], a family of degenerate one dimensional boundary
value problems was studied and the existence of positive (and negative) solutions and solutions
that possess one interior node was shown for a range of values of a parameter, $\lambda$.
It was conjectured that there is a natural extension of these results giving solutions with
any prescribed number of interior nodes. This conjecture will be established here.
Glasner, Eli; Weiss, Benjamin
Three topics in dynamical systems are discussed.
First we deal with cascades and solve two open problems
concerning, respectively, product recurrence, and uniformly rigid actions.
Next we provide a new example that displays some unexpected properties
of strictly ergodic actions of non-amenable groups.
Jo, Jang Hyun; Lee, Jong Bum
Using averaging formulas, we compute the Lefschetz, Nielsen and
Reidemeister numbers of maps on infra-solvmanifolds modeled on Sol, and we study the
Jiang-type property for those infra-solvmanifolds.
Liu, Wenjun; Sun, Yun; Li, Gang
We consider a singular nonlocal viscoelastic problem with a nonlinear source term and a possible damping term.
We prove that if the initial data enter into the stable set, the solution exists globally and
decays to zero with a more general rate, and if the initial data
enter into the unstable set, the solution with nonpositive initial
energy as well as positive initial energy blows up in finite time.
These are achieved by using the potential well theory, the modified convexity method and the perturbed
energy method.
Xiao, Jian-Zhong; Wang, Zhi-Yong; Liu, Juan
A second order semilinear
neutral functional differential inclusion with nonlocal conditions
and multivalued impulse characteristics in a separable Banach space
is considered. By developing appropriate computing techniques for
the Hausdorff product measures of noncompactness, the topological
structure of $C^1$-solution sets is established; and some
interesting discussion is offered when the multivalued nonlinearity
of the inclusion is a weakly upper semicontinuous map satisfying a
condition expressed in terms of the Hausdorff measure.
Zhang, Jian; Tang, Xianhua; Zhang, Wen
In this paper, we study the following
Hamiltonian elliptic system with gradient term and critical growth:
\begin{equation*}
\begin{cases}
-\epsilon^{2}\Delta \psi +\epsilon b\cdot \nabla \psi +\psi=K(x)f(|\eta|)\varphi+W(x)|\eta|^{2^*-2}\varphi &\text{in } \mathbb{R}^{N},\\
-\epsilon^{2}\Delta \varphi -\epsilon b\cdot \nabla \varphi +\varphi=K(x)f(|\eta|)\psi+W(x)|\eta|^{2^*-2}\psi &\text{in } \mathbb{R}^{N},
\end{cases}
\end{equation*}
where $\eta=(\psi,\varphi)\colon \mathbb{R}^{N}\rightarrow\mathbb{R}^{2}$, $K, W\in C(\mathbb{R}^{N}, \mathbb{R})$, $\epsilon$ is a small positive parameter and $b$ is a constant vector.
We require that the nonlinear potentials $K$ and $W$ have
at least one global maximum. Combining this with other suitable assumptions on $f$,
we prove the existence, exponential decay and concentration phenomena of semiclassical ground state
solutions for all sufficiently small $\epsilon> 0$
Castro, Alfonso; Ventura, Ivan
Building on the construction of least energy sign-changing
solutions to variational semilinear elliptic boundary value
problems introduced in [A. Castro, J. Cossio and J.M. Neuberger,
Sign changing solutions for a superlinear
Dirichlet problem, Rocky Mountain J. Math. 27 (1997), 1041--1053],
we prove the existence of a
solution with augmented Morse index at least three when a
sublevel of the corresponding action functional has nontrivial
topology. We provide examples where the set of least energy sign
changing solutions is disconnected, hence has nontrivial topology.
Cassani, Daniele; do Ó, João Marcos; Zhang, Jianjun
We are concerned with the following equation:
$$
-\varepsilon^2\Delta u+V(x)u=f(u),\quad u(x)> 0\quad \mbox{in } \mathbb{R}^2.
$$
By a variational approach, we construct a solution $u_\varepsilon$ which concentrates,
as $\varepsilon \to 0$, around arbitrarily given isolated local
minima of the confining potential $V$: here the nonlinearity $f$ has a quite general Moser's
critical growth, as in particular we do not require the monotonicity
of $f(s)/s$ nor the Ambrosetti-Rabinowitz condition.
Han, Sang-Eon; Yao, Wei
Many properties or formulas related to the ordinary Euler characteristics of topological spaces are well developed under many mathematical
operands, e.g. the product property, fibration property, homotopy axiom, wedge sum property,
inclusion-exclusion principle [E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966],
etc. Unlike these properties, the digital version of the Euler characteristic has its own feature.
Among the above properties, we prove that the digital version of the Euler characteristic has
the wedge sum property which is of the same type as that for the ordinary Euler characteristic.
This property plays an important role in fixed point theory for digital images, digital homotopy
theory, digital geometry and...
Liu, Zhisu; Luo, Chaoliang
We study the existence of positive ground state solutions for the
nonlinear Kirchhoff type equation
$$
\begin{cases}
\displaystyle
-\bigg(a+b\int_{\mathbb R^3}|\nabla u|^2\bigg)\Delta {u}+V(x)u =f(u) & \mbox{in }\mathbb R^3, \\
u\in H^1(\mathbb R^3), \quad u> 0 & \mbox{in } \mathbb R^3,
\end{cases}
$$
where $a,b> 0$ are constants, $f\in C(\mathbb R,\mathbb R)$ has general critical
growth. We generalize a Berestycki-Lions theorem about the
critical case of Schrödinger equation to Kirchhoff type
equation via variational methods. Moreover, some subcritical works
on Kirchhoff type equation are extended to the current critical
case.
Bonotto, Everaldo M.; Gimenes, Luciene P.; Souto, Ginnara M.
Recursive properties on impulsive semidynamical systems are
considered. We obtain results about almost periodic motions and
asymptotically almost periodic motions in the context of impulsive
systems. The concept of asymptotic almost periodic motions is
introduced via time reparametrizations. We also present asymptotic
properties for impulsive systems and for their associated discrete
systems.
Barrientos, Pablo G.; Ghane, Fatemeh H.; Malicet, Dominique; Sarizadeh, Aliasghar
Every quasi-attractor of an iterated function system (IFS) of
continuous functions on a first-countable Hausdorff topological
space is renderable by the probabilistic chaos game. By contrast,
we prove that the backward minimality is a necessary condition to
get the deterministic chaos game. As a consequence, we obtain that
an IFS of homeomorphisms of the circle is renderable by the
deterministic chaos game if and only if it is forward and backward
minimal. This result provides examples of attractors (a forward
but no backward minimal IFS on the circle) that are not renderable
by the deterministic chaos game. We also prove that every
well-fibred quasi-attractor is renderable by the deterministic
chaos game...
Ambrosio, Vincenzo
By using variational methods, we investigate the existence of
$T$-periodic solutions to
\begin{equation*}
\begin{cases}
[(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{in } (0,T)^{N}, \\
u(x+Te_{i})=u(x) &\mbox{for all } x \in \mathbb{R}^N, \ i=1, \dots, N,
\end{cases}
\end{equation*}
where $s\in (0,1)$, $N> 2s$, $T> 0$, $m\geq 0$ and $f$ is a
continuous function, $T$-periodic in the first variable, verifying
the Ambrosetti-Rabinowitz condition, with a polynomial growth at
rate $p\in (1, ({N+2s})/({N-2s}))$.
Papageorgiou, Nikolaos S.; Smyrlis, George
We study a semilinear Robin problem with an indefinite and
unbounded potential and a reaction term which asymptotically at
$\pm \infty $ is resonant with respect to any nonprincipal
nonnegative eigenvalue. We prove two multiplicity theorems
producing three and four nontrivial solutions respectively. Our
approach uses variational methods based on the critical point
theory, truncation and perturbation techniques, and Morse theory
(critical groups).
Artigue, Alfonso
It is a paper about models for isolated sets and the construction
of special hyperbolic Lyapunov functions. We prove that after a
suitable surgery every isolated set is the intersection of an
attractor and a repeller. We give linear models for attractors and
repellers. With these tools we construct hyperbolic Lyapunov
functions and metrics around an isolated set whose values along
the orbits are catenary curves. Applications are given to
expansive flows and homeomorphisms, obtaining, among other things,
a hyperbolic metric on local cross sections for an arbitrary
expansive flow on a compact metric space.
Guimarães, Mateus Balbino; da Silva Rodrigues, Rodrigo
We study a $(p,q)$-Laplacian system of Kirchhoff type equations
with weight and nonlinearity involving a $(p,q)$-superlinear term,
in which $p$ may be different from $q$, and with critical
Caffarelli-Kohn-Nirenberg exponent. Using the Mountain Pass
Theorem, we obtain a nontrivial solution to the problem.
Xia, Shengxiang; Yin, Yanmin
We utilize the methods of computational topology to the
database of optical flow created by Roth and Black from range images, and demonstrate
a qualitative topological analysis of spaces of $3 \times 3, 5 \times 5$ and
$7 \times 7$ optical flow patches. We experimentally prove that there exist
subspaces of the spaces of the three sizes high-contrast patches that are
topologically equivalent to a circle and a three circles model, respectively.
The Klein bottle is the quotient space described as the square $[0,1] \times [0,1]$
with sides identified by the relations $(0, y)\sim (1, y)$ for $y\in [0, 1]$ and
$(x, 0) \sim (1-x, 1)$ for $...
Bugajewska, Daria; Bugajewski, Dariusz; Kasprzak, Piotr; Maćkowiak, Piotr
The main goal of this paper is to give an answer to the main
problem of the theory of nonautonomous superposition operators
acting in the space of functions of bounded variation in the
sense of Jordan. Namely, we give necessary and sufficient
conditions which guarantee that nonautonomous superposition
operators map that space into itself and are locally bounded.
Moreover, special attention is drawn to nonautonomous
superposition operators generated by locally bounded mappings as
well as to superposition operators generated by functions with
separable variables.
Benedetti, Irene; Väth, Martin
An existence result for an abstract nonlocal boundary value
problem $x'\in A(t)x(t)+F(t,x(t))$, $Lx\in B(x)$, is given, where
$A(t)$ determines a linear evolution operator, $L$ is linear, and
$F$ and $B$ are multivalued. To avoid compactness conditions, the
weak topology is employed. The result applies also in nonreflexive
spaces under a hypothesis concerning the De Blasi measure of
noncompactness. Even in the case of initial value problems,
the required condition is essentially milder than previously known
results.