- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 08:08:13
23
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=36&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)
36
2017
1
Approximation by Riesz Means of Hexagonal Fourier Series
Ali
Guven
Balikesir University, BALIKESIR, TURKEY
Hexagonal Fourier series, Hölder class, Riesz mean
Let $f$ be an $H$-periodic (periodic with respect to the hexagon lattice) Hölder continuous function of two real variables. The error $\| f-R_{n}( p_{k};f) \|$ is estimated in the uniform norm and in the Hölder norm, where $(p_{k})$ is a sequence of numbers such that $0 < p_{0} \leq p_{1}\leq \cdots$ and $R_{n} (p_{k};f)$ is the $n$th Riesz mean of hexagonal Fourier series of $f$ with respect to $(p_{k})$.
Fourier analysis
Approximations and expansions
1
16
10.4171/ZAA/1576
http://www.ems-ph.org/doi/10.4171/ZAA/1576
Generalized Morrey Spaces – Revisited
Ali
Akbulut
Ahi Evran University, KIRSEHIR, TURKEY
Vagif Sabir
Guliyev
Ahi Evran University, KIRSEHIR, TURKEY
Takahiro
Noi
Tokyo Metropolitan University, TOKYO, JAPAN
Yoshihiro
Sawano
Tokyo Metropolitan University, TOKYO, JAPAN
Generalized Morrey spaces, decomposition, maximal operators
The generalized Morrey space ${\mathcal M}_{p,\phi}({\mathbb R}^n)$ was defined by Mizuhara 1991 and Nakai in 1994. It is equipped with a parameter $0 < p < \infty$ and a function $\phi:{\mathbb R}^n \times (0,\infty) \to (0,\infty)$. Our experience shows that ${\mathcal M}_{p,\phi}({\mathbb R}^n)$ is easy to handle when $1 < p < \infty$. However, when $0 < p \le 1$, the function space ${\mathcal M}_{p,\phi}({\mathbb R}^n)$ is difficult to handle as many examples show. We propose a way to deal with ${\mathcal M}_{p,\phi}({\mathbb R}^n)$ for $0 < p \le 1$, in particular, to obtain some estimates of the Hardy–Littlewood maximal operator on these spaces. Especially, the vector-valued estimates obtained in the earlier papers are refined. The key tool is the weighted dual Hardy operator. Much is known on the weighted dual Hardy operator.
Fourier analysis
Approximations and expansions
17
35
10.4171/ZAA/1577
http://www.ems-ph.org/doi/10.4171/ZAA/1577
Regularity of Minimizers in the Two-Phase Free Boundary Problems in Orlicz–Sobolev Spaces
Jun
Zheng
Southwest Jiatong University, EMEISHAN, SICHUAN, CHINA
Binhua
Feng
Northwest Normal University, LANZHOU (GANSU), CHINA
Peihao
Zhao
Lanzhou University, LANZHOU (GANSU), CHINA
Free boundary problem, regularity, minimizer, Orlicz spaces
In this paper, we consider the optimization problem of minimizing $\mathcal {J}(u)=\int_{\Omega}(G(|\nabla u|)+\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma}+fu)\text{d}x$ in the class of functions $W^{1,G}(\Omega)$ with $u - \varphi \in W^{1,G}_{0}(\Omega)$ for a given function $\varphi$, where $W^{1,G}(\Omega)$ is the class of weakly differentiable functions with $\int_{\Omega} G(|\nabla u|)\text{d}x
Partial differential equations
37
47
10.4171/ZAA/1578
http://www.ems-ph.org/doi/10.4171/ZAA/1578
Stability with Initial Time Difference of Caputo Fractional Differential Equations by Lyapunov Functions
Ravi
Agarwal
Texas A&M University, KINGSVILLE, UNITED STATES
Donal
O'Regan
National University of Ireland, GALWAY, IRELAND
Snezhana
Hristova
University of Plovdiv, PLOVDIV, BULGARIA
Stability, Caputo derivative, Lyapunov functions, fractional differential equations
The stability with initial data difference for nonlinear nonautonomous Caputo fractional differential equation is introduced. This type of stability generalizes the concept of stability in the literature and it enables us to compare the behavior of two solutions when both the initial times and initial values are different. Our theory is based on a new definition of the derivative of a Lyapunov like function along the given fractional equation. Comparison results for scalar fractional differential equations are presented and sufficient conditions for stability, uniform stability and asymptotic stability with initial time difference are obtained.
Ordinary differential equations
49
77
10.4171/ZAA/1579
http://www.ems-ph.org/doi/10.4171/ZAA/1579
On Local Attractivity and Asymptotic Stability of Solutions of Nonlinear Volterra–Stieltjes Integral Equations in Two Variables
Beata
Rzepka
Rzeszów University of Technology, RZESZÓW, POLAND
Nonlinear Volterra–Stieltjes integral equation, fixed point theorem, measure of noncompactness, uniform local attractivity, asymptotic stability
In this paper we will study the existence of solutions depending on two variables of a nonlinear integral equation of Volterra–Stieltjes type in two variables, in the space of real functions which are continuous and bounded on the set $\mathbb{R}_{+}\times [0,M] $. Moreover, we will give the characterization of those solutions. In our study we will utilize the Darbo type of fixed point theorem and apply a measure of noncompactness. In the last section we present particular cases of the considered equation.
Integral equations
Operator theory
79
98
10.4171/ZAA/1580
http://www.ems-ph.org/doi/10.4171/ZAA/1580
On the Persistence of the Eigenvalues of a Perturbed Fredholm Operator of Index Zero under Nonsmooth Perturbations
Pierluigi
Benevieri
Universita di Firenze, FIRENZE, ITALY
Alessandro
Calamai
Università Politecnica delle Marche, ANCONA, ITALY
Massimo
Furi
Università di Firenze, FIRENZE, ITALY
Maria Patrizia
Pera
Universita di Firenze, FIRENZE, ITALY
Fredholm operators, nonlinear spectral theory, eigenvalues, eigenvectors, degree theory
Let $H$ be a real Hilbert space and denote by $S$ its unit sphere. Consider the nonlinear eigenvalue problem $Lx + \epsilon N(x) = \lambda x$, where $\epsilon, \lambda \in \mathbb R$, $L : H \to H$ is a bounded self-adjoint (linear) operator with nontrivial kernel and closed image, and $N : H \to H$ is a (possibly) nonlinear perturbation term. A unit eigenvector $\bar x \in S \cap \mathrm {Ker} L$ of $L$ (corresponding to the eigenvalue $\lambda = 0$) is said to be persistent if it is close to solutions $x \in S$ of the above equation for small values of the parameters $\epsilon \neq 0$ and $\lambda$. We give an affirmative answer to a conjecture formulated by R. Chiappinelli and the last two authors in an article published in 2008. Namely, we prove that, if $N$ is Lipschitz continuous and the eigenvalue $\lambda = 0$ has odd multiplicity, then the sphere $S\cap \mathrm {Ker} L$ contains at least one persistent eigenvector. We provide examples in which our results apply, as well as examples showing that, if the dimension of $\mathrm {Ker} L$ is even, then the persistence phenomenon may not occur.
Operator theory
Partial differential equations
99
128
10.4171/ZAA/1581
http://www.ems-ph.org/doi/10.4171/ZAA/1581
2
Generalized Hardy–Morrey Spaces
Ali
Akbulut
Ahi Evran University, KIRSEHIR, TURKEY
Vagif Sabir
Guliyev
Ahi Evran University, KIRSEHIR, TURKEY
Takahiro
Noi
Tokyo Metropolitan University, TOKYO, JAPAN
Yoshihiro
Sawano
Tokyo Metropolitan University, TOKYO, JAPAN
Generalized Hardy–Morrey spaces, decomposition, maximal operators, Olsen inequality
This paper is an off -spring of the contribution [Z. Anal. Anwend. 36 (2017)(1), 17–35]. We propose a way to consider the decomposition method of generalized Hardy–Morrey spaces. Generalized Hardy–Morrey spaces emerged from generalized Morrey spaces. By means of the grand maximal operator and the norm of generalized Morrey spaces, we can defi ne generalized Hardy–Morrey spaces. With what we have culminated for the Hardy–Littlewood maximal operator, we can easily refi ne the existing results. As an application, we consider bilinear estimates, which is the "so-called" Olsen inequality. In particular, our results complement the one in the 2014 paper by Iida, the fourth author and Tanaka [Z. Anal. Anwend. 33 (2014)(2), 149–170]; there were two mistakes. One lies in the decomposition result and another lies in the proof of the Olsen inequality.
Approximations and expansions
Fourier analysis
129
149
10.4171/ZAA/1582
http://www.ems-ph.org/doi/10.4171/ZAA/1582
Coproximinality for Quotient Spaces
T.S.S.R.K.
Rao
Indian Statistical Institute, BANGALORE, INDIA
Best coapproximation, Hilbert spaces, quotient spaces
In this paper we study the classical notion of coproximinality, for quotient spaces of Banach spaces. We provide a partial solution to the three space problem, analogous to a classical result of Cheney and Wulbert, by showing that for $ Z \subset Y \subset X$, coproximinality of $Z$ in $X$ and that of $Y/Z$ in $X/Z$ implies the coproximinality of $Y$ in $X$, when $Z$ is an $M$-ideal in $X$. For the space $C(K)$ of continuous functions on a compact extremally disconnected set $K$ we derive the same conclusion under the assumption that $Z$ is an $M$-ideal in $Y$.
Approximations and expansions
Functional analysis
151
157
10.4171/ZAA/1583
http://www.ems-ph.org/doi/10.4171/ZAA/1583
Boundedness of the Generalized Fractional Integral Operators on Generalized Morrey Spaces over Metric Measure Spaces
Yoshihiro
Sawano
Tokyo Metropolitan University, TOKYO, JAPAN
Tetsu
Shimomura
Hiroshima University, Graduate School of Education, HIGASHI-HIROSHIMA, JAPAN
Sobolev's inequality, Morrey space, Riesz potential, non-doubling measure, predual spaces
Our aim in this paper is to deal with the boundedness of the generalized fractional integral operators on generalized Morrey spaces $L_{p,\phi;2}(X;\mu)$ over metric measure spaces. We also discuss a necessary condition for the boundedness of the generalized fractional integral operators. As applications, we establish new results for the predual spaces.
Real functions
Fourier analysis
159
190
10.4171/ZAA/1584
http://www.ems-ph.org/doi/10.4171/ZAA/1584
Two Nontrivial Solutions for the Nonhomogenous Fourth Order Kirchhoff Equation
Ling
Ding
Hubei University of Arts and Science, HUBEI, CHINA
Lin
Li
Chongqing Technology and Business University, CHONGQING, CHINA
Fourth order Kirchhoff equation, variational methods, critical point theorem
In this paper, we consider the following nonhomogenous fourth order Kirchhoff equation $$\Delta^2 u - \left( a + b \int_{\mathbb{R}^N} |\nabla u|^2 dx \right) \Delta u + V(x) u = f(x,u) + g(x), \quad x \in \mathbb{R}^N,$$ where $\Delta^2 := \Delta(\Delta)$, constants $a > 0$, $b \geq 0$, $V \in C(\mathbb{R}^N, \mathbb{R})$, $f \in C(\mathbb{R}^N \times \mathbb{R}, \mathbb{R})$ and $g \in L^2(\mathbb{R}^N)$. Under more relaxed assumptions on the nonlinear term $f$ that are much weaker than those in L. Xu and H. Chen, using some new proof techniques especially the verification of the boundedness of Palais–Smale sequence, a new result is obtained.
Partial differential equations
191
207
10.4171/ZAA/1585
http://www.ems-ph.org/doi/10.4171/ZAA/1585
Nonlinear Dirichlet Problems with no Growth Restriction on the Reaction
Leszek
Gasiński
Jagiellonian University, KRAKÓW, POLAND
Liliana
Klimczak
Jagiellonian University, KRAKÓW, POLAND
Nikolaos
Papageorgiou
National Technical University of Athens, ATHENS, GREECE
Nonlinear regularity, nonlinear maximum principle, constant sign and nodal solutions, $(p, 2)$-equation, critical groups
We consider nonlinear Dirichlet problems driven by the sum of a $p$-Laplacian and a Laplacian and with a Carathéodory reaction which does not satisfy any global growth condition. Instead we assume that it has constant sign $z$-dependent zeros. Using variational methods, truncation techniques and Morse theory, we prove multiplicity theorems providing sign information for all the solutions.
Partial differential equations
Global analysis, analysis on manifolds
209
238
10.4171/ZAA/1586
http://www.ems-ph.org/doi/10.4171/ZAA/1586
Non-Uniform Decay of the Energy of some Dissipative Evolution Systems
Kaïs
Ammari
Université de Monastir, MONASTIR, TUNISIA
Ahmed
Bchatnia
Université de Tunis El Manar, TUNIS, TUNISIA
Karim
El Mufti
Université de Manouba, TUNIS, TUNISIA
Bounded feedback, kind of observability estimate, non-uniform decay
In this paper we consider second order evolution equations with bounded damping. We give a characterization of a non-uniform decay for the damped problem using a kind of observability estimate for the associated undamped problem.
Partial differential equations
239
251
10.4171/ZAA/1587
http://www.ems-ph.org/doi/10.4171/ZAA/1587
3
Superlinear, Noncoercive Asymmetric Robin Problems with Indefinite, Unbounded Potential
Nikolaos
Papageorgiou
National Technical University, Athens, Greece
Vicenţiu
Rădulescu
King Abdulaziz University, Jeddah, Saudi Arabia, and University of Craiova, Romania
Superlinear reaction term, asymmetric nonlinearity, constant sign solutions, critical groups, C-condition, mountain pass theorem, indefinite potential, Robin boundary condition
We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. The reaction term exhibits an asymmetric behavior, namely it is superlinear in the positive direction but without satisfying the Ambrosetti–Rabinowitz condition and it is sublinear but noncoercive in the negative direction. Using variational methods together with suitable truncation and perturbation techniques and Morse theory (critical groups), we prove a multiplicity theorem producing three nontrivial smooth solutions two of which have constant sign (one positive and the other negative).
Partial differential equations
Global analysis, analysis on manifolds
253
281
10.4171/ZAA/1588
http://www.ems-ph.org/doi/10.4171/ZAA/1588
Gelfand Type Elliptic Problem Involving Advection
Baishun
Lai
Henan University, Kaifeng, China
Lulu
Zhang
Henan University, Kaifeng, China
Gelfand problem, stability, nonexistence, advection
We consider the following Gelfand type elliptic problem involving advection $$-\Delta u+a(x) \cdot \nabla u=e^{u}\ \ \mbox{in}\ \mathbb R^{N},$$ where $a(x)$ is a smooth vector field. According to energy estimates, we obtain the nonexistence results of stable solution for this equation under some restrict conditions about $a(x)$ for $N\leq 9$.On the other hand, combining Liapunov–Schmidt reduction method, we prove that it possesses a solution for $N\geq 4$. Besides, if $a$ is divergence free and satisfies a smallness condition, then the equation above admits a stable solution for $N\geq11$.
Partial differential equations
283
295
10.4171/ZAA/1589
http://www.ems-ph.org/doi/10.4171/ZAA/1589
Interior Feedback Stabilization of Wave Equations with Dynamic Boundary Delay
Kaïs
Ammari
University of Monastir, Tunisia
Stéphane
Gerbi
Université Savoie Mont Blanc, Le-Bourget-du-Lac, France
Interior stabilization, dynamic boundary conditions, dynamic boundary delay, wave equations
In this paper we consider an interior stabilization problem for the wave equation with dynamic boundary delay. We prove some stability results under the choice of damping operator. The proof of the main result is based on a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.
Partial differential equations
Systems theory; control
297
327
10.4171/ZAA/1590
http://www.ems-ph.org/doi/10.4171/ZAA/1590
Examples of Plentiful Discrete Spectra in Infinite Spatial Cruciform Quantum Waveguides
Fedor
Bakharev
St. Petersburg State University, St. Petersburg, Russia
Sergey
Matveenko
St. Petersburg State University, and National Research University Higher School of Economics, St. Petersburg, Russia
Sergey
Nazarov
St. Petersburg State University and Institute for Problems of Mechanical Engineering, Saint Petersburg, Russia
Cruciform waveguide, multiplicity of discrete spectrum, asymptotics, localization of eigenfunctions, thin quantum lattices
Spatial cruciform quantum waveguides (the Dirichlet problem for Laplace operator) are constructed such that the total multiplicity of the discrete spectrum exceeds any preassigned number.
Numerical analysis
Operator theory
329
341
10.4171/ZAA/1591
http://www.ems-ph.org/doi/10.4171/ZAA/1591
On Different Types of Stability for Linear Delay Dynamic Equations
Elena
Braverman
University of Calgary, Canada
Bașak
Karpuz
Dokuz Eylul University, Izmir, Turkey
Delay dynamic equations, global stability, uniform exponential stability, global asymptotic stability
We provide explicit conditions for uniform stability, global asymptotic stability and uniform exponential stability for dynamic equations with a single delay and a non-negative coefficient. Some examples on nonstandard time scales are also given to show applicability and sharpness of the new results.
Ordinary differential equations
343
375
10.4171/ZAA/1592
http://www.ems-ph.org/doi/10.4171/ZAA/1592
4
Kernel Estimates for Schrödinger Type Operators with Unbounded Diffusion and Potential Terms
Anna
Canale
Università degli Studi di Salerno, Fisciano, Italy
Abdelaziz
Rhandi
Università degli Studi di Salerno, Fisciano, Italy
Cristian
Tacelli
Università degli Studi di Salerno, Fisciano, Italy
Schrödinger type operator, semigroup, heat kernel estimates
We prove that the heat kernel associated to the Schrödinger type operator $A:=(1+|x|^\alpha)\Delta-|x|^\beta$ satisfies the estimate $$k(t,x,y)\leq c_1e^{\lambda_0t}e^{c_2t^{-b}}\frac{(|x||y|)^{-\frac{N-1}{2}-\frac{\beta-\alpha}{4}}}{1+|y|^\alpha} e^{-\frac{\sqrt{2}}{\beta-\alpha+2}|x|^{\frac{\beta-\alpha+2}{2}}} e^{-\frac{\sqrt{2}}{\beta-\alpha+2}|y|^{\frac{\beta-\alpha+2}{2}}}$$ for $t>0,|x|,|y|\ge 1$, where $c_1,c_2$ are positive constants and $b=\frac{\beta-\alpha+2}{\beta+\alpha-2}$ provided that $N>2,\,\alpha\geq 2$ and $\beta>\alpha-2$. We also obtain an estimate of the eigenfunctions of $A$.
Partial differential equations
Operator theory
377
392
10.4171/ZAA/1593
http://www.ems-ph.org/doi/10.4171/ZAA/1593
Solvability of Hammerstein Integral Equations with Applications to Boundary Value Problems
Daria
Bugajewska
Adam Mickiewicz University, Poznan, Poland
Gennaro
Infante
Universita della Calabria, Cosenza, Italy
Piotr
Kasprzak
Adam Mickiewicz University, Poznan, Poland
Boundary value problem, cone, Hammerstein integral equation, functions of bounded variation
In this paper we present some new results regarding the solvability of nonlinear Hammerstein integral equations in a special cone of continuous functions. The proofs are based on a certain xed point theorem of Leggett and Williams type. We give an application of the abstract result to prove the existence of nontrivial solutions of a periodic boundary value problem. We also investigate, via a version of Krasnosel0ski 's theorem for the sum of two operators, the solvability of perturbed Hammerstein integral equations in the space of continuous functions of bounded variation in the sense of Jordan. As an application of these results, we study the solvability of a boundary value problem subject to integral boundary conditions of Riemann–Stieltjes type. Some examples are presented in order to illustrate the obtained results.
Integral equations
Ordinary differential equations
Operator theory
393
417
10.4171/ZAA/1594
http://www.ems-ph.org/doi/10.4171/ZAA/1594
Existence of Cylindrically Symmetric Ground States to a Nonlinear Curl-Curl Equation with Non-Constant Coefficients
Andreas
Hirsch
Karlsruhe Institute of Technology (KIT), Germany
Wolfgang
Reichel
Karlsruhe Institute of Technology (KIT), Germany
Curl-curl problem, nonlinear elliptic equations, cylindrical symmetry, variational methods
We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U=f(x, |U|^2)U$ in $\mathbb R^3$ related to the nonlinear Maxwell equations with Kerr-type nonlinear material laws. We prove the existence of a symmetric ground-state type solution for a bounded, cylindrically symmetric coefficient $V$ and subcritical cylindrically symmetric nonlinearity $f$. The new existence result extends the class of problems for which ground-state type solutions are known. It is based on compactness properties of symmetric functions due to Lions [J. Funct. Anal. 41 (1981)(2), 236–275], new rearrangement type inequalities by Brock [Proc. Indian Acad. Sci. Math. Sci. 110 (2000), 157–204] and the recent extension of the Nehari-manifold technique from Szulkin and Weth [Handbook of Nonconvex Analysis and Applications (2010), pp. 597–632].
Partial differential equations
Operator theory
Global analysis, analysis on manifolds
419
435
10.4171/ZAA/1595
http://www.ems-ph.org/doi/10.4171/ZAA/1595
Uniform Asymptotic Expansions for the Fundamental Solution of Infinite Harmonic Chains
Alexander
Mielke
Weierstrass Institute für Angewandte Analysis und Stochastik and Humboldt-Universität, Berlin, Germany
Carsten
Patz
Weierstrass Institut für Angewandte Analysis und Stochastik, Berlin, Germany
Asymptotic analysis, method of stationary phase, dispersive decay, oscillatory integrals, Airy function, Fermi–Pasta–Ulam chain
We study the dispersive behavior of waves in linear oscillator chains. We show that for general general dispersions it is possible to construct an expansion such that the remainder can be estimated by $1/t$ uniformly in space. In particlar we give precise asymptotics for the cross-over from the $t^{-1/2}$ decay of nondegenerate wave numbers to the degenerate $t^{-1/3}$ decay of degenerate wave numbers. This involves a careful description of the oscillatory integral involving the Airy function.
Dynamical systems and ergodic theory
Approximations and expansions
Fourier analysis
Mechanics of particles and systems
437
475
10.4171/ZAA/1596
http://www.ems-ph.org/doi/10.4171/ZAA/1596
A Characterization of Circles by Single Layer Potentials
Seyed
Zoalroshd
Vatterott College, Fairview Heights, USA
Circle, single layer potential, polynomial eigenfunction
We give a characterization of circles by polynomial eigenfunctions of single layer potentials.
Potential theory
477
480
10.4171/ZAA/1597
http://www.ems-ph.org/doi/10.4171/ZAA/1597
Multivariate Wave-Packet Transforms
Arash
Ghaani Farashahi
The Johns Hopkins University, Baltimore, USA
Multivariate wavelet (Gabor) transforms, multivariate wave-packet representations, multivariate wave-packet groups, multivariate wave-packet transforms
This paper presents a study for square-integrability of classical multivariate wave-packets in $L^2({\mathbb{R}^d})$ via group representation theory. The abstract notions of multivariate wave-packet groups and multivariate wave-packet representations will be introduced and as the main result, we prove an admissibility condition on closed subgroups of GL$(d, \mathbb R)$, which guarantees the square integrability of classical multivariate wave-packet representations on $L^2({\mathbb{R}^d})$. Finally, we present application of our results in the case of different admissible subgroups.
Fourier analysis
Topological groups, Lie groups
Abstract harmonic analysis
Quantum theory
481
500
10.4171/ZAA/1598
http://www.ems-ph.org/doi/10.4171/ZAA/1598