Home Mathematics Convergence analysis of Kantorovich-type operators in variable exponent Sobolev spaces
Article
Licensed
Unlicensed Requires Authentication

Convergence analysis of Kantorovich-type operators in variable exponent Sobolev spaces

  • Benali Aharrouch and Mustapha Bouallala ORCID logo EMAIL logo
Published/Copyright: December 3, 2025
Become an author with De Gruyter Brill

Abstract

In this work, we establish convergence results in variable exponent Sobolev spaces W 1 , p ( ) ( R ) , where the exponent function p : R R satisfies 1 p p ( x ) p + < + for all x R , for Kantorovich-type operators activated by kernel functions. Furthermore, we prove uniform convergence for functions belonging to appropriate function spaces.

MSC 2020: 41A25; 41A30; 47A58

References

[1] B. Aharrouch, J. Bennouna and B. El Hamdaoui, Existence of weak and renormalized solutions of degenerated elliptic equation, Afr. Mat. 30 (2019), no. 5–6, 755–776. 10.1007/s13370-019-00682-3Search in Google Scholar

[2] B. Aharrouch, M. Boukhrij and J. Bennouna, Existence of solutions for a class of degenerate elliptic equations in P ( X ) -Sobolev spaces, Topol. Methods Nonlinear Anal. 51 (2018), no. 2, 389–411. 10.12775/TMNA.2017.065Search in Google Scholar

[3] C. Bardaro, G. Vinti, P. L. Butzer and R. L. Stens, Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process. 6 (2007), no. 1, 29–52. 10.1007/BF03549462Search in Google Scholar

[4] A. Benali and B. Jaouad, Existence and regularity results for nonlinear and nonhomogeneous elliptic equation, J. Elliptic Parabol. Equ. 7 (2021), no. 2, 961–975. 10.1007/s41808-021-00121-0Search in Google Scholar

[5] M. Cantarini, D. Costarelli and G. Vinti, Approximation of differentiable and not differentiable signals by the first derivative of sampling Kantorovich operators, J. Math. Anal. Appl. 509 (2022), no. 1, Article ID 125913. 10.1016/j.jmaa.2021.125913Search in Google Scholar

[6] D. Costarelli and R. Spigler, Convergence of a family of neural network operators of the Kantorovich type, J. Approx. Theory 185 (2014), 80–90. 10.1016/j.jat.2014.06.004Search in Google Scholar

[7] D. Costarelli and G. Vinti, Convergence for a family of neural network operators in Orlicz spaces, Math. Nachr. 290 (2017), no. 2–3, 226–235. 10.1002/mana.201600006Search in Google Scholar

[8] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011. 10.1007/978-3-642-18363-8Search in Google Scholar

[9] D. E. Edmunds, J. Lang and A. Nekvinda, On L p ( x ) norms, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1981, 219–225. 10.1098/rspa.1999.0309Search in Google Scholar

[10] X. L. Fan and D. Zhao, On the generalised Orlicz–Sobolev Space W k , p ( ) ( Ω ) , J. Gansu Educ. College 12 (1998), no. 1, 1–6. Search in Google Scholar

[11] O. Kováčik and J. Rákosník, On spaces L p ( x ) and W k , p ( x ) , Czechoslovak Math. J. 41(116) (1991), no. 4, 592–618. 10.21136/CMJ.1991.102493Search in Google Scholar

[12] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin, 1983. 10.1007/BFb0072210Search in Google Scholar

[13] J. Musielak and W. Orlicz, On modular spaces, Studia Math. 18 (1959), 49–65. 10.4064/sm-18-1-49-65Search in Google Scholar

[14] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, 1950. Search in Google Scholar

[15] D. Zhao, W. J. Qiang and X. L. Fan, On generalized Orlicz spaces L p ( ) ( Ω ) , J. Gansu Sci. 9 (1997), no. 2, 1–7. Search in Google Scholar

Received: 2025-05-02
Revised: 2025-09-12
Accepted: 2025-09-17
Published Online: 2025-12-03

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 21.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/gmj-2025-2089/html
Scroll to top button