Strong convergence theorems of an iterative scheme for strictly pseudocontractive mappings in Banach spaces

Abstract

The article establishes the sufficient and necessary conditions for the strong convergence of the Ishikawa iterative scheme with random errors in the sense of Xu for a strictly pseudocontractive mapping in an arbitrary real Banach space under certain assumptions. It affirmatively answers the open question that put forth by Zeng. The results in this paper extend and improve the corresponding results of Browder, Petryshyn, Rhoades, Osilike and Zeng.

Keywords:

• Modified Ishikawa iterative process with errors
• strong convergence
• strictly pseudocontractive mapping
• fixed point
• demi-compact

AMS Subject Classifications:

• 47H09
• 47H10
• 47J25

Acknowledgements

The author thanks the referees for many valuable suggestions to improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 61374081]; the Humanity and Social Science Planning (Youth) Foundation of Ministry of Education of China [grant number 14YJAZH095], [grant number 16YJC630004)]; the Natural Science Foundation of Guangdong Province [grant number 2015A030313485] and the Guangzhou Science and Technology Project [grant number 201707010494].