1.
Generation theory for semigroups of holomorphic mappings
in Banach spaces - Reich, Simeon; Shoikhet, David
We study nonlinear semigroups of holomorphic mappings in Banach spaces and their infinitesimal generators. Using resolvents, we characterize, in particular, bounded holomorphic generators on bounded convex domains and obtain an analog of the Hille exponential formula. We then apply our results to the null point theory of semi-plus complete vector fields. We study the structure of null point sets and the spectral characteristics of null points, as well as their existence and uniqueness. A global version of
the implicit function theorem and a discussion of some open problems are
also included.
2.
Iterative solution of unstable variational inequalities on
approximately given sets - Alber, Y. I.; Kartsatos, A. G.; Litsyn, E.
The convergence and the stability of the iterative regularization method for solving variational inequalities with bounded nonsmooth properly
monotone (i.e., degenerate) operators in Banach spaces are studied.
All the items of the inequality (i.e., the operator $A$ , the right hand
side $f$ and the set of constraints $\Omega$ ) are to be perturbed.
The connection between the parameters of regularization and perturbations
which guarantee strong convergence of approximate solutions is established.
In contrast to previous publications by Bruck, Reich and the first author, we do not suppose
here that the approximating sequence is a priori bounded. Therefore the present
results are new even for operator equations in...
4.
Inertial manifolds and stabilization of nonlinear beam equations
with Balakrishnan-Taylor damping - You, Yuncheng
In this paper we study a hinged, extensible, and elastic nonlinear beam equation
with structural damping and Balakrishnan-Taylor damping with the full exponent $2(n+\beta)+ 1$ . This strongly nonlinear equation, initially proposed by Balakrishnan
and Taylor in 1989, is a very general and useful model for large aerospace
structures. In this work, the existence of global solutions and the existence
of absorbing sets in the energy space are proved. For this equation, the feature
is that the exponential rate of the absorbing property is not a global constant, but
which is uniform for the family of trajectories starting from any given bounded set in the
state space....
5.
The generalized Conley index and multiple solutions of semilinear elliptic problems - Dancer, E. N.; Du, Yihong
We establish some framework so that the generalized Conley
index can be easily used to study the multiple solution problem of semilinear elliptic
boundary value problems. Both the parabolic flow and the gradient
flow are used. Some examples are given to compare our approach here with
other well-known methods. Our abstract results with parabolic flows may
have applications to parabolic problems as well.
6.
An Ambrosetti-Prodi-type problem for an elliptic system of
equations via monotone iteration method and Leray-Schauder degree theory - Filho, D. C. de Morais
In this paper we employ the Monotone Iteration Method and the Leray-Schauder
Degree Theory to study an $\mathbb{R}^2$ -parametrized system of elliptic
equations. We obtain a curve dividing the plane into two regions. Depending
on which region the parameter is, the system will or will not have solutions.
This is an Ambrosetti-Prodi-type problem for a system of equations.
8.
Attractors of semigroups associated with nonlinear systems for
diffusive phase separation - Kenmochi, Nobuyuki
We consider a model for diffusive phase transitions, for
instance, the component separation in a binary mixture. Our model is
described by two functions, the absolutete temperature $\theta :=\theta(t,x)$ and the order parameter $w := w(t,x)$ , which are governed by a
system of two nonlinear parabolic PDEs. The order parameter
$w$ is constrained to have double obstacles $\sigma_* \le w \le \sigma^*$ (i.e., $\sigma_*$ and $\sigma^*$ are the threshold values of $w$ ). The objective of this paper is to discuss the semigroup $\{S(t)\}$ associated with the phase separation model, and construct its global attractor.
9.
Carleson embeddings - Heiming, Helmut J.
In this paper we discuss several operator ideal properties for so
called Carleson embeddings of tent spaces into specific $L^q(\mu)$ -spaces, where $\mu$ is a Carleson measure on the complex unit disc. Characterizing absolutely
$q$ -summing, absolutely continuous and
$q$ -integral Carleson embeddings in terms of the underlying measure is our main topic. The presented results extend and integrate results especially known for composition operators on Hardy spaces as well as embedding theorems for function spaces of similar kind.
10.
The exponential stability of a coupled hyperbolic/parabolic
system arising in structural acoustics - Avalos, George
We show here the uniform stabilization of a coupled system of hyperbolic and parabolic PDEs which describes a particular fluid/structure interaction system. This system has the wave equation, which is satisfied on the
interior of a bounded domain $\Omega$ , coupled to a paraboliclike
beam equation holding on $\partial\Omega$ , and wherein the coupling is accomplished through velocity terms on the boundary. Our result is an analog of a recent result by Lasiecka and Triggiani which shows the exponential stability of the wave
equation via Neumann feedback control, and like that work, depends upon a
trace regularity estimate for solutions of hyperbolic equations.
11.
Interior point control and observation for the wave equation - Khapalov, A. Y.
This paper is concerned with the approximate and exact controllability
properties of the wave equation with interior point controls entering
via the concentrated force, the velocity of the displacement and the moment.
The emphasis is given to the moving point controls and their dual
observations whose advantages and disadvantages, versus the static ones,
are analyzed with respect to the space dimension, the duration of the
control time interval and the function spaces involved.
12.
Invariance dun convexe fermé par un semi-groupe associé à une forme non-linéaire - Barthélemy, Louise
The invariance of a closed and convex set under a semigroup $S(t)$ associated with a nonlinear form is investigated. Other properties of increase and domination
of the semigroup $S(t)$ are also derived. Examples are also given to demonstrate the power of the theoretical results.
13.
Global solutions of semilinear heat equations in Hilbert spaces - Iancu, G. Mihai; Wong, M. W.
The existence, uniqueness, regularity and asymptotic
behavior of global solutions of semilinear heat equations in Hilbert spaces
are studied by developing new results in the theory of one-parameter strongly
continuous semigroups of bounded linear operators. Applications to special
semilinear heat equations in $L^2(\mathbb{R}^n)$ governed by pseudo-differential
operators are given.
14.
Existence of multiple critical points for an asymptotically
quadratic functional with applications - Li, Shujie; Su, Jiabao
Morse theory for isolated critical points at infinity is used for
the existence of multiple critical points for an asymptotically quadratic
functional. Applications are also given for the existence of multiple
nontrivial periodic solutions of asymptotically Hamiltonian systems.
15.
Bifurcation of the equivariant minimal interfaces in a
hydromechanics problem - Borisovich, A. Y.; Marzantowicz, W.
In this work we study a deformation of the minimal interface of two fluids in a vertical tube under the presence of gravitation.
We show that a symmetry of the base of tube let us to apply
a method developed earlier by the first author and based
on the Crandall-Rabinowitz bifurcation theorem.
Using the natural symmetry of the corresponding variational problem
defined by a symmetry of region and restricting the functional
to spaces of invariant functions we show the existence of bifurcation,
and describe its local picture,
for interfaces parametrized by the square and disc.
16.
Embedding functions and their role in interpolation theory - Pustylnik, Evgeniy
The embedding functions of an intermediate space $A$ into a Banach couple
$(A_0,A_1)$ are defined as its embedding constants into the couples
$(\frac{1}{\alpha}A_0,\frac{1}{\beta}A_1)$ , $\forall\alpha,\beta > 0$ . Using these functions,
we study properties and interrelations of different intermediate spaces, give a new description of all real
interpolation spaces, and generalize the concept of weak-type interpolation to
any Banach couple to obtain new interpolation theorems.
17.
Stability of coupled systems - Khodja, Farid Ammar; Benabdallah, Assia; Teniou, Djamel
The exponential and asymptotic stability are studied for certain coupled systems
involving unbounded linear operators and linear infinitesimal semigroup generators.
Examples demonstrating the theory are also given from the field of partial differential equations.
18.
On the Mann and Ishikawa iteration processes - Haiyun, Zhou; Yuting, Jia
It is shown that a result of Chidume, involving the strong convergence of the Mann iteration process for continuous strongly accretive
operators, is actually a corollary to a result by Nevanlinna and Reich. It is then
shown that the Nevanlinna and Reich result can be extended to the case of an Ishikawa iteration process.
19.
Nonlinear semigroups and the existence and stability of solutions
of semilinear nonautonomous evolution equations - Aulbach, Bernd; Minh, Nguyen Van
This paper is concerned with the existence and stability of solutions
of a class of semilinear nonautonomous evolution equations. A procedure
is discussed which associates to each nonautonomous equation the
so-called evolution semigroup of (possibly nonlinear) operators.
Sufficient conditions for the existence and stability of solutions
and the existence of periodic oscillations are given in terms of the
accretiveness of the corresponding infinitesimal generator. Furthermore,
through the existence of integral manifolds for abstract evolutionary
processes we obtain a reduction principle for stability questions of mild
solutions. The results are applied to a class of partial functional
differential equations.
20.
On a local degree for a class of multi-valued vector fields in
infinite dimensional Banach spaces - Benkafadar, N. M.; Gelman, B. D.
This paper is devoted to the development of a local degree for multi-valued vector
fields of the form $f - F$ . Here, $f$ is a single-valued,
proper, nonlinear, Fredholm, $C^1$ -mapping
of index zero and $F$ is a multi-valued upper semicontinuous, admissible, compact
mapping with compact images. The mappings $f$ and $F$ are acting from a subset of a Banach space $E$ into another Banach space $E_1$ . This local degree is used to
investigate the existence of solutions of a certain class of operator
inclusions.
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