Total asymptotically nonexpansive mappings and generalized variational-like inclusion problems in semi-inner product spaces

ABSTRACT

This paper focuses on investigating the problem of finding a common element of the set of solutions of a generalized nonlinear implicit variational-like inclusion problem involving an $(A,\eta )$-maximal m-relaxed monotone mapping in the sense of L.-G.-s.i.p. and the set of fixed points of a total asymptotically nonexpansive mapping. To achieve such a purpose, a new iterative algorithm is constructed. Applying the concepts of graph convergence and generalized resolvent operator associated with an $(A,\eta )$-maximal m-relaxed monotone mapping in the sense of L.-G.-s.i.p. As an application of the obtained equivalence relationship, the strong convergence of the sequence generated by our proposed iterative algorithm to a point belonging to the intersection of the two sets mentioned above is proved.

KEYWORDS:

• Total asymptotically nonexpansive mapping
• fixed point problem
• variational-like inclusions
• semi-inner product spaces
• resolvent operator technique
• graph convergence
• $(A,\eta )$-maximal m-relaxed monotone mapping
• iterative algorithm
• convergence analysis

Acknowledgements

The second author is grateful to King Fahd University of Petroleum and Minerals for providing excellent research facilities.

Disclosure statement

No potential conflict of interest was reported by the author(s)