Various eigenvalue and range results are given for perturbations of <mml:math alttext="$m$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>-accretive and maximal monotone operators. The eigenvalue results improve and extend some recent results by Guan and Kartsatos, while the range theorem gives an affirmative answer to a recent problem of Kartsatos.

Zhou  Haiyun

Athanassios G. Kartsatos

Eigenvalues and ranges for perturbations of nonlinear accretive
and monotone operators in Banach spaces

The integral wavelet transform is defined in weighted Sobolev
spaces, in which some properties of the transform as well as its
asymptotical behaviour for small dilation parameter are studied.

Nguyen Minh Chuong

Ta Ngoc Tri

The integral wavelet transform in weighted Sobolev spaces

The Czochralski method of the industrial production of a silicon single crystal consists of pulling up the single crystal from the silicon melt. The flow of the melt during the production is called the Czochralski flow. The mathematical description of the flow consists of a coupled system of six P.D.E. in cylindrical coordinates containing Navier-Stokes equations (with the stream function), heat convection-conduction equations, convection-diffusion equation for oxygen impurity and an equation describing magnetic field effect.This paper deals with the analysis of the system in the form used
for numerical simulation. The weak formulation is derived and the existence of the weak solution to the stationary and the evolution problem is proved.

Jan  Franc

Modelling of the Czochralski flow

We investigate here the properties of extremal solutions for
semilinear elliptic equation <mml:math alttext="$-\Delta u=\lambda f(u)$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x2212;</mml:mo><mml:mi>&#x394;</mml:mi><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>&#x3BB;</mml:mi><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> posed on a bounded smooth domain of <mml:math alttext="${\mathbb R}^n$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>&#x211D;</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math> with Dirichlet boundary condition and with <mml:math alttext="$f$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> exploding at a finite positive value <mml:math alttext="$a$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi></mml:math>.

Lamia Ben Chaabane

On the extremal solutions of semilinear elliptic problems

The problem <mml:math alttext="$\dot{x} = f(t,x),x(-\infty)= x(+\infty)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>&#729;</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mtext>&#8201;</mml:mtext><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>&#8722;</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>+</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, where <mml:math alttext="$x(\pm\infty):=\lim_{t\rightarrow\pm\infty}x(t) \in\mathbb{R}^{n}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>&#177;</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mo>lim</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#8594;</mml:mo><mml:mo>&#177;</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow></mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8712;</mml:mo><mml:msup><mml:mi>&#8477;</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math>, is considered. Some existence results for this
problem are established using the fixed point method and
topological degree theory.

Cezar  Avramescu

Existence problems for homoclinic solutions

A technique, based on the investigations of the images of maps,
for obtaining fixed-point and coincidence results in a new class of maps and domains is described. In particular, we show that the problem concerning the existence of fixed points of expansive set-valued maps and inner set-valued maps on not necessarily convex or compact sets in Hausdorff topological vector spaces has a solution. As a consequence, we prove a new intersection theorem concerning not necessarily convex or compact sets and its applications. We also give new coincidence and section theorems for maps defined on not necessarily convex sets in Hausdorff topological vector spaces. Examples and counterexamples show a fundamental difference between our results and the well-known ones.

Kazimierz  W&#322;odarczyk

Dorota  Klim

Fixed-point and coincidence theorems for set-valued maps with nonconvex or noncompact domains in topological vector spaces

On the basis of <mml:math alttext="$G$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math>-convergence we prove an averaging result for nonlinear abstract parabolic equations, the operator coefficient of which is a stationary stochastic process.

G.  Bruno

A.  Pankov

T.  Pankova

Time averaging for random nonlinear abstract parabolic equations

We propose another extension of Orlicz-Sobolev spaces to metric spaces based on the concepts of the <mml:math alttext="$\Phi$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#934;</mml:mi></mml:math>-modulus and <mml:math alttext="$\Phi$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#934;</mml:mi></mml:math>-capacity. The resulting space <mml:math alttext="$N_{\Phi}^{1}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>N</mml:mi><mml:mi>&#934;</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> is a Banach space. The relationship between <mml:math alttext="$N_{\Phi}^{1}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>N</mml:mi><mml:mi>&#934;</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> and <mml:math alttext="$M_{\Phi}^{1}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>M</mml:mi><mml:mi>&#934;</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> (the first extension defined in A&#239;ssaoui (2002)) is studied. We also explore and compare different definitions of capacities and give a criterion under which <mml:math alttext="$N_{\Phi}^{1}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>N</mml:mi><mml:mi>&#934;</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> is strictly smaller than the Orlicz space <mml:math alttext="$\mathbf{L}_{\Phi}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="bold">L</mml:mi><mml:mi>&#934;</mml:mi></mml:msub></mml:math>.

Noureddine  A&#239;ssaoui

Another extension of Orlicz-Sobolev spaces to metric spaces

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.

Simeon Reich

David Shoikhet

Generation theory for semigroups of holomorphic mappings
in Banach spaces

We establish several results concerning the asymptotic behavior of random infinite products of generic sequences of affine uniformly continuous operators on bounded closed convex subsets of a Banach space. In addition to weak ergodic theorems we also obtain convergence to a unique common fixed point and more generally, to an affine retraction.

S. Reich

A. J. Zaslavski

Convergence of generic infinite products of affine operators

We obtain results of existence of weak solutions in the Hopf sense
of the initial-boundary value problem for the generalized Navier-Stokes equations containing perturbations of retarded type. The degree theory for maps <mml:math alttext="$A - g$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:mo>&#8722;</mml:mo><mml:mi>g</mml:mi></mml:mrow></mml:math>, where <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> is invertible and <mml:math alttext="$g$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>g</mml:mi></mml:math> is <mml:math alttext="$\mathcal{A}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#119964;</mml:mi></mml:math>-condensing, is used.

V. T. Dmitrienko

V. G. Zvyagin

The topological degree method for equations of the Navier-Stokes type

We give several examples of Douglas algebras that do not have any maximal subalgebra. We find a condition on these algebras that
guarantees that some do not have any minimal superalgebra. We also
show that if <mml:math alttext="$A$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>
is the only maximal subalgebra of a Douglas
algebra <mml:math alttext="$B$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math>, then the algebra <mml:math alttext="$A$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> does 
not have any maximal subalgebra.

Carroll Guillory

Douglas algebras without maximal subalgebra and without
           minimal superalgebra

About this journal

Abstract and Applied Analysis publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.

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Chief Editor, Dr Wong, is an associate professor at Nanyang Technological University, Singapore. Her research interests include differential equations, difference equations, integral equations, and numerical mathematics.

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Abstract and Applied Analysis - Table of contents

title

Abstract and Applied Analysis publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.

Abstract and Applied Analysis