bluuebunny/crossref_metadata_2025 huggingface dataset:
On the Decomposition of Unitary Operators into a Product of Finitely Many Positive Operators
Authors: Peltri G.
Published: September 2013

We will show that in an infinite-dimensional separable Hilbert space

\mathcal H

, there exist constants

N \in \mathbb N

and

c,d \in \mathbb R

such that every unitary operator can be written as the product of at most

N

positive invertible operators

\{a_k\} \subseteq B(\mathcal H)

with

\| a_k \| ≤ c

and

\|a^{–1}_k \| ≤ d

for all

k

. Some consequences of this result in the context of von Neumann algebras are discussed.
Non-Symmetric Matrix Riccati Equations
Authors: Freiling Gerhard , Jank G.
Published: September 2013

We prove a fundamental representation formula for all solutions of the matrix Riccati differential equation and of the corresponding algebraic Riccati equation. This formula contains the complete information on the phase portrait of the matrix equation and on the structure of the set F of all solutions of the corresponding algebraic equation. In particular we describe all constant, periodic and almost-periodic solutions of the matrix Riccati differential equation. Further we give an application of the fundamental representation formula to the investigation of non-autonomous Riccati equations.
Variational Bounds to Eigenvalues of Self-Adjoint Eigenvalue Problems with Arbitrary Spectrum
Authors: Zimmermann S. , Mertins U.
Published: September 2013

In the present paper a method by Lehmann-Maehly and Goerisch is extended to self-adjoint eigenvalue problems with arbitrary essential spectrum. This extension is obtained by consequently making use of the local character of the method. In this way, upper and lower bounds to all isolated eigenvalues are derived. In our proofs, the close relationship to Wielandt’s inverse iteration becomes quite obvious.
On the Sharpness of Error Bounds for the Numerical Solution of Initial Boundary Value Problems by Finite Difference Schemes
Authors: Esser H. , Goebbels Steffen J. , Nessel Rolf Joachim
Published: September 2013

The present paper studies the sharpness of error bounds obtained for approximate solutions of initial boundary value problems by finite difference schemes. Whereas the direct estimates in terms of partial moduli of continuity for partial derivatives of the (exact) solutions follow by standard methods (stability inequality plus Taylor expansion of the truncation error), the sharpness of these bounds is established by an application of a quantitative extension of the uniform boundedness principle. To verify the relevant resonance condition a general procedure is suggested, in contrast to our previous investigations which were based on rather specific properties of the discrete Green’s functions associated. Exemplarily, details are worked out in connection with Crank–Nicolson, Du Fort-Frankel and Saulyev schemes.
On a Fixed Point Theorem by Brosowski and Singh
Authors: Carbone Antonio
Published: September 2013

We generalize a fixed point theorem of B. Brosowski and S. P. Singh to multi-valued non-expansive maps in Banach spaces with convex structure. Such maps have applications, for example, in game theory and in the mathematical modelling of some economical problems.
On some Subclasses of Nevanlinna Functions
Authors: Hassi Seppo , de Snoo Hendrik S. V.
Published: September 2013

For any function

Q = Q(\ell)

belonging to the class

\mathbb N

of Nevanlinna functions, the function

Q_\tau = Q_{\tau}(\ell)

defined by

Q_{\tau}(\ell) = \frac{Q (\ell) –\tau(lmQ(\mu))^2}{\tau Q(\ell)+1}

belongs to

\mathbb N

for all values of

\tau \in \mathbb R \bigcup {\infty}

. The class

\mathbb N

possesses subclasses

\mathbb N_0 \subset \mathbb N_1

, each defined by some additional asymptotic conditions. If a function

Q

belongs to such a subclass, then for all but one value of

\tau \in \mathbb R \bigcup {\infty}

the function

Q_{\tau}

belongs to the same subclass and the corresponding exceptional function can be characterized (cf. [4]). In this note we introduce two subclasses

\mathbb N_{–2} \subset \mathbb N_{–1}

of

\mathbb N_0

which can be described in terms of the moments of the spectral measures in the associated integral representations. We characterize the corresponding exceptional function in a purely function-theoretic way by suitably estimating certain quadratic terms. The behaviour of the exceptional function connects the subclasses

\mathbb N_{–2}

and

\mathbb N_{–1}

to the classes

\mathbb N_0

and

\mathbb N_{–1}

, respectively, as studied in [4]. In operator-theoretic terms these notions have a translation in terms of

Q

-functions of selfadjoint extensions of a symmetric operator with defect numbers (1, 1). In this sense the exceptional function has an interpretation in terms of a generalized Friedrichs extension of the symmetric operator.
Nonregular Pseudo-Differential Operators
Authors: Marschall Jürgen
Published: September 2013

We study the boundedness properties of pseudo-differential operators

a(x, D)

and their adjoints

a(x, D)*

with symbols in a certain vector-valued Besov space on Besov spaces

B^s_{p,q}

and Triebel spaces

F^s_{p,q} (0 < p,q < \infty)

. Applications are given to multiplication properties of Besov and Triebel spaces. We show that our results are best possible for both pseudo-differential estimates and multiplication. Denoting by

(\cdot,\cdot)

the duality between Besov and between Triebel spaces we derive general conditions under which

(a(x, D)f, g) = (f a(x, D)*g)

holds. This requires a precise definition of

a(x, D)f

and

a(x, D)*f

for

f \in F^s_{p,q}

and

f \in B^s_{p,q}

.
A Numerically Rigorous Proof of Curve Veering in an Eigenvalue Problem for Differential Equations
Authors: Behnke H.
Published: September 2013

We consider parameter-dependent self-adjoint eigenvalue problems for differential equations. Frequently the eigenvalue curves show the interesting phenomenon of curve veer-ing. We propose a numerically rigorous procedure for proving this phenomenon in concrete situations.
On the Condition of Orthogonal Polynomials via Modified Moments
Authors: Fischer H.-J.
Published: September 2013

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.
A Note on the Bonnet-Myers Theorem
Authors: Boju Valentin , Funar Louis
Published: September 2013

The aim of this note is to derive a compactness result for complete manifolds whose Ricci curvature is bounded from below. The classical result, usually stated as Bonnet-Myers theorem, provides an estimation of the diameter of a manifold whose Ricci curvature is greater than a strictly positive constant. Weaker assumptions that the Ricci curvature function tends slowly to zero (when the distance from a fixed point goes to infinity) were already considered in [2, 3]. We shall improve here their results.
On the Dirichlet Problem for the Ekman Equation
Authors: Frischmuth Kurt , Rossmann Jürgen
Published: September 2013

The Ekman partial differential equation for the stream function of turbulent mass flow in shallow and small-sized surface waters are discussed. The Dirichlet problem for the Ekman 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 are studied.
On $x$-Analytic Solutions to the Cauchy Problem for Partial Differential Equations with Retarded Variables
Authors: Augustynowicz A. , Leszczyński H.
Published: September 2013

We prove some existence results for solutions analytic with respect to the spatial variables to first-order equations with a delay and some deviations not only at the function, 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.
Regularity for a Variational Inequality with a Pseudo differential Operator of Negative Order
Authors: Schumann Rainer
Published: September 2013

We prove that the solution of a variational inequality on a submanifold in

mathbb R^n

involving a pseudodifferential operator of order -1 is bounded.
On the Application of the Newton–Kantorovich Method to Nonlinear Partial Integral Equations
Authors: Appell Jürgen , De Pascale Espedito , Kalitvin Anatolij S. , Zabrejko Petr P.
Published: September 2013

We discuss the applicability of the Newton–Kantorovich method to a nonlinear equation which contains partial integrals with Uryson type kernels. A basic ingredient of this method consists in verifying a local Lipschitz condition for the Fréchet derivatives of the nonlinear partial integral operators generated by such kernels. The abstract results are illustrated in the space

C

of continuous functions and the Lebesgue space

L_p

for

1 ≤ p ≤ \infty

. In particular, it is shown that a local Lipschitz condition for the derivative in the space

L_p

for

p < \infty

leads to a degeneracy of the corresponding kernels. For ordinary integral operators, such a degeneracy occurs for

p ≤ 2

only.
On Absolute Summability Factors
Authors: Bor H.
Published: September 2013

By using for

\delta ≥ 0

so-called

|\bar{N}, p_n; \delta |_k

-boundedness of series

\sum^{\infty}_{n=1} a_n

and sequences

(\lambda _n)^{\infty}_{n=1}

we prove

| \bar{N}, p_n; \delta |_k

-summability of the series

\sum^{\infty}_{n=1} a_n \lambda _n

. This result generalizes a known one related to

|\bar{N}, p_n |_k

-summability of series.
Compactness of an Integro-Differential Operator of Cauchy-Kovalevskaya Theory
Authors: Tutschke Wolfgang , Vasudeva Harkrishan L.
Published: September 2013

The integral operator connected with the Cauchy-Kovalevskaya. initial value prob-lem has been shown to be compact between suitable Frechét spaces.
Fréchet Differentiability of the Solution of the Heat Equation with Respect to a Nonlinear Boundary Condition
Authors: Rösch Arnd
Published: September 2013

We consider the heat equation

\frac{\partial u}{\partial t} (t, x) = \Delta _x u(t,x)

with a nonlinear function a in the boundary condition

\frac{\partial u}{\partial n} (t, x) =\alpha ((u(t,z))(\vartheta – u(t,x))

depending on the boundary values

x

of the solution u of the initial-boundary value problem only and belonging to a set of admissible differentiable or uniformly Lipschitz continuous functions. For this problem Lipschitz continuity and Fréchet differentiability of the mapping

\Phi : \alpha \mapsto x

under different assumptions are derived.
A Non-Degeneracy Property for a Class of Degenerate Parabolic Equations
Authors: Ebmeyer Carsten
Published: September 2013

We deal with the initial and boundary value problem for the degenerate parabolic equation

u_t = \Delta \beta (u)

in the cylinder

\Omega \times (0,T)

, where

\Omega \subset \mathbb R^n

is bounded,

\beta (0) = \beta’(0) = 0

, and

\beta' ≥ 0

(e.g.,

\beta (u) = u |u|^{m–1} \ (m > 1))

. We study the appearance of the free boundary, and prove under certain hypothesis on

\beta

that the free boundary has a finite speed of propagation, and is Holder continuous. Further, we estimate the Lebesgue measure of the set where

u > 0

is small and obtain the non-degeneracy property

|\{ 0 < \beta' (u(x,t)) < \epsilon \} | ≤ c \epsilon^{\frac{1}{2}}

.
On Linear Integro-Differential Equations with Integral Impulsive Conditions
Authors: Akça Haydar , Berezansky Leonid , Braverman Elena
Published: September 2013

Linear integro-differential equations with linear integral impulsive conditions are considered. Existence results and representations of solutions are obtained. Stability of these equations is investigated.
Asymptotic Inequalities Related to the Maximum Modulus of a Polynomial
Authors: Frappier C. , Qazi M.A.
Published: September 2013

Let

\mathcal P_n

be the class of all polynomials of degree at most

n

. If

\| \cdot \|

denotes the supremum norm on

| z | =1

and

M_p(R) = max_{|x|=R} | P(z) |

, then for an arbitrary polynomial

P(z) = \sum ^n_{v=0} a_v z^v

in

\mathcal P_n

the inequality

M_P(R) ≤ R^n \| P \|

holds, with equality if and only if

a_0 = … = a_{n–1} = 0

. Given

n,k \in \mathbb N

with

0 ≤ k ≤ n–1

, let

\varphi _{n,k} (R)

be the largest number such that

M_P (R)+ \varphi_{n,k}(R)|a_k| ≤ R^n \|P\| (R ≥ 1)

for all

P \in \mathcal P_n

. Values of

\varphi_{n,k} (R)

for

k=0

and

k = 1

are known since some time. We study the case

k ≥ 2

.
On the Method of Backward Steps of Carathéodory-Tonelli
Authors: De Pascale Espedito , Marino Giuseppe , Metafune Giorgio
Published: September 2013

We study the convergence of a sequence of functions which has been introduced by Carathéodory and Tonelli in connection with the solvability of the Cauchy problem for ordinary differential equations.
Fourier Multipliers between Weighted Anisotropic Function Spaces. Part II: Besov-Triebel Spaces
Authors: Dintelmann P.
Published: September 2013

We determine certain classes

M(X^{s_0}_{p_0, q_0} (w_0)), Y^{s_1}_{p_1, q_1} (w_1)

of Fourier multipliers between weighted anisotropic Besov and Triebel spaces

X^{s_0}_{p_0, q_0} (w_0)

and

Y^{s_1}_{p_1, q_1} (w_1)

where

p_0 ≤ 1

and

w_0, w_1

are weight functions of polynomial growth. To this end we refine a method based on discrete characterizations of function spaces which was introduced in Part I of the paper. Thus widely generalized versions of known results of Bui, Johnson and others are obtained in a unified way.
Approximation of Solutions of Stochastic Differential Equations by Discontinuous Galerkin Methods
Authors: Grecksch Wilfried , Wadewitz Annett
Published: September 2013

The generalized solution of a system of Stratonovich equations is approximated by a discontinuous Galerkin method. A piecewise polynomial approximation is introduced. 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.
On a Class of Nonlinear Neumann Problems of Parabolic Type: Blow-Up of Solutions
Authors: Pozio M.A. , Tesei Alberto
Published: September 2013

We investigate large time behaviour of solutions for a class of nonlinear Neumann parabolic problems of indefinite type, possibly degenerate. 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.
Stability of Linear Evolution Equations in Lattice Normed Spaces
Authors: Gil' Michael
Published: September 2013

Linear evolution equations in a space with a generalized norm are considered. Stability conditions are obtained. In particular, the "freezing" method for ordinary differential equations is extended to equations in Banach spaces.
A Variant of the Mountain Pass Theorem and its Application to Hammerstein Integral Equations
Authors: Moroz V. , Zabrejko P. P.
Published: September 2013

A new variant of the mountain pass theorem based on a ’strong’ deformation lemma is presented. Some applications to the existence of non-trivial solutions of nonlinear Hammerstein integral equations are given too.
Conformal Completion of $\mathbb U(n)$-invariant Ricci-Flat Kãhler Metrics at Infinity
Authors: Kühnel Wolfgang , Rademacher Hans-Bert
Published: September 2013

For every

n≥2

we give an example of a complete

\mathbb U(n)

-invariant cohomogeneity one metric on

\mathbb R^{2n}

which is not conformally flat and which carries twistor spinors with zeros. The construction uses a conformal completion at infinity of a

\mathbb U(n)

-invariant Ricci-flat Kähler metric on

\mathbb R^{2n} \backslash \lbrace 0 \rbrace

given by Calabi [2] and by Freedman and Gibbons [4]. This extends our results in [6] for

n = 2

to all even dimensions.
On Some Interrelations between the Generalized Nehari Problem for the Carathéodory Class and Prediction Theory
Authors: Fritzsche B. , Kirstein B.
Published: September 2013

This paper is a continuation of the authors’ former studies on generalized Nehari problems (see [40, 44, 47]). We indicate the stochastic background of the generalized Nehari problem for the Carathéodory class. Moreover, we discuss some intimately related questions of prediction theory for stationarily cross-correlated stationary sequences.
Characterization of the Exponential Distribution by Properties of the Difference $X_{k+s:n} – X_{k:n}$ of Order Statistics
Authors: Rossberg H.-J. , Riedel M. , Ramachandran B. B.
Published: September 2013

Let

X_1 , X_2, \dots, X_n

be independent and identically distributed random variables subject to a continuous distribution function

F

, let

X_{1:n}, X_{2:n}, \dots, X_{n:n}

be the corresponding order statistics, and write

\tag{0}
P(X_{k+s:n} – X_{k:n} ≥ x) = P(X_{s:n–k} ≥ x) \ \ \ (x≥0)

where

n,k

and

s

are fixed integers with

k + s ≤ n

. It is an old question if condition (0) implies that

F

is of exponential type. In [8] we showed among others that condition (0) can be greatly relaxed; namely, it can be replaced by asymptotic relations (either as

x \to \infty

or

x \downarrow 0

) to derive this very result. Using a theorem on integrated Cauchy functional equations and in essential way a result of [8] we find now a more elegant and deeper theorem on this subject. The case of lattice distributions is also considered and some new problems are stated.
Nonlinear Vibration Systems with Two Parallel Random Excitations
Authors: vom Scheidt J. , Wöhrl U.
Published: September 2013

Systems of nonlinear vibration differential equations are investigated where the non-linearities are given by polynomials of any degree. The random excitations are induced by two parallel processes. These random excitations of an often applied type are expressed by linear functionals of weakly correlated processes with correlation length

\epsilon

. The moments of the solutions and their first and second derivatives are expanded with respect to

\epsilon

where all terms up to order

\epsilon^2

are included. Approximations of the correlation functions are given explicitely. Only the quadratic and cubic non-linearities have an influence on the correlation functions in this approximation order.
On the $\mathcal L$-Characteristic of the Superposition Operator in Lebesgue Spaces with Mixed Norm
Authors: Chur-jen Cheng , Väth Martin
Published: September 2013

We consider the superposition operator

Fx(t,s) = f(t,s,x(t,s))

of functions of two variables in spaces with mixed norm

[L_p \to L_q]

. After establishing a necessary and sufficient acting condition, we get some conclusions on the

\mathcal L

-characteristic of

F

. We also prove some theorems, which imply that

F

is uniformly continuous on balls in the interior of its

\mathcal L

-characteristic.
A General Approach to the Min-Max Principle
Authors: Corvellec J.-N.
Published: September 2013

Following the abstract approach of [11], we give a general mm-max principle in critical point theory which covers the classical results and applies to a variety of settings. Especially, thanks to the notion of weak slope introduced in [16] and results of [12], this principle applies to continuous functionals and some classes of lower semicontinuous functionals considered in the literature.
On the Mixed Problem for Quasilinear Partial Differential-Functional Equations of the First Order
Authors: Człapiński Tomasz
Published: September 2013

We consider the mixed problem for the quasilinear partial differential-functional equation of the first order

D_z(x,y) = \sum^n_{i=1}f_i(x, y, z_{(x,y)} D_{y,i}z(x, y) + G(x, y, z_{(x,y)})

z(x,y) = \phi(x,y) \ \ \ ((x,y) \in [-r,a] x [-b,b + h] \backslash [0,a] \times [-b,b])

where

z_{(x,y)} : [-r,0] \times [0,h] \to \mathbb R

is a function defined by

z_{(x,y)}(t,s) = z(x + t,y + s)

for

(t,s) \in [-r,0] \times [0,h]

. Using the method of characteristics and the fixed-point method we prove, under suitable assumptions, a theorem on the local existence and uniqueness of solutions of the problem.
On some Applications of Theorems on the Spectral Radius to Differential Equations
Authors: Bugajewski Dariusz
Published: September 2013

In this paper we obtain uniqueness theorems for the Darboux problem of neutral type in an implicit form. Our proofs are based on a fixed point theorem and theorems on the spectral radius of the sum of operators.
A Multiplier Approach to the Lance-Blecher Theorem
Authors: Frank Michael
Published: September 2013

A new approach to a theorem of E. C. Lance and D. P. Blecher in Hilbert

C*

- module theory and to two extensions of it is presented resting on a reinterpretation of key structural elements in terms of multiplier theory of operator

C*

-algebras. In the course of proving further facts are obtained.
Finite Section Method in some Algebras of Multiplication and Convolution Operators and a Flip
Authors: Roch Steffen , Santos P.A. , Silbermann Bernd
Published: September 2013

This paper is concerned with the applicability of the finite section method to operators belonging to the closed subalgebra of

\mathcal L(L^2(\mathbb R))

generated by operators of multiplication by piecewise continuous functions in

\mathbb R

, convolution operators - also with piecewise continuous generating functions - and the flip operator

(Ju)(x) = u(–x)

. For this, a larger algebra of sequences is introduced, which contains the special sequences we are interested in. There is a direct relationship between the applicability of the finite section method for a given operator and the invertibility of the corresponding sequence in this algebra. Exploring this relationship, the methods of essentialization, localization and identification of the local algebras through construction of locally equivalent representations are used and so useful invertibility criteria are derived. Finally, examples are presented, including explicit conditions for the applicability of the finite section method to a Wiener-Hopf plus Hankel operator with piecewise continuous symbols, and some relations between the approximation operators and the limit operator are discussed.
Nonlinear Geometric Optics for Shock Waves Part I: Scalar Case
Authors: Wang Ya-Guang
Published: September 2013

In this paper we study the nonlinear geometric optics of the shock wave for a scalar conservation law in one space variable. The existence of the oscillatory shock wave and its leading terms are obtained. Meanwhile, we rigorously justify the asymptotic properties of the shock wave as well as the shock front.
About the Existence of Solutions of a Boundary Value Problem for a Carathéodory Differential System
Authors: Anichini G. , Conti Giuseppe
Published: September 2013

A boundary value problem of the form

x’ = f(t,x), x \in S

is considered where

f

is a Carathéodory function and

S

is a suitable boundary constraint. A related integral equation with an exact number of solutions is studied, in order to show the existence of solutions of the boundary value problem through a fixed point theorem for

w

-maps in the Darbo sense.
Marcinkiewicz-Zygmund-Type Inequalities Trigonometric Interpolation on Non-Uniform Grids and Unconditional Schauder Bases in Besov Spaces on the Torus
Authors: Runovski Konstantin V. , Sickel Winfried
Published: September 2013

We investigate extensions of the classical Marcinkiewicz-Zygmund inequality to non-uniform grids. Then we show how this inequality can be used to characterize Besov spaces on the torus by means of approximation by corresponding interpolatory polynomials. One of these characterizations can be applied to the construction of unconditional Schauder bases.
Peano Kernels of Non-Integer Order
Authors: Diethelm K.
Published: September 2013

We consider the representation of error functionals in numerical quadrature by the Peano kernel method. It is easily observed that the usual expressions for Peano kernels of order s still make sense if s is not a natural number. In this paper, we discuss how to interpret these Peano kernels, we state their main properties, and we compare them to the (classical) Peano kernels of integer order.
The Skew Derivative Problem in the Exterior of Open Curves in a Plane
Authors: Krutitskii P.A.
Published: September 2013

The skew derivative problem for the Laplace equation in the exterior of open curves in the plane is reduced to the Fredholm integral equation of the second kind, which is uniquely solvable.
Weighted Norm Inequalities for Riemann-Liouville Fractional Integrals of Order Less than One
Authors: Rakotondratsimba Y.
Published: September 2013

Necessary and sufficient condition on weight functions

u(\cdot)

and

v(\cdot )

are derived in order that the Riemann-Liouville fractional integral operator

R_{\alpha} (0 < \alpha < 1)

is bounded from the weighted Lebesgue spaces

L^p((0, \infty),v(x)dx)

into

L^q((0, \infty),u(x)dx)

whenever

1 < p ≤ q < \infty

or

1 < q < p < \infty

. As a consequence for monotone weights then a simple characterization for this boundedness is given whenever

p ≤ q

. Similar problems for convolution operators, acting on the whole real axis

(–\infty, \infty)

, are also solved.
A Note on Fixed Point Theorems in Shells of Banach Spaces for Weakly Condensing Operators
Authors: O'Regan Donal
Published: September 2013

New fixed point results are presented for weakly condensing maps. The analysis relies on a continuation theorem [3), recently presented for such operators.
A Nonlinear Beam Equation Arising in the Theory of Elastic Bodies
Authors: Doppel Karl , Herfort W. , Pflüger K.
Published: September 2013

We study the global solvability of a nonlinear Cauchy problem, which arises in the theory of oscillations in elastic bodies. We show that the linearized problem defines a contraction semigroup, which is then used to transform the Cauchy problem into an integral equation. Finally, it is shown that the corresponding integral operator has a unique fixed point, which gives rise to a global solution of the original nonlinear problem.
On Ill-Posedness Measures and Space Change in Sobolev Scales
Authors: Hofmann Bernd , Tautenhahn Ulrich
Published: September 2013

The degree of ill-posedness of a linear inverse problem is an important knowledge base to select appropriate regularization methods for the stable approximate solution of such a problem. In this paper, we consider ill-posedness measures for a linear ill-posed operator equation

Ax = y

, where the compact linear operator

A : X \to Y

maps between infinite dimensional Hilbert spaces. Using the decay rate of singular values of

A

tending to zero we define an interval of ill-posedness and motivate its meaning by considering lower and upper bounds for the rates of the condition numbers occurring in the numerical solution process of the discretized problem. An equivalent interval information is obtained when compactness measures as

\epsilon

-entropy or

\epsilon

-capacity are exploited alternatively. For the specific case

X := L^2(0, 1)

, the space change problem of shifting the space

X

along a Sobolev scale is treated. In detail, we study the change of the interval of ill-posedness if the solutions are restricted to the Sobolev space

W^1_2 [0, 1)

. The results of these considerations are a warning to characterize the ill-posedness of a problem superficial. Moreover, the interdependences between ill-posedess measures, embedding operators, Hilbert and Sobolev scales are discussed.
On Bernstein’s Theorem for Quasiminimal Surfaces Part II
Authors: Hoy Erich
Published: September 2013

Bernstein-type theorem is proved for surfaces with a quasiconformal Gauss map and with a growth condition for the total curvature.
On Symbol Analysis of Periodic Pseudo differential Operators
Authors: Turunen Ville , Vainikko Gennadi
Published: September 2013

This paper deals with periodic pseudodifferential operators on the unit circle

\mathbb T^1 = \mathbb R^1/ \mathbb Z^1

. The main results are asymptotic expansions for adjoints and products of periodic pseudodifferential operators.
An Integral Estimate for the Gradient for a Class of Nonlinear Elliptic Equations in the Plane
Authors: Softova Lubomira G.
Published: September 2013

An a priori estimate is established for the gradient of the solution to Dirichlet’s problem for a class of nonlinear differential equations on a convex domain in the plane. The nonlinear operator is assumed to be elliptic in the sense of Campanato. By virtue of the Leray-Schauder fixed point theorem an existence result for the problem under consideration is derived.
An Inverse Problem for a Viscoelastic Timoshenko Beam Model
Authors: Cavaterra Cecilia
Published: September 2013

We consider the Timoshenko model for a viscoelastic beam. This model consists in a system of two coupled Volterra integrodifferential equations describing the evolution of the mean displacement

w

and of the mean angle of rotation

\varphi

. The damping mechanism is characterized by two time-dependent memory kernels,

a

and

b

, which are a priori unknown. Provided that

(w, \varphi)

solves a suitable initial and boundary value problem for the evolution system, the inverse problem of determining a and b fçom supplementary information is analyzed. A result of existence and uniqueness on a given bounded time interval is proved. In addition, Lipschitz continuous dependence of the solution

(w, \varphi,a,b)

on the data is shown.
Weighted Inequalities of Weak Type for the Fractional Integral Operator
Authors: Rakotondratsimba Y.
Published: September 2013

Sufficient conditions on weights

u(\cdot)

and

v(\cdot)

are given in order that the usual fractional integral operator

I_{\alpha} (0 < \alpha < n)

is bounded from the weighted Lebesgue space

L^p(v(x)dx)

into weak-

L^p(u(x)dx)

, with

1 ≤ p < \infty

. As a consequence a characterization for this boundedness is obtained for a large class of weight functions which particularly contains radial monotone weights.
On the Solution of an Integral-Functional Equation with a Parameter
Authors: Berg Lothar , Krüppel Manfred
Published: September 2013

For a homogeneous integral-functional equation containing a parameter, we show existence and uniqueness of a compactly supported solution with given value for its integral. The solution is infinitely often differentiable, symmetric with respect to the point 1/2, monotonous at both sides of 1/2 and satisfies further functional equations. The Fourier series of the periodic continuation is determined. We also investigate spectral properties of the integral equation and find surprising connections between the Laplace transform of the eigenfunction and the eigenfunctions of the adjoint equation, and also directly between different eigenfunc-tions both in the compact and in the non-compact case. Moreover, asymptotic considerations are made.
Convergence of the Newton-Kantorovich Method under Vertgeim Conditions: a New Improvement
Authors: De Pascale Espedito , Zabreiko Petr P.
Published: September 2013

Let

f: B(x_0, R) \subset X \to Y

be an operator from a closed ball of a Banach space

X

to a Banach space

Y

. We give new conditions to ensure the convergence of Newton-Kantorovich approximations toward a solution of the equation

f(x) = 0

, under the hypothesis that

f'

be Hölder continuous. The case of

f'

being Hölder continuous in a generalized sense is analyzed as well.
Spectral Properties of Partial Integral Operators of Volterra and Volterra-Fredholm Type
Authors: Kalitvin A.S.
Published: September 2013

We give sufficient conditions under which the spectral radius of a partial integral operator of Volterra type is zero, or a partial integral operator of Volterra-Fredholm type has a trivial essential spectrum.
On Persson’s Theorem in Local Dirichiet Spaces
Authors: Grillo Gabriele
Published: September 2013

Given a strongly local, regular and irreducible Dirichlet form

\mathcal E

, we prove a version of Persson’s theorem concerning the variational characterization of the bottom of the essential spectrum of the generator

H

of

\mathcal E

. Such a result is then used to prove

L^p

-exponential decay of the "small eigenfunctions" of

H

.
Blow-Up in Exterior Domains: Existence and Star-Shapedness
Authors: Francini E. , Greco Antonio
Published: September 2013

For some nonlinear elliptic equations in divergence form, we consider the solutions, defined in the exterior of a contractible bounded domain

\mathcal G

, which become infinite at the boundary. We prove the existence of such solutions and study the shape of their level sets when

\mathcal G

is star-shaped.
An Extremal Problem Related to the Maximum Modulus Theorem for Stokes Functions
Authors: Kratz Werner
Published: September 2013

There are considered classical solutions

\nu

of the Stokes system in the ball

B = \{Ix \in \mathbb R^n : |x| < 1\}

, which are continuous up to the boundary. We derive the-optimal constant

c = c_n

, such that, for all

x \in B

,

| \nu (x) | ≤ c \ \mathrm {max}_{\xi \in \partial B} | \nu(\xi)| \ \ \ (\ast)

holds for all such functions. We show that

c_n = \mathrm {max}_{x \in \partial B} c_n (x)

exists, where

c_n(x)

is the minimal constant in (

\ast

) for any fixed

x \in B

. The constants

c_n(x)

are determined explicitly via the Stokes–Poisson integral formula and via a general theorem on the norm of certain linear mappings given by some matrix kernel. Moreover, the asymptotic behaviour of the

c_n(x)

as

x \to \partial B

and as

n \to \infty

is derived.In the concluding section the general result on the norm of linear mappings is used to prove two inequalities: one for linear combinations of Fourier coefficients and the other from matrix analysis.
On the Stabilization of the Inversion of Some Kontorovich-Lebedev Like Integral Transforms
Authors: Glaeske Hans-Jürgen , Yakubovich Semyon B.
Published: September 2013

In this paper we construct a special type of regularization operators for the Kontorovich-Lebedev type integral transforms to stabilize their inversion in weighted $L_{\nu, p} spaces. Some estimates of norms in these spaces are obtained.
The Riemann Problem for a Two-Dimensional Hyperbolic System of Nonlinear Conservation Laws: Multiplication of Distribution Solutions
Authors: Hu Jiaxin
Published: September 2013

In the papers [6-8], the author has constructed the Riemann solutions to a two-dimensional hyperbolic system of nonlinear conservation laws for any piecewise constant initial data having two discontinuity rays with origin as vertex. It has been found that, for some initial data, the Riemann solutions no longer lie in

L^{\infty}_{loc} (\mathbb R^2 \times \mathbb R_+)

, and the non-classical waves (labelled as Dirac-contact waves) have arisen. But it remains open in [6-8] to verify that the non-classical solutions constructed satisfy the system considered. In the present paper, we borrow the new mathematical theory of generalized functions, chiefly initiated by J. F. Colombeau and Rosinger, to deal with the diffculty of the multiplication of distribution solutions. The non-classical Riemann solutions we constructed in [6-8] satisfy the system in the sense of association. The present paper provides a good example of applications for this new mathematical theory in powerfully handling the product of generalized functions.
Solutions and Periodic Solutions for Nonlinear Evolution Equations with Non-Monotone Perturbations
Authors: Avgerinos E.P. , Papageorgiou Nikolaos S.
Published: September 2013

In this paper we solve periodic and Cauchy problems for nonlinear evolution equa-tions driven by time-dependent, pseudomonotone operators and a non-monotone perturbation term. Our proof produces as a by-product a useful property of the solution map for maximal monotone problems. Two examples of nonlinear parabolic problems illustrate the applicability of our work.
Note on the Fourier-Laplace Transform of $\overline{\partial}$-Cohomology Classes
Authors: Hatziafratis T.
Published: September 2013

We construct the inverse of the Fourier-Laplace transform of

\overline{\partial}

-cohomology classes (of

(n, n–1)

-forms) in the complement of a convex compact set in

\mathbb C^n

, thus giving an analogue of the Borel transform (and its Polya representation) of entire functions of exponential type in several variables. The construction is based on a formula of Berndtsson.
Cantor Sets and Integral-Functional Equations
Authors: Berg Lothar , Krüppel Manfred
Published: September 2013

In this paper, we continue our considerations in [1] on a homogeneous integral-functional equation with a parameter

a > 1

. In the case of

a > 2

the solution

\phi

satisfies relations containing polynomials. By means of these polynomial relations the solution can explicitly be computed on a Cantor set with Lebesgue measure 1. Thus the representation of the solution

\phi

is immediately connected with the exploration of some Cantor sets, the corresponding singular functions of which can be characterized by a system of functional equations depending on

a

. In the limit case

a = 2

we get a formula for the explicit computation of

\phi

in all dyadic points. We also calculate the iterated kernels and approximate

\phi

by splines in the general case

a > 1

.
A Multi-Dimensional Hausdorff Moment Problem: Regularization by Finite Moments
Authors: Ang Dang Dinh , Gorenflo Rudolf , Trong Dang Duc
Published: September 2013

We consider the multi-dimensional Hausdorff moment problem over the unit cube: to reconstruct an unknown function from the (inaccurately) given values of the integrals of the unknown function multiplied by all power-products of the independent variables. We describe a regularization scheme using orthogonalization by the tensor product of (shifted) Legendre polynomials and "approximation" of the unknown function by a finite sum, the dimension of the space of approximation playing the role of the regularization parameter. For the case of square integrability of the unknown function we present an estimate of the regularization error that implies convergence if the data error tends to zero.
On Morozov’s Method for Tikhonov Regularization as an Optimal Order Yielding Algorithm
Authors: Nair M. Thamban
Published: September 2013

It is shown that Tikhonov regularization for an ill-posed operator equation

Kx = y

using a possibly unbounded regularizing operator

L

yields an order-optimal algorithm with respect to certain stability set when the regularization parameter is chosen according to Morozov’s discrepancy principle. A more realistic error estimate is derived when the operators

K

and

L

are related to a Hilbert scale in a suitable manner. The result includes known error estimates for ordininary Tikhonov regularization and also estimates available under the Hilbert scales approach.
Frequent Oscillation of a Class of Partial Difference Equations
Authors: Tian C.J. , Zhang Binggen
Published: September 2013

This paper is concerned with the partial difference equation

A_{m+l,n} + A_{m,n+l} - a_{m,n}A_{m,n} + p_{m,n}A_{m–k,n–l} = 0

where

k

and

l

are non-negative integers, {

a_{m,n}

} and {

p_{m,n}

} are real double sequences. Frequent oscillation criteria of this equation are obtained.
On the Autonomous Nemytskij Operator in Hölder Spaces
Authors: Goebel Manfred , Sachweh F.
Published: September 2013

The paper is devoted to the autonomous Nemytskij operator (superposition operator) in Hölder spaces

H^{k+\alpha}[a,b], (k, \alpha) \in \mathbb Z_+ \times [0, 1]. We study acting, continuity, Lipschitz continuity, and Fréchet differentiability conditions. For

k = 0

,

\alpha \in [0, 1]

and

k \in \mathbb N, \alpha = 1

the respective conditions are both necessary and sufficient. For

k \in \mathbb N, \alpha \in (0,1)$ only the acting condition is both necessary and sufficient; the other investigated properties are characterized by necessary and sufficient conditions different from each other.
Determining the Relaxation Kernel in Nonlinear One-Dimensional Viscoelasticity
Authors: Grasselli Maurizio
Published: September 2013

We consider a viscoelastic string whose mechanical behavior is governed by a non-linear stress-strain relationship. This constitutive law is characterized by a time-dependent relaxation kernel

k

which is assumed to be unknown. The resulting motion equation is then associated with initial and Dirichlet boundary conditions. We show that the traction measurement at one end allows to identify

k

. More precisely, we prove an existence and uniqueness result on a small time interval. Also, we show how the solution continuously depends on the data.
On Operators and Elementary Functions in Clifford Analysis
Authors: Sprössig Wolfgang
Published: September 2013

In this paper a survey on the construction of Clifford regular elementary func-tions.by Fueter’s mapping is given. Furthermore, using a suitable decomposition of the Dirac operator an application of the

\bar{\partial}

-problem is lined out.
Iterated Integral Operators in Clifford Analysis
Authors: Begehr Heinrich
Published: September 2013

Integral representation formulas of Cauchy-Pompeiu type expressing Clifford-algebra-valued functions in domains of

\mathbb R^m

through its boundary values and its first order derivatives in form of the Dirac operator are iterated in order to get higher order Cauchy-Pompeiu formulas. In the most general representation formulas obtained the Dirac operator is replaced by products of powers of the Dirac and the Laplace operator. Boundary values of lower order operators are involved too. In particular the integral operators provide particular solutions to the inhomogeneous equations

\partial ^k w = f, \Delta ^k w = g

and

\partial \Delta ^k w = h

. The main subject of this paper is to develop the representation formulas. Properties of the integral operators are not studied here.
Nonlinear Singular Integral Equations Involving the Hilbert Transform in Clifford Analysis
Authors: Bernstein Swanhild
Published: September 2013

We apply operator-theoretical methods for monotone and maximal monotone operators to prove the existence of solutions for nonlinear singular integral and integro-differential equations involving the Hilbert transform in the Clifford algebra

C \mathcal l_{n, 0}

. Properties of the Hilbert transform are proved using Clifford analysis. We generalize well-known results concerning the complex Hilbert transform and the singular Cauchy integral operator to higher dimensions.
The Discretization of Neutral Functional Integro-Differential Equations by Collocation Methods
Authors: Brunner Hermann
Published: September 2013

We study the approximate solution of certain neutral functional integro- differentialequations describing the aeroelastic motions of certain airfoils, and of related Volterra equations with a delay argument. The focus of the paper is on questions concerning the (uniform) con-vergence and local superconvergence properties of discretization methods based on collocation techniques for equations with smooth kernels.
A Multidimensional Identification Problem Related to a Hyperbolic Integro-Differential Equation
Authors: Lorenzi Alfredo
Published: September 2013

We prove a global in time existence and uniqueness theorem for the identification of a relaxation kernel h entering a hyperbolic integro- differential equation, related to a convex cylinder with a smooth lateral surface, when the coefficient

h

is assumed to depend on time and one space variable and general additional conditions are provided. A continuous dependence result for the identification problem is also stated. Finally, a separate proof concerning the existence and uniqueness of the solution to the related direct integro-differential problem is also given in a suitable functional space. Moreover, the dependence of such a solution with respect to the relaxation kernel is fully analysed.
On the Controllability of a Slowly Rotating Timoshenko Beam
Authors: Krabs Werner , Sklyar Grigory M.
Published: September 2013

We consider a slowly rotating Timoshenko beam in a horizontal plane whose move-ment is controlled by the angular acceleration of the disk of a driving motor into which the beam is clamped. The problem to be solved is to transfer the beam from a position of rest into a position of rest under a given angle within a given time. We show that this problem is solvable, if the time of rotation prescribed is large enough.
Lipschitz Stability of Solutions to Parametric Optimal Control Problems for Parabolic Equations
Authors: Malanowksi K. , Tröltzsch Fredi
Published: September 2013

A class of parametric optimal control problems for semilinear parabolic equations is considered. Using recent regularity results for solutions of such equations, sufficient condi-tions are derived under which the solutions to optimal control problems are locally Lipschitz continuous functions of the parameter in the

L^\infty

-norm. It is shown that these conditions are also necessary, provided that the dependence of data on the parameter is sufficiently strong.
On Uniqueness Conditions for Decreasing Solutions of Semilinear Elliptic Equations
Authors: Tadie
Published: September 2013

For

f\in C((0, \infty)) \cap C^1 ((0, \infty))

and

b > 0

, existence and uniqueness of radial solutions

u = u(r)

of the problem

\Delta u + f(u_+) = 0

in

\mathbb R^n (n > 2), u(0) = b

and

u’(0) = 0

are well known. The uniqueness for the above problem with boundary conditions

u(R) = 0

and

u’(0) = 0

is less known beside the cases where

lim_{r \to \infty} u(r) = 0

. It is our goal to give some sufficient conditions for the uniqueness of the solutions of the problem

D_{\alpha}u + f(u_+) = 0 (r > 0), u(p) = 0

and

u’(0) = 0

based only on the evolution of the functions

f(t)

and

\frac{d}{dt} \frac{f(t)}{t}

.
Compactness and Existence Results for Ordinary Differential Equations in Banach Spaces
Authors: Appell Jürgen , Väth Martin , Vignoli A.
Published: September 2013

We prove that the Picard-Lindelöf operator

Hx(t) = \int^t_{t_0} f(s, x(s)) ds

with a vector function

f

is continuous and compact (condensing) in

C

, if

f

satisfies only a mild boundedness condition, and if

f(s,\cdot)

is continuous and compact (resp. condensing). This generalizes recent results of the second author and immediately leads to existence theorems for local weak solutions of the initial value problem for ordinary differential equations in Banach spaces.
The Generalized Riemann-Hilbert Boundary Value Problem for Non-Homogeneous Polyanalytic Differential Equation of Order $n$ in the Sobolev Space $W_{n,p}(D)$
Authors: Mshimba Ali Seif
Published: September 2013

Given is a nonlinear non-homogeneous polyanalytic differential equation of order

n

in a simply-connected domain

D

in the complex plane. Initially we prove (under certain conditions) the existence of its general solution in

W_{n,p}(D)

by first transforming it into a system of integro-differential equations. Next we prove the solvability of a generalized Riemann-Hilbert problem for the differential equation. This is effected by first reducing the boundary value problem posed to a corresponding one for a polyanalytic function. The latter is then transformed into

n

classical Riemann-Hilbert problems for holomorphic functions, whose solutions are known in the literature.
On Products of Admissible Liftings and Densities
Authors: Macheras N. D. , Musiał K. , Strauss W.
Published: September 2013

We introduce the classes of admissible densities and show that there exist densities respecting coordinates, and product densities with marginals prescribable within these classes. For liftings there is the corresponding class of admissibly generated liftings. We apply these results to improve theorems on product liftings and liftings respecting coordinatei and to provide them a unifying approach.