Zeitschrift für Analysis und ihre Anwendungen
Journal for Analysis and its Applications
Volume 21 (2002)
Abstracts
T. Constantinescu, J. L. Johnson:
Tensor Algebras and Displacement Structure. I: The Schur Algorithm, 21 (2002) 003--020
- We explore the connection between tensor algebras and displacement structure.
Thus, we describe a scattering experiment in this framework, we obtain a
realization of the elements of the tensor algebra as transfer maps of a
certain class of non-stationary linear systems, and we describe a Schur
type algorithm for the Schur elements of the tensor algebra.
- We propose a simple quaternionic reformulation of Maxwell equations for
inhomogeneous media and use it in order to obtain new solutions in a static
case.
- We present and discuss to some extent a number of first order
systems of partial differential operators with constant coefficients which
arise naturally within the language of Clifford analysis. We also present
resolutions for certain examples.
- We consider systems of quasilinear partial differential equations of second
order in two- and three-dimensional domains with corners and edges. The
analysis is performed in weighted Sobolev spaces with attached asymptotics
generated by the asymptotic behaviour of the solutions of the corresponding
linearized problems near boundary singularities. Applying the Local Invertibility
Theorem in these spaces we find conditions which guarantee existence of small
solutions of the nonlinear problem having the same asymptotic behaviour as
the solutions of the linearized problem. The main tools are multiplication
theorems and properties of composition (Nemytskij) operators in weighted
Sobolev spaces. As application of the general results a steady-state
drift-diffusion system is explained.
- We consider the pair diffusion process in more than two spatial
dimensions. In this case we are able to prove just a local existence result,
since it is not possible to deduce global a priori estimates for the equations
as it can be done in the two-dimensional case. The model includes a nonlinear
system of reaction-drift-diffusion equations, a nonlinear ordinary differential
equation in Banach spaces and an elliptic equation for the electrostatic
potential. The local existence result is based on the fixed point theorem
of Schauder.
- We study a transmission problem with a fractal interface K, where a second
order transmission condition is imposed. We consider the case in which the
interface K is the Koch curve and we prove existence and uniqueness of the
weak solution of the problem in V (W, K), a
suitable "energy space". The link between the variational formulation and
the problem is possible once we recover a version of the Gauss-Green formula
for fractal boundaries, hence a definition of "normal derivative".
- We prove Sobolev-type and Morrey-type inequalities for Sobolev spaces
related to a family of non-smooth vector fields which formally satisfy
the Hörmander condition of step 2. The coefficients of the vector fields
are not regular enough to define the Carnot-Caratheodory distance.
Thus the result is proved by developing a real analysis technique which
is based on an approximation procedure of Lipschitz continuous vector fields
with a family of left-invariant first order operators on a nilpotent Lie
group.
- This article deals with asymptotic estimates of strong solutions of Stokes equations
in aperture domains. An aperture domain is a domain, which outside a bounded set is
identical to two half spaces separated by a wall and connected inside the bounded set
by one or more holes in the wall. It is known that the corresponding Stokes operator
generates a bounded analytic semigroup in the closed subspace Jq(W) of divergence free vector fields of Lq(W)n. We deal with Lq-Lr-estimates
for the semigroup, which are known for Rn, the half space and exterior domains.
- We study the set of pointwise multipliers in the Lizorkin-Triebel space
F (p; s, q) and of the corresponding multiplier set in the Besov space
B (p; s, q), where we give sufficient conditions on the parameters s,
p and p1 such that the embeddings of the intersection of
F (p1; n/p1, infinity) and Linfinity into
M (F (p; s, q)) and of B (p1; n/p1, infinity) into
M (B (p; s, q)) hold.
- We deal with the study of limits of solutions of a class of
fully nonlinear elliptic problems at nearly critical growth. By means of
P. L. Lions' concentration-compactness principle, we prove an alternative
result for the existence of non-trivial solutions of the limit problem.
- We show that the set of stably solvable maps from an infinite dimensional Banach
space E into itself is not open in the topological space C(E) of the continuous
selfmaps of E. The question of whether or not this set is open is related to
nonlinear spectral theory and was posed in a previous paper of the author,
M. Martelli and A. Vignoli [Ann. Mat. Pura Appl. 118 (1978) 229--294].
- We consider a singularly perturbed convection-diffusion problem.
The existence of certain decompositions of the solution into a regular
solution component and a layer component is studied.
Such decompositions are useful for the convergence analysis of numerical
methods.
Our aim is to show that such decompositions exist under less restrictive
assumptions on the data of the problem than those required in earlier publications.
- We study passive scattering systems in the framework
introduced by Arov. The main purpose is to find conditions for conserving
the optimality of a cascade connection of passive scattering systems in
terms of the best minorant outer function and to characterize optimal
passive scattering systems which have the same transfer function.
- This paper deals with obstacle problems on a bounded open subset of Rn.
To state some sufficient criteria for determining parts of the coincidence set
and of the non-coincidence set of the optimal solution u to this obstacle
problem, optimal solutions to some particular auxiliary problems without obstacle
are used as orienting tool. For this purpose, we do not assume any coercive
assumption, but only the uniqueness of the optimal solution to auxiliary
problems, which is ensured if e.g. the performance index is strictly convex.
- By making use of the Leray-Schauder fixed point theorem we prove the global
existence of solutions to some integro-differential equations with delay
subject to non-local conditions, and this problem is considered in an
arbitrary Banach space.
- It is shown that a Hilbert-type inequality with weight w(n)
= p - (q / SquareRoot (2n + 1)) can be established where
q = 17 / 20. As application, a quite sharp result of the
Hardy-Littlewood inequality is obtained and some further extensions are obtained.
- It is well known that, when f(z) is an entire function of order
r and r <
infinity, then the limit limsup T(r,f') / T(r,f) is finite as r approaches
infinity through all values or outside a set E of finite measure. But for
r = infinity, Hayman has shown that the
assertion does not hold by constructing an entire function f (z) and an
exceptional set E of even infinite measure. In this paper, we will further
extend his result to the case where f (z) is an algebroid function of order
r = infinity.
- We investigate the consequences of the lower frame condition and the lower
Riesz basis condition without assuming the existence of the corresponding upper
bounds. We prove that the lower frame bound is equivalent to an expansion property
on a subspace of the underlying Hilbert space H, and that the lower frame condition
alone is not enough to obtain series representations on all of H. We prove that the
lower Riesz basis condition for a complete sequence implies the lower frame condition
and w-independence; under an extra condition the statements
are equivalent.
- This paper introduces, by way of constructing, specific finite and infinite integral
transforms with Bessel functions Jn and
Yn in their kernels. The infinite transform
and its reciprocal look deceptively similar to the known Weber transform and its
reciprocal, respectively, but fundamentally differ from them. The new transform
enjoys an operational property that makes it useful for applications to some problems
in differential equations with non-constant coefficients. The paper gives a
characterization of the image of some spaces of square integrable functions
with respect to some measure under the infinite and finite transforms.
- This note deals with a nonlinear system of partial differential equations
accounting for phase transition phenomena. The existence of solutions to a
Cauchy-Neumann problem is established in the one-dimensional space setting,
using a regularization -- a priori estimates -- passage to limit procedure.
- We study the behavior of eigenfunctions of the Schrödinger operator
-D + v with potential having power, exponential
or super-exponential growth at infinity and discontinuities on manifolds in
Rn. We use a connection between the domain of analyticity of the
main symbol (|x|2 + v(x))-1
of the parametrix -D + v at infinity or near
singularities of v and the behavior of eigenfunctions at infinity or near
singularities of potentials. Our approach is based on a general calculus of
pseudodifferential operators with analytic symbols.
- We apply a fixed point theorem for increasing operators in ordered Banach
spaces to prove the existence of extremal (i.e. maximal or minimal) solutions
for the variational inequality <Av, w - v> >= IntegralW f (x, v)(w - v) dx where A is the p-Laplacian and
f (x, u) = F(x, u, u) with F(x, u, v) being a function, non-decreasing in u and
non-increasing in v.
- An integral equation and a related integral-differential equation
of first order over R+ with a quadratic integral term
representing the so-called autocorrelation of the unknown function
is dealt with. For both equations the general solution is
constructed and estimated in the L2-norm. Further, the
asymptotic behaviour and the stability of the solution are
investigated.
- The problem of recovering a degenerate operator kernel in a hyperbolic
integro-differential operator equation is studied. Existence,
uniqueness and stability for the solution are proved. A
conditional convergence of a sequence of solutions corresponding
to degenerate kernels to a solution corresponding to a non-degenerate
kernel is shown. Such results are applied to determine space- and
time-dependent relaxation kernels in a multi-dimensional
viscoelastic wave equation with given boundary observations of
traction type on the assumption that the kernels to be determined
are representable as a finite or infinite sum of products of known
space-dependent and unknown time-dependent functions.
- We give a unified approach to a class of nonlinear parabolic inverse problems
involving kernels of convolution type. Our main tools are optimal regularity
results, in Sobolev and Hölder spaces, for parabolic equations and analytic
semigroup theory. We apply the main abstract results (Theorems 2.1 -- 2.2) to
a model of population dynamics, to the theory of combustion of a material
with memory and, finally, to a parabolic equation with elliptic part of order
2m, which for m = 1 is the heat equation with memory and with non-linearity
containg derivatives up to order 2m - 1.
- A general inverse problem for the identification of a memory kernel in
viscoelasticity in one space dimension is dealt with, where the kernel is
represented by a finite sum of products of known spatially dependent
functions and unknown time-dependent functions. Using the Laplace transform
method an existence and uniqueness theorem for the memory kernel is proved.
- We introduce the class of hyper-solvable equations whose concept may be
regarded as an extension to the context of topological spaces of the
known notion of 0-epi maps. After collecting some notation, definitions
and preliminary results we give a homotopy principle for hyper-solvable
equations. We provide examples showing how these equations arise in the
framework of Leray-Schauder degree, Lefschetz number theory and essential
compact vector fields in the sense of A. Granas.
- Certain stability properties for meromorphic solutions w(z) = u(x, y) +
i v(x, y) of partial differential equations of the form Sumt=0m
ft (w')m-t = 0 are considered. Here the coefficients ft
are functions of x, y, of u, v and the partial derivatives of u, v. Assuming
that certain growth conditions for the coefficients ft are valid in the
preimage under w of five distinct complex values, we find growth estimates,
in the whole complex plane, for the order r(w) and
the unintegrated Ahlfors-Shimizu characteristic A(r, w).
- We investigate some nonlinear difference equations with
continuous variable. A linearized oscillation result is established and
oscillation criteria for some forced difference equations are obtained.
- The report deals with the equation (r(t) u'(t))' + p(t) u(t) = 0 and renders
effective sufficient conditions for its oscillation and non-oscillation in
the case Integralinfinity (1/r(t)) dt < infinity.
- We present sharp rational bounds for I(n-1, x) I(n+1, x) /
(I(n, x))2, where I(n, x) = (-x)(n+1)/(n+1)!
+ (-x)(n+2)/(n+2)! + ... Our result improves inequalities
published by M. Merkle in 1997.
- Some minor corrections to a previous paper of the author [Z. Anal. Anw. 19 (2000), 339--357]
are given.
- We prove that the topological singular set of a map in W1,3(M, S3)
is the boundary of an integer-multiplicity rectifiable current in M, where M is a closed
smooth manifold of dimension greater than 3 and S3 is the three-dimensional
sphere. Also, we prove that the mass of the minimal integer-multiplicity rectifiable
current taking this set as the boundary is a strongly continuous functional on W1,3
(M, S3).
- Various products of distributions with coinciding point singularities are
derived when the products are 'balanced' so that their sum is a generalized
function which is associated to a distribution. These products follow the
idea of a known result on distributional products published by Jan Mikusinski
in 1966. The results in the present paper are obtained in the Colombeau
algebra of generalized functions, which provides an efficient tool for
dealing with nonlinear problems of Schwartz distributions.
- We discuss several essentially different formulas for the general derivatives
qn(z) of the fundamental solution of the Cauchy-Riemann operator in
Clifford Analysis, upon which -- among other important applications -- the theory
of monogenic Eisenstein series is based. Using Fourier and plane wave decomposition
methods, we obtain a compact integral representation formula over a half-space, which
also lends itself to establish upper bounds on the values |qn(z)|. A second
formula that we discuss is a recurrence formula involving permutational products of
hypercomplex variables by which these estimates can be obtained immediately. We further
prove several formulas for qn(z) in terms of explicit, non-recurrent finite
sums, leading themselves to further representations in terms of permutational products
but using different and fewer hypercomplex variables than used in the recurrence relations.
Summing up a fixed qn over a given discrete lattice leads to a variant of the
Riemann zeta function. We apply one of the closed representation formulas for qn(z)
to express this variant of the Riemann zeta function as a finite sum of real-valued
Dirichlet series.
- Characterizations of weighted Hölder continuity and weighted Lipschitz
continuity are obtained for the hyperbolic Bloch functions on the unit ball
of Rn. Similar results are extended to hyperbolic little Bloch and Besov
spaces.
- We continue to explore the connection between tensor algebras
and displacement structure. We focus on recursive orthonormalization and
we develop an analogue of the Szegö-type theory of orthogonal polynomials
in the unit circle for several non-commuting variables. Thus we obtain
recurrence equations and Christoffel-Darboux formulas for Szegö polynomials
in several non-commuting variables, as well as a Favard type result. Also,
we continue to study a Szegö-type kernel for the N-dimensional unit ball
of an infinite-dimensional Hilbert space.
- We extend a variety of index integral transforms (i.e. integral
transforms over an index as integration variable) with Bessel and Lommel
functions as kernels by considering mapping properties of the related
integral operators. This class of transforms includes, for instance,
operators of Titchmarsh type. Useful integral representations of the
considered kernels are deduced and boundedness properties, Parseval
equalities, Plancherel type theorem and inversion formula are given.
- We present a method to study asymptotically linear degenerate problems with
sublinear unbounded non-linearities. The method is based on the uniform
convergence to zero of projections of non-linearity increments onto some
finite-dimensional spaces. Such convergence was used for the analysis of
resonant equations with bounded non-linearities by many authors. The
unboundedness of nonlinear terms complicates essentially the analysis of
most problems: existence results, approximate methods, systems with
parameters, stability, dissipativity, etc. In this paper we present
statements on projection convergence for unbounded non-linearities and
apply them to various resonant asymptotically linear problems: existence of
forced periodic oscillations and unbounded sequences of such oscillations,
existence of unbounded solutions, sharp analysis of integral equations with
simple degeneration of the linear part (a scalar two-point boundary value
problem is considered as an example), existence of non-trivial cycles for
higher order autonomous ordinary differential equations, and Hopf
bifurcations at infinity.
- We construct explicit solutions for initial value problem for
a system of first order equations. When n = 1, this system is just the
standard Hopf equation in conservative form. When n > 1, the system is
non-conservative. We use the vanishing viscosity method to construct
solutions. As the system is non-conservative we use Volpert product and the
algebra of generalized Colombeau functions to make sense of the products
which appear in the equations.
- We derive weight inequalities for one-sided and Riesz potentials
in Lp(x) spaces under the condition that p satisfies a weak Lipschitz
condition. Compactness of these operators in Lp(x) spaces is also
established.
- we investigate some connections between Hausdorff measures, Hölder
functions and analytic sets in terms of images of zero-derivative sets
and level sets. We characterize in terms of Hausdorff measures and
descriptive complexity subsets M of R which are
(1) the image under some Cn, a
function f of the set of points where the derivatives of first n orders
are zero
(2) the set of points where the level sets of some Cn, a
function are perfect
(3) the set of points where the level sets of some Cn, a
function are uncountable.
- In the theory of p-summing operators studied by Pietsch we know that
p2(C(K), H) = pp
(C(K), H) for any Hilbert space H and any p such that 2 < p < +infinity. We prove that
this equality is not true in the same notion generalized by Junge and Pisier to operator
spaces, i.e. pl2(B(l2), OH)
( = p20(B(l2), OH)) is not equal
to plp(B(l2), OH).
- Let X(x) be the distribution of convex sets over a
domain D, subset of Rn and let f: partial D
to R be a function. We consider the existence problem of locally Lipschitz functions
f defined in the domain D so that f restricted to partial D =
f and nabla f(x) in X(x) almost
everywhere in D. These questions are related to the existence problem for space-like
surfaces of arbitrary codimension with prescribed boundary in Minkowski space.
- A continuation principle is obtained for maps defined on a closed, convex
subset which may have empty interior in a Frechet space, and satisfying a
condition of Mönch type. An application to first order systems of differential
equations is presented to illustrate our theory.
- We generalize the usual Babuska-Brezzi theory to a class of nonlinear
variational problems with constraints. The corresponding operator equation
has a dual-dual type structure since the nonlinear operator involved has
itself a dual structure with a strongly monotone and Lipschitz-continuous
main operator. We provide sufficient conditions for the existence and
uniqueness of solution of the continuous and Galerkin formulations, and
derive a Strang-type estimate for the associated error. An application to
the coupling of mixed-FEM and BEM for a nonlinear transmission problem in
potential theory is also described.
- Instead of a single matrix occurring in the standard setting, the leading
term of the linear differential algebraic equation is composed of a pair of
well matched matrices. An index notion is proposed for the equations. The
coefficients are assumed to be continuous and only certain subspaces have to
be continuously differentiable. The solvability of lower index problems is
proved. The solution representations are based on the solutions of certain
inherent regular ordinary differential equations that are uniquely determined
by the problem data. The assumptions allow for a unified treatment of the
original equation and its adjoint. Both equations have the same index and
are solvable simultaneously. Their fundamental solution matrices satisfy a
relation that generalizes the classical Lagrange identity.
- For a scalar delay differential equation
dot x(t) + Sumk=1infinity ak(t) x(hk(t)) = 0 ,
(hk(t) <= t)
a connection between the following four properties is established: (1) non-oscillation
of this equation, (2) non-oscillation of the corresponding differential inequality, (3)
positiveness of the fundamental function, (4) existence of a non-negative solution for
a certain explicitly constructed nonlinear integral inequality.
Explicit non-oscillation and oscillation conditions, comparison theorems and a criterion
of the existence of a positive solution are presented for this equation.
- A parametric class of series generated by integration of complete elliptic integrals
Sum-r not equal to k = 0infinity (2k over k) devided by (k + r)16k
is valuated in closed form. Alternative proofs to results of Ramanujan and
others are given. Also, a particular case of the Saalschützian hypergeometric
series 4F3(1) is derived.
- Properties of the Minkowski-Pontryagin subtraction of closed
bounded convex sets are investigated and four criteria for summands
of closed bounded convex sets are given.
- We investigate integration with respect to a finitely additive measure of
integrands with closed, convex values and we obtain a closedness result for
the Aumann integral.
- Let K be a bounded closed convex subset of a Banach space. We study several
convergence properties of infinite products of non-expansive self-mappings of
K. In our recent work we have considered several spaces of sequences of
such self-mappings. Endowing them with appropriate topologies, we have shown
that the infinite products corresponding to generic sequences converge. In
the present paper we prove that the subsets consisting of all sequences of
mappings with divergent infinite products are not only of the first Baire
category, but also s-porous.
- Convolution dilation operators with non-compactly supported kernels are
considered and effective formulae for their spectral radii are found. The
formulae depend on the behaviour of the eigenvalues of the dilation matrix.
- A l-Hankel operator X is a bounded operator
on Hilbert space satisfying the operator equation S*X - XS =
lX, where S is the (unilateral) forward shift
and S* is its adjoint. We prove that there are non-compact l-Hankel operators for l a
complex number of modulus less than 2, by first exhibiting a way to obtain
bounded solutions to the above equation by associating to it a Carleson measure.
We then show that an interpolating sequence can be given such that the l-Hankel operator associated with the Carleson measure given
by the interpolating sequence is non-compact.
- We give a general construction in arbitrary normed spaces to produce
fixed-point free continuous maps with a large minimal displacement,
contractions of the sphere, and retractions onto the sphere such that
the corresponding maps have small measures of non-compactness.
- [Abstract-pdf]
We describe convex sets on the $({1 \over p},{1 \over q})$-plane for
which the well-known Bochner-Riesz operator with the symbol $(1-|\xi|^2)_+
^{-\alpha} \ \ (0 < {\rm Re}\,\alpha < {n + 1 \over 2})$ is bounded from
$L_p$ into $L_q$.
- [Abstract-pdf]
The purpose of this paper is to study the autonomous third order nonlinear
differential equation
f ''' + [ (m + 1) / 2 ] f f '' - m (f ')2 = 0 on
(0, infinity),
subject to the boundary conditions f (0) = a in R, f '(0) = 1 and f '(t)
converges to 0 as t approaches infinity. This problem arises when looking for
similarity solutions to problems of boundary-layer theory in some contexts
of fluids mechanics, as free convection in porous medium or flow adjacent
to a stretching wall. Our goal here is to investigate by a direct approach
this boundary value problem as completely as possible, say studying existence
or non-existence and uniqueness or non-uniqueness of solutions according to the
values of the real parameter m. In particular, we will emphasize similarities
and differences between the cases a = 0 and a not equal to 0 in the boundary
condition f(0) = a.
- We consider the Cauchy problem for a nonlinear equation on the Haar pyramid.
By using a discretization with respect to spatial variables, the partial
functional-differential equation is transformed into a system of ordinary
functional-differential equations. We investigate the question of under what
conditions the classical solutions of the original problem are approximated
by solutions of associated systems of ordinary functional-differential
equations. The proof of the convergence of the method of lines is based on
the differential-inequalities technique. A numerical example is given.
Differential equations with retarded variables and differential-integral
equations are particular cases of a general model considered in the paper.
- We examine nonlinear periodic evolution inclusions of the subdifferential
type and prove two existence theorems: one for the "non-convex, lower
semicontinuous" problem and the other for the "convex, h-upper semicontinuous"
problem. Our method of proof is based on the theory of nonlinear operators
of monotone type and on multi-valued analysis. We also present three examples
from partial and ordinary differential inclusions, illustrating the
applicability of our work.
- [Abstract-pdf]
We consider the Hamiltonian system
Ju'(x) + Mu(x) - nablau F(x, u(x)) = l u(x).
Using variational methods obtained by Stuart on the one hand and by Giacomoni and
Jeanjean on the other, we get bifurcation results for homoclinic solutions by
imposing conditions on the function F. We study both the case where F is defined
globally with respect to u and the case where F is defined locally only.
- We present a new approach to the analysis of solvability properties for
complementarity problems in a Hilbert space. This approach is based on the
Skrypnik degree which, in the case of mappings in a Hilbert space, is
essentially more general in comparison with the classical Leray-Schauder
degree. Namely, the Skrypnik degree allows us to obtain some new results
about solvability of complementarity problems in the infinite-dimensional
case. The case of generalized solutions is also considered.
- [Abstract-pdf]
Consider the non-autonomous logistic model
$$
\Delta x_n = p_nx_n\Big({1 - x_{n-k_n} \over 1 + \lambda x_{n-k_n}}\Big)^r
\qquad (n \ge 0)
$$
where $\Delta x_n = x_{n+1} - x_n$, $\{p_n\}$ is a sequence of positive real
numbers, $\{k_n\}$ is a sequence of non-negative integers such that $\{n -
k_n\}$ is non-decreasing, $\lambda \in [0,1]$, and $r$ is the ratio of two
odd integers. We obtain new sufficient conditions for the attractivity of the
equilibrium $x = 1$ of the model and conditions that guarantee the solution
to be positive, which improve and generalize some recent results established
by Ch. G. Phios [Proc. Edinburgh Math. Soc. 35 (1992) 121--131] and by Zh. Zhou
and Q. Q. Zhang [Comp. Math. Appl. 38 (1999) 57--64].
- Two Wiener-type Tauberian theorems concerning Fourier hyperfunctions are
proved and commented. It is shownt that the shift asymptotics (S-asymptotics)
of a hyperfunction f is determined by the ordinary asymptotics of (f * K) (x)
as x approaches infinity, where K is Hörmander's kernel. Moreover, Wiener-type
theorems are used for the asympthotic analysis of solutions to some (pseudo-)
differential equations.
- We establish estimates for normal K-quasiconformal mappings z = g(w) of
any finitely-connected domain in the extended w-plane onto the interior or
exterior of the unit circle or the extended z-plane with n (>= 0)
slits
on the circles |z| = Rj (j = 1, ... , n). The bounds in the
estimates for Rj, |g(w)|, etc. are explicitly given. They are
sharp or asymptotically sharp and deduced mainly from estimates for the
inverse mappings of g in our previous paper [Rev. Roum. Math. Pures Appl. 38
(1993) 369--378] based on Carleman's and Grötzsch's inequalities and partly
improved here. A generalization of the Schwarz lemma and improvements of some
classical inequalities for conformal mappings are shown.
- The classical Cauchy-Kovalevskaya problem with holomorphic intial functions
is uniquely solvable provided the right-hand sides of the differential
equations are holomorphic in their variables, i.e., they transform holomorphic
functions into holomorphic functions. Moreover, the solutions depend
holomorphically on the space-like variables. A far-reaching generalization
of the Cauchy-Kovalevskaya Theorem is its abstract version which considers
an abstract operator equation in a scale of Banach spaces where the behaviour
of complex derivatives at the boundary is expressed by a certain mapping
property of the operator under consideration in the underlying scale.
Another generalization of the Cauchy-Kovalevskaya Theorem replaces the space
of holomorphic functions by another so-called associated space which is
defined by an elliptic operator. Making use of this second approach, the
present short note solves an extended Cauchy-Kovalevskaya problem in which
an initial value problem is combined with an implicit equation.
- Solutions of nonlinear difference equations of second order are investigated
with respect to their asymptotic behaviour. In particular, seven conjectures
of Kulenovic and Ladas concerning rational difference equations are verified.