Zeitschrift für Analysis und ihre Anwendungen
Journal for Analysis and its Applications
Volume 15 (1996)
Abstracts
A. Carbone: On a Fixed Point Theorem by Brosowski and Singh, 15 (1996) 003--006
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- Let n denote a strictly positive integer, R the field of real numbers
and E = Rn. We construct a complex differential algebra G of
so-called 2pi-periodic generalized functions on E. We show that the space
D' of 2pi-periodic distributions on E can be canonically embedded into G.
Next we lay the foundation for calculation in G. This algebra G enables one
to solve, in a canonical way, differential problems with strong singular
data which have no solution in D'.
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- The fundamental matrix of the 5-by-5 system of partial differential
operators describing linear thermodiffusion inside elastic media is -- by
a standard procedure -- expressible through the fundamental solution of its
determinant. This determinant is equal to the square of a wave operator
multiplied by the so-called operator of dynamic linear thermodiffusion,
which is of the fourth order with respect to the time variable. In this paper,
we deduce, by means of a variant of Cagniard-de Hoop's method, a representation
of the fundamental solution of this operator by simple definite integrals. This
formula allows the explicit computation of thermal and diffusion effects which
result from instantaneous point forces or heat sources.
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- We consider the condition of orthogonal polynomials, encoded by
the coefficients of their three-term recurrence relation, if the measure
is given by modified moments (i.e. integrals of certain polynomials forming
a basis). The results concerning various polynomial bases are illustrated by
simple examples of generating (possibly shifted) Chebyshev polynomials of first
and second kind.
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- We show that compactness in Lebesgue measure of sets, i.e.
compactness in the topology induced by convergence in measure,
can be reduced to the equality of two numerical characteristics
of sets, namely the measures of noncompactness.
It is well-known, that the inequality by Ehrling-Nirenberg needs
not to be true in the cases of Sobolev spaces on a domain with an
irregular boundary. Here an analog to the inequality of Ehrling-Nirenberg
is obtained, which holds always. To prove the solvability of the
Neumann problem we require -- instead of assuming that the Ehrling-Nirenberg
inequality shall be fulfilled -- that the characteristic of a degree of
noncompactness of the embedding map from Sobolev spaces into Lebesgue spaces
is reasonably small.
An extension of some results by V. G. Maz'ya is proved.
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- The Ekman partial differential equation for the stream function of
turbulent mass flow in shallow and small-sized surface waters is
discussed. The Dirichlet problem for the Eknan equation is shown to be
well-posed in a weighted Sobolev space. Conditions for the existence of
classical solutions are given. The dependence of regularity and asymptotics
of the solution on the properties of the depth profile is studied.
- We consider some existence results for solutions analytic with
respect to the spatial variables to the first-order equations with
a delay and some deviations not only at the functions, but also at
its derivative. We construct a natural Banach space and a norm which
make an adequate integral operator contractive. Due to a useful
relation of partial order in this space the main problem is also
placed in the theory of monotone iterative techniques.
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- The paper deals with quasilinear parabolic boundary value problems where
the nonlinear coefficients and right-hand side are defined w.r.t. u
only in a neighbourhood of the initial function. The quasilinear problem is
approximated by linear elliptic problems by means of semidiscretization
in time. It is proved that the approximations converge uniformly although
the data are not continuous functions, and the limit turns out to be the
weak solution of the parabolic problem for sufficiently small time t.
The crucial points of the paper are Linfty-estimates to
ensure that the approximations belong to the domain of nonlinearities and
uniform estimates of the discrete time derivatives in a Lebesgue space in
order to obtain uniform convergence.
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- The purpose of this note is to define a numerical range for nonlinear
operators in smooth Banach spaces and to use this numerical range to
localize certain spectral sets of Lipschitz continuous operators.
- We are concerned with generalizations of the results of A. Douady and J.
Oesterlé on estimates for the Hausdorff dimension of sets on Riemannian
manifolds being negatively invariant with respect to a map. The main
theorem that we derive for maps allows a number of corollaries which
generalize several other results of A. V. Boichenko, F. Ledrappier and
G. A. Leonov. We extend the concept on differential equations and the
corresponding vector fields on Riemannian manifolds. To obtain upper bounds
for the Hausdorff dimension we formulate conditions for the eigenvalues of the
symmetric part of the covariant derivative of the vector field. Modifications
of the eigenvalues by the choice of an apropriate Riemannian metric will be
of great importance. Besides the investigation of dimension of negatively
invariant sets we are interested in the convergence behaviour of autonomous
differential equations on Riemannian manifolds. We propose also a general
form of the Bendixson-Dulac criterion for the non-existence of non-trivial
periodic orbits of vector fields on compact Riemannian manifolds.
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- The paper deals - under the viewpoint of topology - with discrete Cauchy spaces,
which are spaces where a discrete Cauchy structure (t,C) (with t being a discrete
convergence and C being a discrete pre-Cauchy structure) is defined. More precisely,
let E1, E2, ..., and E be arbitrary sets and let S denote the
set of all discrete sequences (xn, n in N') with xn in
En (n in N') and with N' being an infinite subset of the natural numbers.
Then t and C are certain subsets of (S,E) respectively of S, which in a certain sense
are assumed to be compatible. The paper gives properties of t and C and, among others
is devoted to the problem of completion of discrete Cauchy spaces (((E1,
E2, ...), E); (t,C)). The construction of a completion of a discrete
Cauchy space differs (in some sense essentially) from the construction of a completion
of a usual sequential Cauchy space and is even more simple.
An essential part of the paper is devoted to certain metric discrete Cauchy spaces.
It turns out that such a metric discrete Cauchy space is complete if and only if (E,d)
is complete and that also the completion is metric.
A further subject of the paper are metric discrete Cauchy spaces of mappings between
metric discrete Cauchy spaces, where simple characterizations of the corresponding
discrete convergence and discretee pre-Cauchy structure of such a discrete Cauchy
space as well as a necessary and sufficient condition for its completeness are given.
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- Originally the solution of the classical Cauchy-Kovalevskaya
problem (=initial-value problem with holomorphic initial data and
holomorphic right-hand sides) was constructed by power series.
The functional-analytic approach initiated by M. Nagumo (Japan.
Journ. Math. 18 (1941), 41-47) is based on an integral rewriting
of that problem. Using this integral rewriting of the classical
Cauchy-Kovalevskaya problem, W. Walter gave an elementary proof
of the Cauchy-Kovalevskaya Theorem (Amer. Math. Monthly 92
(1985), 115-125) by the contraction-mapping principle applied to
a Banach space of holomorphic functions which is equipped with a
weighted supremum norm.
The present paper investigates the compactness of the operator
under consideration. Result: While the operator is not compact
as operator mapping the Banach space into itself, it is compact
as operator from the Banach space into the Frechet space of
locally uniform convergence.
The paper investigates also the convergence behaviour of the
successive approximations. The non-compactness is proved with
the help of the Fredholm alternative. The paper ends with hints to
initial value problems with generalized analytic initial
functions where analogous considerations can be carried out.
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- We deal with the initial and boundary value problem for the degenerate
parabolic equation ut=\Delta\beta(u) on multidimensional bounded
domains. We assume that \beta'(0)=0 (e.g., \beta(u)=u|u|m-1
and m>1). We study the appearance of the free boundary. We prove
that the free boundary has a finite speed of propagation and is Hölder
continuous. Further, we estimate the Lebesgue measure of the set where
u>0 is small. This yields a non-degeneracy property.
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- Application of solution representations is one of basic methods in stability
of functional differential equations. The present paper is aimed to obtain such
a formula for an integro-differential equation with integral impulsive conditions
and to apply it in stability research.
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- For an arbitrary polynomial with restricted coefficients, we obtain an
asymptotic inequality for the maximum modulus over a circle of radius R, as
R goes to infinity. The basic idea of proof is a method of convolution.
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- The authors study the convergence of a sequence of functions which
has been introduced by Caratheodory and Tonelli in connection with the
solvability of the Cauchy problem for ordinary differential equations.
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- The generalized solution of a system of Stratonovich equations
is approximated by a discontinuous Gelerkin method. A piecewise
polynomial approximation is intoduced. The convergence and error estimates
are proved. The solution of Galerkin equations can be approximated by
the solution of a system of equations with an inhomogeneous random part
and the simulation of a stochastic integral.
- We investigate large time behaviour of solutions for a class of nonlinear
homogeneous Neumann parabolic problems, possibly degenerate, in a bounded
domain. The reaction term is a power of the solution u times a changing sign
spatially dependent coefficient. Depending on the features of the problem,
several parameters play a role to establish global boundedness or finite time
blow-up of solutions. The occurrence of either situation is related with the
existence of stationary solutions. Proofs make extensive use of monotonicity
methods.
- By means of a contraction principle in a Banach space E with a scale of norms
| . |s (s>=0) existence, uniqueness and stability of solutions are
proved for a general class of operator equations u+G0u+G1u=g
including multilinear ones where G0, G1 are operators from E
to E. The theorems are applicable to equations with operators of generalized
convolution type.
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- The paper is a contribution to convex analysis in ordered linear topological
spaces. For any convex function f from a Banach space X into a partially
ordered one Y endowed with a convex cone K some properties of the
εk0-subdifferential J
ek0≤ f(x) of f
are examined. The nonemptiness of J
ek0≤ f(x) is proved,
whenever Y is a normal order complete vector lattice and f belongs to the
class of functions which are continuous and convex with respect to the
cone K. For the real-valued case Bronsted and Rockafellar have proved that
the set of subgradients of a lower semicontinuous function f on a Banach
space X is dense in the set of e-subgradients.
We deduce a similar result for a class of ek0-subdifferentials of functions which takes values in an ordered
linear topological space Y.
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