@article{ea2a2e4d84ba4e72aa91dc1b4db365fa,
title = "A Halpern-Type Iteration Method for Bregman Nonspreading Mapping and Monotone Operators in Reflexive Banach Spaces",
abstract = "In this paper, we introduce an iterative method for approximating a common solution of monotone inclusion problem and fixed point of Bregman nonspreading mappings in a reflexive Banach space. Using the Bregman distance function, we study the composition of the resolvent of a maximal monotone operator and the antiresolvent of a Bregman inverse strongly monotone operator and introduce a Halpern-type iteration for approximating a common zero of a maximal monotone operator and a Bregman inverse strongly monotone operator which is also a fixed point of a Bregman nonspreading mapping. We further state and prove a strong convergence result using the iterative algorithm introduced. This result extends many works on finding a common solution of the monotone inclusion problem and fixed-point problem for nonlinear mappings in a real Hilbert space to a reflexive Banach space.",
author = "Ogbuisi, {F. U.} and Jolaoso, {L. O.} and Isiogugu, {F. O.} and Jianhua Chen",
note = "Funding Information: https://orcid.org/0000-0002-8255-5522 Ogbuisi F. U. ferdinard.ogbuisi@unn.edu.ng 1 2 Jolaoso L. O. 216074984@stu.ukzn.ac.za 1 https://orcid.org/0000-0002-3959-9562 Isiogugu F. O. felicia.isiogugu@unn.edu.ng 1 2 3 Chen Jianhua 1 School of Mathematics Statistics and Computer Science University of Kwazulu-Natal Durban South Africa ukzn.ac.za 2 Department of Mathematics University of Nigeria Nsukka Nigeria unn.edu.ng 3 DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) Johannesburg South Africa 2019 2 9 2019 2019 14 05 2019 22 07 2019 2 9 2019 2019 Copyright {\textcopyright} 2019 F. U. Ogbuisi et al. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we introduce an iterative method for approximating a common solution of monotone inclusion problem and fixed point of Bregman nonspreading mappings in a reflexive Banach space. Using the Bregman distance function, we study the composition of the resolvent of a maximal monotone operator and the antiresolvent of a Bregman inverse strongly monotone operator and introduce a Halpern-type iteration for approximating a common zero of a maximal monotone operator and a Bregman inverse strongly monotone operator which is also a fixed point of a Bregman nonspreading mapping. We further state and prove a strong convergence result using the iterative algorithm introduced. This result extends many works on finding a common solution of the monotone inclusion problem and fixed-point problem for nonlinear mappings in a real Hilbert space to a reflexive Banach space. National Research Foundation 111992 Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS) BA 2018/012 Funding Information: The work of the first author is based on the research supported wholly by the National Research Foundation (NRF) of South Africa (Grant no. 111992). the third author acknowledges the financial support from the Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF CoEMaSS) (postdoctoral fellowship) (Grant no. BA 2018/012). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF and CoE-MaSS. Publisher Copyright: {\textcopyright} 2019 F. U. Ogbuisi et al.",
year = "2019",
doi = "10.1155/2019/8059135",
language = "English",
volume = "2019",
journal = "Journal of Mathematics",
issn = "2314-4629",
publisher = "Hindawi Limited",
}