Korean J. Math.  Vol 25, No 4 (2017)  pp.483-494
DOI: https://doi.org/10.11568/kjm.2017.25.4.483

On singular integral operators involving power nonlinearity

Sevgi Esen Almalı, Gumrah Uysal, Vishnu Narayan Mishra, Ozge Ozalp Guller


In the current manuscript, we investigate the pointwise convergence of the singular integral operators involving power nonlinearity given in the following form:
}(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) ,
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 $f$ $\in L_{1}\left( \mathbb{R}\right) .$


Pointwise convergence, Nonlinear integral operators, $\mu-$generalized Lebesgue point

Subject classification

41A35, 41A25, 47G10, 47A58.


Full Text:



S. E. Almali, On approximation properties for non-linear integral operators, New Trends Math. Sci. 5 (4) (2017), 123–129. (arXiv:1702.04190v1) (Google Scholar)

S. E. Almali and A. D. Gadjiev, On approximation properties of certain multidimensional nonlinear integrals, J. Nonlinear Sci. Appl. 9 (5) (2016), 3090–3097. [3] C. Bardaro, On approximation properties for some classes of linear operators of convolution type, Atti Sem. Mat. Fis. Univ. Modena 33 (2) (1984), 329–356. (Google Scholar)

C. Bardaro and G. Vinti, On approximation properties of certain nonconvolution integral operators, J. Approx. Theory, 62 (3) (1990), 358–371. (Google Scholar)

C. Bardaro, J. Musielak and G. Vinti, Approximation by nonlinear integral op- erators in some modular function spaces, Ann. Polon. Math. 63 (2) (1996), 73–182. (Google Scholar)

C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applications, de Gruyter Series in Nonlinear Analysis and Applications, 9., Walter de Gruyter & Co., Berlin, 2003. (Google Scholar)

P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, vol. I. Academic Press, New York, London, 1971. (Google Scholar)

A. D. Gadjiev, On nearness to zero of a family of nonlinear integral operators of Hammerstein, Izv. Akad. Nauk Azerba ̆ıdˇzan. SSR Ser. Fiz.-Tehn. Mat. Nauk, (1966), no. 2, 32–34. (Google Scholar)

A. D. Gadjiev, The order of convergence of singular integrals which depend on two parameters, Special Problems of Functional Analysis and their Appl. to the Theory of Diff. Eq. and the Theory of Func., Izdat. Akad. Nauk Azerba ̆ıdaˇzan. SSR., (1968), 40–44. (Google Scholar)

H. Karsli and E. Ibikli, On convergence of convolution type singular integral operators depending on two parameters, Fasc. Math. 38 (2007), 25–39. (Google Scholar)

R. G. Mamedov, On the order of convergence of m-singular integrals at generalized Lebesgue points and in the space Lp(−∞,∞), Izv. Akad. Nauk SSSR Ser. Mat. 27 (2) (1963), 287–304. (Google Scholar)

J. Musielak, On some approximation problems in modular spaces, Constructive Function Theory 1981, (Proc. Int. Conf., Varna, June 1-5, 1981), Publ. House Bulgarian Acad. Sci., Sofia, (1983), 455–461. (Google Scholar)

J. Musielak, Approximation by nonlinear singular integral operators in generalized Orlicz spaces, Comment. Math. Prace Mat. 31 (1991), 79–88. (Google Scholar)

B. Rydzewska, Approximation des fonctions par des int ́e grales singuli`eres ordinaires, Fasc. Math. 7 (1973), 71–81. (Google Scholar)

M. D. Spivak, Calculus (3rd ed.), Publish or Perish, Inc., Houston, Texas, 1994. (Google Scholar)

T. Swiderski and E. Wachnicki, Nonlinear singular integrals depending on two parameters, Comment. Math. 40 (2000), 181–189. (Google Scholar)

R. Taberski, Singular integrals depending on two parameters, Prace Mat. 7 (1962), 173–179. (Google Scholar)

R. Taberski, Exponential approximation on the real line, Approximation and function spaces (Warsaw, 1986), Banach Center Publ., PWN, Warsaw, 22 (1989), 449–464. (Google Scholar)

G. Uysal, V. N. Mishra, O. O. Guller and E. Ibikli, A generic research on nonlinear non-convolution type singular integral operators, Korean J. Math. 24 (3) (2016), 545–565. (Google Scholar)


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