A Mann Type Iterative Algorithm for Bounded and Strongly Monotone Mappings in Certain Banach Spaces

Let E be a uniformly smooth and uniformly convex real Banach space with dual space \(E^*\) . Let \(A: E\rightarrow E^*\) be a bounded and strongly monotone mapping such that \(A^{-1}(0)\neq \emptyset \) . For a given \(x_1\in E\) , let \(\{x_n\}\) be generated iteratively by the algorithm: \(\displaystyle x_{n+1}=x_n-\lambda _nJ^{-1}Ax_n,\;\;\; n\geq 1, \) where J is the normalized duality mapping from E into \(E^*\) and \(\{ \lambda _n\} \) is a real sequence in \((0,1)\) satisfying some suitable conditions. Then it is proved that the sequence \(\{x_n\}\) converges strongly to \(x^*\) , the unique zero of A. Our results are applied to the convex minimization problem. Futhermore, the technique of proof used is of independent of interest.