- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 13:32:58
6
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=36&iss=1&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