Titles and Abstracts
Oscillation criterea for second order matrix dynamic equations
on a time scale
Lynn Erbe and Allan Peterson
University of Nebraska-Lincoln, USA
Abstract: We obtain oscillation criteria for a
second order self-adjoint
matrix equation on a time scale in terms of the coefficient matrices
and
the graininess function. Our proofs use the associated Riccati equation.
We illustrate our results with several examples.
----------------------------------------------------------------------------------------------------
Spectrum of Positively Homogeneous Operators and Applications
Wenying Feng
Departments of Computer Science and Mathematics, Trent University
Peterborough, Ontario, Canada K9J 7B8
Abstract: The spectral theory for nonlinear operators
has been extensively studied by many authors. After the theory of Furi,
Martelli and Vignoli, a new definition was inroduced by the author. Later,
the work was generated to semilinear operators.
In this paper, some results on the relationship between the
eigenvalues and the spectrum of a positively homogeneous operators were
obtained. Applying the results, we prove a theorem that gives a condition
for a compact, positive operator to have a positive eigenvalue and eigenvector.
The theorem can be used in the study of a second order differential equations
with a three point boundary value conditions that has been studied recently
by some authors.
-------------------------------------------------------------------------------------------------------------------------------
On the Solvability of Implicit Complementarity problem
and Implicit Variational Inequalities
A Unified Approach and Implicit
Projected Dynamical System
G. Isac
Department of Mathematics
Royal Military College of Canada
P. O. Box 17000 STN Forces
Kingston, Ontario, Canada K7K 7B4
Abstract: In this first part of this paper we
will present a unified approach of the study of Implicit Complementary
Problems and Implicit Variational Inequalities. This study is based on
the concept of ''Exceptional Family of Elements'' for a function. This
concept is obtained in this case using a kind of implicit Leray-Schauder
alternative.
In the second part of this paper we will present a study of
solutions of Implicit Complementary Problems and Implicit Variational
Inequalities, from the dynamical point of view. This study is obtained
using an implicit global projected dynamical system. This paper will be
finished by comments and open problems.
-------------------------------------------------------------------------------------------------------------------------------
Singular Nonlinear Boundary Value Problems with Multiple
Positive Solutions
John V. Baxley and Philip T. Carroll
Department of Mathematics
Wake Forest University
Winston-Salem, NC 27109 USA
Abstract: We extend recent results of Henderson
and Thompson, Baxley and Haywood, and Graef, Qian, and Yang, which provided
conditions on the nonlinear function f(y) in order that the boundary value
problem (-1)^n y^{(2n)} = f(y), y^{(2k)}
(0) = 0, y^{(2k)} (1) = 0, for k=0,... ,n-1 have multiple symmetric
positive solutions. Since such solutions must satisfy y^{(2k+1)}
(1/2) =0, for k = 0, ..., n-1, we consider the more general problem
L(y) = f(y), where L is the nth iterate of the Sturm-Liouville operator
\dis -\frac{1}{w} (p y')', with boundary conditions y^{(2k)} (0)= 0, y^{(2k+1)}
(b) = 0, b>0, for k = 0,...,n-1. The conditions we obtain allow singular
behavior in the operator L at x=0. In the case w \equiv p \equiv 1, b=1/2,
our conclusions reduce to the earlier results mentioned above. Previous
work using the Leggett-Williams fixed point theorem or a fixed point theorem
of Krasnosel'ski\u{i} has used properties of relevant Green's functions.
Here we use a refined version of the same theorem of Krasnosel'ski\u{i},
but need no use of Green's functions to obtain the necessary estimates.
-------------------------------------------------------------------------------------------------------------------------------
Permanence under strong competition is possible
Santiago Cano-Casanova
Departamento de Matem\'atica Aplicada y Computaci\'on
Universidad Pontificia Comillas de Madrid
28015-Madrid, Spain
and
Juli\'an L\'opez-G\'omez
Departamento de Matem\'atica Aplicada
Universidad Complutense de Madrid
28040-Madrid, Spain
Abstract: We analyze the limiting behavior of
the positive solutions of a general class of sublinear elliptic weighted
mixed boundary value problems as the amplitude of the positive part of
the lower order terms of the differential operator blows up to infinity.
The main result establishes that the positive solutions approximate zero
within the support of the positive part of the potential, whereas
they stabilize to the positive solution of a certain elliptic mixed boundary
value problem on its complement. Further, we use this result for deriving
some general principles in competing species dynamics. Precisely, we show
that in the presence of a refuge region two competing species must coexist
if their reproduction rates are sufficiently large, independently of the
strength of the competition. It should be emphasized that these results
allow measuring how large the reproduction rates should be for being permanent,
providing us, simultaneously, with the limiting behavior of each of the
species separately. Basically, when the pressure from the competitor grows
the tackled species concentrates within its refuge. These results are substantial
extensions of some pioneer results found by J. L\'opez-G\'omez in \cite[Section
4]{Lo95}. The main ingredients in deriving the above results are the continuous
dependence of the principal eigenvalue with respect to a general class
of perturbations of the domain around its Dirichlet boundary, recently
proved in \cite{CL99}, and the continuous dependence of the positive solutions
of the sublinear problem, coming from \cite{CL00}.
\begin{thebibliography}{99}
\bibitem{Lo95} J. L\'opez-G\'omez, {\it Permanence under strong competition},
World Scientific Series in Applied Analysis 4 (1995), 473-488.
\bibitem{CL99} S. Cano-Casanova and J. L\'opez-G\'omez, {\it Properties
of the principal eigenvalues of a general class of non-classical mixed
boundary value problems}, J. Diff. Eqns. In Press.
\bibitem{CL00} S. Cano-Casanova and J. L\'opez-G\'omez, {\it Varying
domains in a general class of sublinear elliptic problems}. Submited.
------------------------------------------------------------------------------------------------------------------------------
An Existence Result on Positive Solutions for a Class
of Semilinear Elliptic Systems
Ratnasingham Shivaji
W.L.Giles Distinguished Professor
Dept. of Mathematics and Statistics
Mississippi State University
Mississippi State, MS 39762,USA.
Abstract: We consider the system
\begin{equation*} \Delta u+\lambda f(v)&=&0,
\quad x\in \Omega\\ \Delta v+\lambda g(u)&=&0, \quad x\in \Omega\\
u=&0&=v, \quad x\in \partial\Omega,
\end{equation*}
where $\lambda$ is a positive parameter and $\Omega$ is a bounded domain.
Assuming $f(x)>L$, $g(x)>L$ for $x>K$ for some $L>0$ and $K>0$,
and $\lim_{x\rightarrow\infty}\frac{f( Mg(x))}{x}=0$ for every $M>0$, we
establish the existence of a large positive solution $(u,v)$
for $\lambda$ large. In particular, we do not assume any monotonicty
assumptions on $f$ or $g$ nor any sign conditions on $f(0)$ or $g(0)$.
-----------------------------------------------------------------------------------------------------------------------
Positive solutions of differential equations with nonlinear
boundary conditions
Gennaro Infante
Dipartimento Di Matematica, Universita Della Calabria
87036 Arcavacata Di Rende (CS), Italy
and
Department of Mathematics, University of Glasgow
Glasgow G12 8QW, UK
Abstract: Using the theory of fixed point index,
we establish new results for some differential equations subject to nonlinear
boundary conditions. We obtain existence of at least one or several
positive solutions.
------------------------------------------------------------------------------------------------------------------------------
Multiple Positive Solutions of Conjugate Boundary Value Problems
with Singularities
K. Q. Lan
Department of Mathematics, Physics and Computer Science
Ryerson University
350 Victoria Street
Toronto, Ontario
Canada, M5B 2K3
Abstract: We consider the existence of one or
several nonzero positive solutions for a higher order nonlinear ordinary
differential equation with conjugate boundary conditions. The conjugate
boundary value problems can be changed into a Hammerstein integral
equation with a suitable kernel. We shall show that the kernel has
upper and lower bounds. This enables us not only to exhibit a new property
of positive solutions for the conjugate boundary value problems but also
to derive new results on the conjugate boundary value problems from
the well-known results on the existence of one or several positive solutions
of Hammerstein integral equations with singularities obtained by the author
recently. Our results generalize some known results where stronger conditions
were imposed and the theory of fixed point index for compact maps defined
on cones was used directly.
--------------------------------------------------------------------------------------------------------------------------------
Multiple positive solutions for nonlinear
$m$-point boundary value problems
Ruyun Ma
Department of Mathematics, Northwest Normal University
Lanzhou 730070, Gansu, People's Republic of China
Abstract: In this paper, we consider the existence
and multiplicity of positive solutions for the $m$-boundary value problems
(p(t)u')' -q(t) u+ f(t, u)=0, 0<t<1
a u(0)- b p(0) u'(0)=\sum^{m-2}_{i=1}\alpha_i u(\xi_i),
c u(1)+ d p(1) u'(1)=\sum^{m-2}_{i=1}\beta_i u(\xi_i),
where $p\in C([0,1],(0,\infty))$, $q\in C([0,1], [0,\infty))$
$a, b, c, d\in [0,\infty)$, $\xi_i\in (0,1)$, $\alpha_i, \; \beta_i\in
[0,\infty)$ (for $i\in \{1,\cdots m-2\}$) are given constants satisfying
some suitable conditions. Our results extend some of the existing literature
on two-point
boundary value problem. Our proofs are based on fixed point theorem
in cones.
-------------------------------------------------------------------------------------------------
On the
existence of explosive solutions for semilinear elliptic problems
Zhijun Zhang
Department of Mathematics and Information Science,
Yantai University, Yantai, Shandong, 264005,
People' Republic of China
Abstract: TBA
----------------------------------------------------------------------------------------------
Remarks on positive solutions
of some 3-point boundary value problems
J. R. L. Webb
Department of Mathematics
University of Glasgow
Glasgow G12 8QW, UK
Abstract: Some recent work onexistence of one
or of multiple solutions of a nonlinear second order differential equation
with nonlocal boundary conditions will be discussed by the method
of fixed point index. An optimal value will be given for a constant
that appears in the definition of the cone being used and in some
of the other hypotheses.
------------------------------------------------------------------------------------------------------------------------
Kelvin-Helmholtz Instability
Waves and Upstream Propagating Acoustic Waves
in Supersonic Multiple Jets
Joshua Z. Du
Department of Mathematics
Kennesaw State University
Kennesaw, GA 30144 USA
and
Kenneth Dye
Lockheed Martin Aeronautical Company
Learning and Development
D/RT4M, USA
Abstract: Jet aircraft were introduced right after that the Second World War. Shortly after that, jet noise prediction and reduction became an important research topic. Because of the need for large thrust, many high performance military aircraft are propelled by two or more jet engines housed close to each other.
Three Physics Laws, Conservation of Mass, Momentum and Energy, in differentiation form are employed to formulate the Kelvin-Helmholtz Instability problem of supersonic triple jets. The general solution of the system about pressure of the jets and the dispersion relation for instability waves about jet noise are derived.
-------------------------------------------------------------------------------------------------------------------------------
Half-eigenvalues --- an alternative approach to the Fucik spectrum
for semilinear problems with jumping non-linearities
Bryan Rynne
Department of Mathematics
Heriot-Watt University
Edinburgh EH14 4A, UK
Abstract: We consider the boundary value problem
\begin{equation}
Lu(x)=f(x,u(x))+h(x),\quad x\in(0,\pi),
\end{equation}
where $L$ is a second order Sturm-Liuouville operator and $f$
is a Caratheodory function with limits
$\zeta_{\pm}(x)=\lim_{\xi\to\pm\infty} f(x,\xi)/\xi$
(and $\zeta_{\pm}\in L^\infty(0,\pi)$).
These limits may differ, so $f$ is termed {\em jumping}.
Existence and non-existence conditions for (1) are usually expressed
in terms of the Fucik spectrum
($(\alpha,\beta) \in R2$ is in the {\em Fucik spectrum} if $L
u=\alpha u^+ -\beta u^-$ has a solution $u\ne 0$
(where $u^{\pm}(x)=\max\{\pm u(x),0\}$)). However, more general results
can be obtained using a
different spectrum.
A number $\lambda\in R$ is a {\em half-eigenvalue} if $Lu=\zeta_+
u^+ -\zeta_- u^- +\lambda u$
has a solution $u\ne 0$. Now, the equation
\begin{equation} \label{hlslvble.eq}
L u=\zeta_+ u^+ -\zeta_- u^- +h,
\end{equation}
is a `limiting' form of (1), and the solvability properties of (2)
depend on the location of the point $\lambda=0$ relative to the set of
half-eigenvalues. Corresponding solvability properties of (1) can then
be obtained by perturbation.
These solvability results can also be extended to some cases where
the limits $\zeta_{\pm}$ do not exist.
If $\zeta_+\equiv\alpha$, $\zeta_-\equiv\beta$ then $(\alpha,\beta)$
lies in the Fucik spectrum if and only if $0$ is
a half-eigenvalue, and the solvability results obtained from these
concepts coincide. However, if $\zeta_{\pm}$ are not constant then the
half-eigenvalue concept yields more general solvability results than the
Fucik spectrum, which cannot easily
incorporate the behaviour of $\zeta_{\pm}$.