A new Krasnoselskii’s type algorithm for zeros of strongly monotone and Lipschitz mappings


 Sene, M., Ndiaye, M. and Djitte, N.


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For q>1, let E be a q-uniformly smooth real Banach space with dual space E^*. Let A: E\rightarrow E^* be a Lipschitz and strongly monotone mapping such that A^{-1}(0)\neq \emptyset. For given x_1\in E, let \{x_n\} be generated iteratively by the algorithm :

    \[x_{n+1}=x_n-\lambda J^{-1}(Ax_n),\;n\geq1,\]

where J is the normalized duality mapping from E into
E^* and \lambda is a positive real number choosen in a suitable interval. Then it is proved that the sequence \{x_n\} converges strongly to x^*, the unique point of A^{-1}(0). Our theorems are applied to the convex minimization problem. Futhermore, our technique of proof is of independent interest.

 

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 Sene, M., Djitte, N., Ndiaye, M.