# The implicit midpoint rule of nonexpansive mappings and applications in uniformly smooth Banach spaces

Volume 11, Issue 12, pp 1374--1391 Publication Date: September 21, 2018       Article History
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### Authors

M. O. Aibinu - School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal , Durban, South Africa. - DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa. P. Pillay - School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa. J. O. Olaleru - Department of Mathematics, Faculty of Science, University of Lagos, Akoka, Yaba, Lagos, Nigeria. O. T. Mewomo - School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa.

### Abstract

Let $K$ be a nonempty closed convex subset of a Banach space $E$ and $T: K\rightarrow K$ be a nonexpansive mapping. Using a viscosity approximation method, we study the implicit midpoint rule of a nonexpansive mapping $T.$ We establish a strong convergence theorem for an iterative algorithm in the framework of uniformly smooth Banach spaces and apply our result to obtain the solutions of an accretive mapping and a variational inequality problem. The numerical example which compares the rates of convergence shows that the iterative algorithm is the most efficient. Our result is unique and the method of proof is of independent interest.

### Keywords

• Viscosity technique
• implicit midpoint rule
• nonexpansive
• accretive
• variational inequality problem

•  47H06
•  47J05
•  47J25
•  47H10
•  47H17

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