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Creating slowly oscillating sequences and slowly oscillating continuous functions
R. W. Vallin
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE , 2011,
Abstract: This paper is a further study of slowly oscillating convergence and slowly oscillating continuity as introduced by Cakalli in [3]. We answer an open question from that paper and relate the slowly oscillating continuous functions to metric preserving functions, thus infusing this subject with a ready made collection of pathological examples.
On $Δ$-quasi-slowly oscillating sequences  [PDF]
Huseyin Cakalli
Mathematics , 2011,
Abstract: A sequence $(x_{n})$ of points in a topological group is called $\Delta$-quasi-slowly oscillating if $(\Delta x_{n})$ is quasi-slowly oscillating, and is called quasi-slowly oscillating if $(\Delta x_{n})$ is slowly oscillating. A function $f$ defined on a subset of a topological group is quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillating continuous if it preserves quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillating sequences, i.e. $(f(x_{n}))$ is quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillating whenever $(x_{n})$ is. We study these kinds of continuities, and investigate relations with statistical continuity, lacunary statistical continuity, and some other types of continuities in metrizable topological groups.
Creating slowly oscillating sequences and slowly oscillating continuous functions  [PDF]
R. W. Vallin
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE , 2011,
Abstract: This paper is a further study of slowly oscillating convergence and slowlyoscillating continuity as introduced by Cakalli in [3]. Weanswer an open question from that paper and relate the slowly oscillatingcontinuous functions to metric preserving functions, thus infusing thissubject with a ready made collection of pathological examples.
Functions preserving slowly oscillating double sequences  [PDF]
Huseyin Cakalli,Richard F. Patterson
Mathematics , 2013,
Abstract: A double sequence $\textbf{x}=\{x_{k,l}\}$ of points in $\textbf{R}$ is slowly oscillating if for any given $\varepsilon>0$, there exist $\alpha=\alpha(\varepsilon)>0$, $\delta=\delta (\varepsilon) >0$, and $N=N(\varepsilon)$ such that $|x_{k,l}-x_{s,t}|<\varepsilon$ whenever $k,l\geq N(\varepsilon)$ and $k\leq s \leq (1+\alpha)k$, $l\leq t \leq (1+\delta)l$. We study continuity type properties of factorable double functions defined on a double subset $A\times A$ of $\textbf{R}^{2}$ into $\textbf{R}$, and obtain interesting results related to uniform continuity, sequential continuity, and a newly introduced type of continuity of factorable double functions defined on a double subset $A\times A$ of $\textbf{R}^{2}$ into $\textbf{R}$.
Slowly Oscillating Solutions for Differential Equations with Strictly Monotone Operator  [cached]
Zhang Chuanyi,Guo Yali
Journal of Inequalities and Applications , 2007,
Abstract: The authors discuss necessary and sufficient conditions for the existence and uniqueness of slowly oscillating solutions for the differential equation with strictly monotone operator. Particularly, the authors give necessary and sufficient conditions for the existence and uniqueness of slowly oscillating solutions for the differential equation , where denotes the gradient of the convex function on .
Slowly Oscillating Solutions of a Parabolic Inverse Problem: Boundary Value Problems  [cached]
Yang Fenglin,Zhang Chuanyi
Boundary Value Problems , 2010,
Abstract: The existence and uniqueness of a slowly oscillating solution to parabolic inverse problems for a type of boundary value problem are established. Stability of the solution is discussed.
Slowly Oscillating Solutions for Differential Equations with Strictly Monotone Operator
Chuanyi Zhang,Yali Guo
Journal of Inequalities and Applications , 2007, DOI: 10.1155/2007/60239
Abstract: The authors discuss necessary and sufficient conditions for the existence and uniqueness of slowly oscillating solutions for the differential equation u'+F(u)=h(t) with strictly monotone operator. Particularly, the authors give necessary and sufficient conditions for the existence and uniqueness of slowly oscillating solutions for the differential equation u'+ ¢ |(u)=h(t), where ¢ | denotes the gradient of the convex function | on ¢ N.
Sufficient Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data  [PDF]
Alexei Yu. Karlovich,Yuri I. Karlovich,Amarino B. Lebre
Mathematics , 2010,
Abstract: Suppose $\alpha$ is an orientation preserving diffeomorphism (shift) of $\mR_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$. We establish sufficient conditions for the Fredholmness of the singular integral operator \[ (aI-bW_\alpha)P_++(cI-dW_\alpha)P_- \] acting on $L^p(\mR_+)$ with $1
Necessary Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data  [PDF]
Alexei Yu. Karlovich,Yuri I. Karlovich,Amarino B. Lebre
Mathematics , 2010,
Abstract: Suppose $\alpha$ is an orientation-preserving diffeomorphism (shift) of $\mR_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$. In \cite{KKLsufficiency} we found sufficient conditions for the Fredholmness of the singular integral operator with shift \[ (aI-bW_\alpha)P_++(cI-dW_\alpha)P_- \] acting on $L^p(\mR_+)$ with $1
On regularization of Mellin PDO's with slowly oscillating symbols of limited smoothness  [PDF]
Alexei Yu. Karlovich,Yuri I. Karlovich,Amarino B. Lebre
Mathematics , 2014,
Abstract: We study Mellin pseudodifferential operators (shortly, Mellin PDO's) with symbols in the algebra $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ of slowly oscillating functions of limited smoothness introduced in \cite{K09}. We show that if $\mathfrak{a}\in\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ does not degenerate on the "boundary" of $\mathbb{R}_+\times\mathbb{R}$ in a certain sense, then the Mellin PDO ${\rm Op}(\mathfrak{a})$ is Fredholm on the space $L^p$ for $p\in(1,\infty)$ and each its regularizer is of the form ${\rm Op}(\mathfrak{b})+K$ where $K$ is a compact operator on $L^p$ and $\mathfrak{b}$ is a certain explicitly constructed function in the same algebra $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ such that $\mathfrak{b}=1/\mathfrak{a}$ on the "boundary" of $\mathbb{R}_+\times\mathbb{R}$. This result complements a known Fredholm criterion from \cite{K09} for Mellin PDO's with symbols in the closure of $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$.
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