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TOPOLOGICAL METHODS
IN
NONLINEAR ANALYSIS
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| Vol. 37, No. 1 March 2011 |
TABLE OF CONTENTS
| Title and Author(s) |
Page |
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Localized singularities and the Conley index
Maria C. Carbinatto and Krzysztof P. Rybakowski
| ABSTRACT.
We establish some abstract convergence and Conley index continuation
principles for families of singularly perturbed semilinear parabolic
equations and apply them to reaction-diffusion equations with nonlinear
boundary conditions and localized large diffusion. This extends and refines
previous results of [Ro] and [ACR].
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1
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Competition systems with strong interaction on a subdomain
Elaine C. M. Crooks and E. Norman Dancer
| ABSTRACT.
We study the large-interaction limit of an elliptic system modelling
the steady states of two species $u$ and $v$ which compete to some
extent throughout a domain $\Omega$ but compete strongly on a subdomain
$A \subset \Omega$. In the strong-competition limit, $u$ and $v$ segregate
on $A$ but not necessarily on $\Omega \setminus A$. The limit problem is
a system on $\Omega \setminus A$ and a scalar equation on $A$ and in general
admits an interesting range of types of solution, not all of which can be
the strong-competition limit of coexistence states of the original system.
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37
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On global regular solutions to the Navier-Stokes equations in cylindrical domains
Wojciech M. Zajaczkowski
| ABSTRACT.
We consider the incompressible fluid motion described by the
Navier-Stokes equations in a cylindrical domain $\Omega\subset\R^3$ under
the slip boundary conditions. First we prove long time existence of regular
solutions such that $v\in W_2^{2,1}(\Omega\times(0,T))$,
$\nabla p\in L_2(\Omega\times(0,T))$, where $v$ is the velocity of the fluid
and $p$ the pressure. To show this we need smallness of
$\|v_{,x_3}(0)\|_{L_2(\Omega)}$ and $\|f_{,x_3}\|_{L_2(\Omega\times(0,T))}$,
where $f$ is the external force and $x_3$ is the axis along the cylinder.
The above smallness restrictions mean that the considered solution remains
close to the two-dimensional solution, which, as is well known, is regular.
Having $T$ sufficiently large and imposing some decay estimates on
$\|f(t)\|_{L_2(\Omega)}$ we continue the local solution step by step up to
the global one.
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55
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Positive solutions for a 2nth-order boundary value problem involving all derivatives of odd orders
Zhilin Yang and Donal O'Regan
ABSTRACT.
We are concerned with the existence, multiplicity and uniqueness
of positive solutions for the $2n$-order boundary value problem
\cases
(-1)^nu^{(2n)}=f(t,u,u',-u''',\ldots,
(-1)^{i-1}u^{(2i-1)},\ldots, (-1)^{n-1}u^{(2n-1)}),
u^{(2i)}(0)=u^{(2i+1)}(1)=0, \quad i=0,\ldots,n-1.
where $n\geq 2$ and $f\in C([0,1]\times \Bbb R_+^{n+1},\Bbb R_+)$
$(\Bbb R_+:=[0,\infty))$ depends on $u$ and all derivatives
of odd orders. Our main hypotheses on $f$ are formulated in terms of
the linear function $g(x):=x_1+2\sum_{i=2}^{n+1}x_i$. We use fixed
point index theory to establish our main results, based on a priori
estimates achieved by utilizing some integral identities and an
integral inequality. Finally, we apply our main results to establish
the existence, multiplicity and uniqueness of positive symmetric
solutions for a Lidostone problem involving an open question posed
by P. W. Eloe in 2000.
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87
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Existence of positive solutions for a semilinear elliptic system
Zhitao Zhang and Xiyou Cheng
| ABSTRACT.
In this paper, we are concerned with the existence of
(component-wise) positive solutions for a semilinear elliptic
system, where the nonlinear term is superlinear in one equation and
sublinear in the other equation. By constructing a cone $K_1 \times
K_2$ which is the Cartesian product of two cones in space
$C(\overline{\Omega})$ and computing the fixed point index in $K_1
\times K_2$, we establish the existence of positive solutions for
the system. It is remarkable that we deal with our problem on the
Cartesian product of two cones, in which the features of two
equations can be exploited better.
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103
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Differential inclusions with nonlocal conditions: existence results and topological properties of solution sets
John R. Graef, Johnny Henderson and Abdelghani Ouahab
ABSTRACT.
In this paper, we study the topological structure of solution sets
for the first-order differential inclusions with nonlocal
conditions:
y'(t) \in F(t,y(t)) &\text{a.e. } t\in [0,b],
y(0)+g(y)=y_0,
where $F\colon [0,b]\times\R^n\to{\Cal P}(\R^n)$ is a multivalued map.
Also, some geometric properties of solution sets, $R_{\delta}$,
$R_\delta$-contractibility and acyclicity, corresponding to
Aronszajn-Browder-Gupta type results, are obtained. Finally, we
present the existence of viable solutions of differential
inclusions with nonlocal conditions
and we investigate the topological properties of the set constituted
by these solutions.
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117
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Systems of first order inclusions on time scales
Marlene Frigon and Hugues Gilbert
| ABSTRACT.
This paper presents existence results for systems of first order inclusions on
time scales with an initial or a periodic boundary value condition.
The method of solution-tube is developed for this system.
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147
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Modified Swift-Hohenberg equation
Maria B. Kania
| ABSTRACT.
We consider the initial-boundary value problem for a modified
Swift-Hohenberg equation in space dimension $n\leq 7 $. Based
on the semigroup theory, we formulate this problem as an abstract
evolutionary equation with sectorial operator in the main part.
We show that the semigroup generated by this problem admits a global
attractor in the phase space $H^2(\Omega)\cap H^{1}_{0}(\Omega)$ and
characterize the contents of the attractor.
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165
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An extension of Leggett-Williams norm-type theorem for coincidences and its application
Aijun Yang
| ABSTRACT.
In this paper, several versions extension of
Leggett-Williams norm-type theorem for coincidences are given and
proved to obtain the positive solutions of the operator equation
$Mx=Nx$, where $M$ is a quasi-linear operator and $N$ is nonlinear.
Moreover, as an application, the existence of positive solutions for
multi-point boundary value problem with a $p$-Laplacian is obtained
by one of those theorems.
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177
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Best proximity points of cyclic $\varphi$-contractions in ordered metric spaces
Sh. Rezapour, M. Derafshpour and N. Shahzad
| ABSTRACT.
In this paper, we shall give some results about best proximity points
of cyclic $\varphi$-contractions in ordered metric spaces.
These results generalize some known results.
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193
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