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European Mathematical Society Publishing House
2024-03-29 10:26:18
8
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=25&iss=2&update_since=2024-03-29
Zeitschrift für Analysis und ihre Anwendungen
Z. Anal. Anwend.
ZAA
0232-2064
1661-4534
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
10.4171/ZAA
http://www.ems-ph.org/doi/10.4171/ZAA
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2006)
25
2006
2
Nonexistence of Solutions to a Hyperbolic Equation with a Time Fractional Damping
Mokhtar
Kirane
Lab. et Dép. de Mathématiques, LA ROCHELLE CEDEX 1, FRANCE
Nasser-edine
Tatar
Dept. of Mathematics, DHAHRAN, SAUDI ARABIA
Fractional damping, non-existence, nonlinear hyperbolic equations
We consider the nonlinear hyperbolic equation \begin{align*} u_{tt}-\Delta u+D_{+}^{\alpha }u=h(t,x)\left| u\right| ^{p} \end{align*} posed in $Q:=(0,\infty )\times \mathbb{R}^{N},$ where $D_{+}^{\alpha }u$, $% 0
Partial differential equations
Real functions
Ordinary differential equations
General
131
142
10.4171/ZAA/1281
http://www.ems-ph.org/doi/10.4171/ZAA/1281
A Gauss-Bonnet Formula for Metrics with Varying Signature
Michael
Steller
Universität Stuttgart, STUTTGART, GERMANY
Gauss-Bonnet formula, singular metric, pseudo-geodesic, generic metric
A Gauss-Bonnet formula for compact orientable connected Riemannian or Lorentz\-ian 2-manifolds is well-known. We investigate singular metrics on 2-manifolds with varying signature. Such metrics are necessarily degenerate at some points of $M$ where most of the usual definitions for geometric quantities break down. We prove that under some additional assumptions there is a Gauss--Bonnet formula for compact orientable connected 2-manifolds with a singular metric. Some examples are given.
Differential geometry
General
143
162
10.4171/ZAA/1282
http://www.ems-ph.org/doi/10.4171/ZAA/1282
Approximative Compactness and Full Rotundity in Musielak-Orlicz spaces and Lorentz-Orlicz spaces
Henryk
Hudzik
Adam Mickiewicz University, POZNAN, POLAND
Wojciech
Kowalewski
Adam Mickiewicz University, POZNAN, POLAND
Grzegorz
Lewicki
Jagiellonian University, KRAKOW, POLAND
Musielak-Orlicz spaces, Lorentz-Orlicz spaces, Luxemburg norm, Amemyia norm, approximative compactness, reflexivity, Kadec-Klee property, drop property, full rotundity
We prove that approximative compactness of a Banach space $X$ is equivalent to the conjunction of reflexivity and the Kadec-Klee property of $X$. This means that approximative compactness coincides with the drop property defined by Rolewicz in {\it Studia Math.} 85 (1987), 25 -- 35. %\cite{RO}. Using this general result we find criteria for approximative compactness in the class of Musielak--Orlicz function and sequence spaces for both (the Luxemburg norm and the Amemiya norm) as well as critria for this property in the class of Lorentz--Orlicz spaces. Criteria for full rotundity of Musielak-Orlicz spaces are also presented in the case of the Luxemburg norm. An example of a reflexive strictly convex K\"othe function space which is not approximatively compact and some remark concerning the compact faces property for Musielak--Orlicz spaces are given.
Functional analysis
General
163
192
10.4171/ZAA/1283
http://www.ems-ph.org/doi/10.4171/ZAA/1283
The Plancherel and Hausdorff–Young Type Theorems for an Index Transformation
Semyon
Yakubovich
Faculdade de Ciências do Porto, PORTO, PORTUGAL
Bessel functions, index transform, Mellin transform, Kontorovich-Lebedev transform, Plancherel theorem, Hausdorff-Young inequality, Parseval equality
The Plancherel and Hausdorff-Young type theorems are proved for an integral transformation, which is associated with the product of the modified Bessel functions of different arguments. The transform essentially generalizes Lebedev's transformation involving squares of the modified Bessel functions as kernels.
Integral transforms, operational calculus
Special functions
General
193
204
10.4171/ZAA/1284
http://www.ems-ph.org/doi/10.4171/ZAA/1284
Conditions for Correct Solvability of a Simplest Singular Boundary Value Problem of General Form. I
N.A.
Chernyavskaya
Ben Gurion University of the Negev, BEER-SHEBA, ISRAEL
L.A.
Shuster
Bar-Ilan University, RAMAT-GAN, ISRAEL
First order linear differential equation, correct solvability
We consider the singular boundary value problem %\eqref{1} -- \eqref{2} %\begin{equation}\label{1} %$$-r(x)y'(x)+q(x)y(x)=f(x),\quad x\in R$$ %\end{equation} %\begin{equation}\label{2} %$$\lim_{|x|\to\iy}y(x)=0,$$ %\end{equation} \begin{align*} -r(x)y'(x)+q(x)y(x)&=f(x),\quad x\in R \\ \lim_{|x|\to\iy}y(x)&=0, \end{align*} where $f \in L_p(\mathbb R),$\ $p\in[1,\iy]$ $(L_\iy(\mathbb R):=C(\mathbb R)),$\ $r $ is a continuous positive function on $\mathbb R$, \ $ 0\le q \in L_1^{\loc}.$ A solution of this problem is, by definition, any absolutely continuous function $y $ satisfying the limit condition and almost everywhere the differential equation. This problem is called correctly solvable in a given space $L_p(\mathbb R)$ if for any function $f\in L_p(\mathbb R)$ it has a unique solution $y\in L_p(\mathbb R)$ and if the following inequality holds with an absolute constant $c_p\in (0,\iy):$ %\begin{equation}\label{3} $$\|y\|_{L_p(\mathbb R)}\le c_p\|f\|_{L_p(\mathbb R)},\quad \ f\in L_p(\mathbb R) . %\end{equation} $$ We find minimal requirements for $r $ and $q$ under which the above problem is correctly solvable in $L_p(\mathbb R).$
Ordinary differential equations
General
205
235
10.4171/ZAA/1285
http://www.ems-ph.org/doi/10.4171/ZAA/1285
Existence of Periodic Solutions of a Class of Planar Systems
Xiaojing
Yang
Tsinghua University, BEIJING, CHINA
Periodic solutions, resonance, planar systems
In this paper, we consider the existence of periodic solutions for the following planar system: $$ J u'=\D H(u)+ G(u)+h(t)\,, $$ where the function $H(u)\in C^3(\R^2\backslash \{0\},\,\R)$ is positive for $u\ne 0$ and positively $(q,\,p)$-quasi-homogeneous of quasi-degree $pq,\, \,G: \R^2\to \R^2$ is local Lipschitz and bounded, $h\in L^\infty(0,\,2\pi)$ is $2\pi$-periodic and $J$ is the standard symplectic matrix.
Ordinary differential equations
General
237
248
10.4171/ZAA/1286
http://www.ems-ph.org/doi/10.4171/ZAA/1286
Fewer Convergence Conditions for the Halley Method
José Antonio
Ezquerro
Universidad de la Rioja, LOGRONO - LA RIOJA, SPAIN
M.A.
Hernández
Universidad de la Rioja, LOGRONO - LA RIOJA, SPAIN
Nonlinear equations in Banach spaces, Halley's method, semilocal convergence theorem, nonlinear integral equation
We present a new semilocal convergence result of Newton-Kantorovich type for Halley's method, where fewer convergence conditions are required than all the existing ones until now.
Operator theory
Integral equations
Numerical analysis
General
249
255
10.4171/ZAA/1287
http://www.ems-ph.org/doi/10.4171/ZAA/1287
An Existence Result for a Class of Extended Inclusion Problems with Applications to Equilibrium Problems
Ya-ping
Fang
Sichuan University, SICHUAN, CHINA
Nan-jing
Huang
Sichuan University, SICHUAN, CHINA
Extended inclusion problem, equilibrium problem, variational inequality, existence
Let $X$ be a real reflexive Banach space, $K\subset X$ a nonempty, closed and convex set, and $F:K\times K\to 2^X$ (the family of all the subsets of $X$) be a multi-valued mapping. In this paper, we consider the following extended inclusion problem: find $x^*\in K$ such that $K\subset F(x^*,x^*)$. Under suitable conditions, we establish an existence result for the extended inclusion problem. As applications, we give some existence theorems for equilibrium problems.
Calculus of variations and optimal control; optimization
Operator theory
General
257
264
10.4171/ZAA/1288
http://www.ems-ph.org/doi/10.4171/ZAA/1288