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A GENERIC RESEARCH ON NONLINEAR NON-CONVOLUTION TYPE SINGULAR INTEGRAL OPERATORS
Uysal, Gumrah,Mishra, Vishnu Narayan,Guller, Ozge Ozalp,Ibikli, Ertan The Kangwon-Kyungki Mathematical Society 2016 한국수학논문집 Vol.24 No.3
In this paper, we present some general results on the pointwise convergence of the non-convolution type nonlinear singular integral operators in the following form: $$T_{\lambda}(f;x)={\large\int_{\Omega}}K_{\lambda}(t,x,f(t))dt,\;x{\in}{\Psi},\;{\lambda}{\in}{\Lambda}$$, where ${\Psi}$ = <a, b> and ${\Omega}$ = <A, B> stand for arbitrary closed, semi-closed or open bounded intervals in ${\mathbb{R}}$ or these set notations denote $\mathbb{R}$, and ${\Lambda}$ is a set of non-negative numbers, to the function $f{\in}L_{p,{\omega}}({\Omega})$, where $L_{p,{\omega}}({\Omega})$ denotes the space of all measurable functions f for which $\|{\frac{f}{\omega}}\|^p$ (1 ${\leq}$ p < ${\infty}$) is integrable on ${\Omega}$, and ${\omega}:{\mathbb{R}}{\rightarrow}\mathbb{R}^+$ is a weight function satisfying some conditions.
SOME WEIGHTED APPROXIMATION PROPERTIES OF NONLINEAR DOUBLE INTEGRAL OPERATORS
Uysal, Gumrah,Mishra, Vishnu Narayan,Serenbay, Sevilay Kirci The Kangwon-Kyungki Mathematical Society 2018 한국수학논문집 Vol.26 No.3
In this paper, we present some recent results on weighted pointwise convergence and the rate of pointwise convergence for the family of nonlinear double singular integral operators in the following form: $$T_{\eta}(f;x,y)={\int}{\int\limits_{{\mathbb{R}^2}}}K_{\eta}(t-x,\;s-y,\;f(t,s))dsdt,\;(x,y){\in}{\mathbb{R}^2},\;{\eta}{\in}{\Lambda}$$, where the function $f:{\mathbb{R}}^2{\rightarrow}{\mathbb{R}}$ is Lebesgue measurable on ${\mathbb{R}}^2$ and ${\Lambda}$ is a non-empty set of indices. Further, we provide an example to support these theoretical results.
On singular integral operators involving power nonlinearity
Sevgi Esen Almali,Gumrah Uysal,Vishnu Narayan Mishra,Ozge Ozalp Guller 강원경기수학회 2017 한국수학논문집 Vol.25 No.4
In the current manuscript, we investigate the pointwise convergence of the singular integral operators involving power nonlinearity given in the following form: \begin{equation*} T_{\lambda }(f;x)=\int \limits_{a}^{b}\sum \limits_{m=1}^{n}f^{m}(t)K_{\lambda ,m}(x,t)dt,\text{ }\lambda \in \Lambda ,\text{ }x\in \left( a,b\right) , \end{equation*} where $\Lambda $ is an index set consisting of the non-negative real numbers, and $n\geq 1$ is a finite natural number, at $\mu -$generalized Lebesgue points of integrable function $f$ $\in L_{1}\left( a,b\right) .$ Here, $f^{m}$ denotes $m-th$ power of the function $f$ and $\left( a,b\right)$ stands for arbitrary bounded interval in $ \mathbb{R} $ or $\mathbb{R}$ itself. We also handled the indicated problem under the assumption $f$ $\in L_{1}\left( \mathbb{R}\right) .$