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Computation of Zeros of Monotone Type Mappings : On Chidume's Open Problem
For p≥2, let E be a 2-uniformly smooth and p-uniformly convex real Banach space and let A:E→E∗ be a Lipschitz and strongly monotone mapping such that A−1(0)≠∅. For given x1∈E, let {xn} be generated by the algorithm xn+1=J−1(Jxn−λAxn), n≥1, where J is the normalized duality mapping from E into E∗ and λ is a positive real number in (0,1) satisfying suitable conditions. Then it is proved that {xn} converges strongly to the unique point x∗∈A−1(0). Furthermore, our theorems provide an affirmative answer to the Chidume et al. open problem [‘Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical Banach spaces’, SpringerPlus4 (2015), 297]. Finally, applications to convex minimization problems are given.
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