A Perturbed Mann-Type Algorithm for Zeros of Maximal Monotone Mappings

Let E be a uniformly convex and uniformly smooth real Banach space and \(E^*\) its dual. Let \(A:E\to E^*\) be a bounded maximal monotone mapping such that \(A^{-1}(0)\neq \emptyset \) . We first introduce the algorithm: For given \(x_1\in E\) , let \(\{x_n\}\) be generated by the formula: \(x_{n+1}= x_n -\lambda _n J^{-1}Ax_n -\lambda _n\theta _n(x_n-x_1),\,n\geq 1\) , where J is the normalized duality mapping from E into \(E^*\) and \(\lambda _n\) and \(\theta _n\) are positive real numbers in \((0,1)\) satisfying suitable conditions. Next, we obtain the strong convergence of the sequence \(\{x_n\}\) to the solution of the equation \(Au=0\) closest to the initial point \(x_1\) . Using this result, we deal with the convex minimization problem. Our results improve and unify most of the ones that have been proved in this direction for this important class of nonlinear mappings. Furthermore, our new technique of proof is of independent interest.