Computation of zeros of monotone type mappings: on Chidume’s open problem. (English) Zbl 1435.47058
Summary: For \(p\geq 2\), let \(E\) be a 2-uniformly smooth and \(p\)-uniformly convex real Banach space and let \(A:E\to E^{\ast }\) 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 by the algorithm \(x_{n+1}=J^{-1}(Jx_n- \lambda Ax_n)\), \(n\geq 1\), where \(J\) is the normalized duality mapping from \(E\) into \(E^{\ast }\) and \(\lambda\) is a positive real number in \((0,1)\) satisfying suitable conditions. Then it is proved that \(\{x_n\}\) converges strongly to the unique point \(x^{\ast }\in A^{-1}(0)\). Furthermore, our theorems provide an affirmative answer to the open problem formulated by C. E. Chidume, A. U. Bello and B. Usman [“Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical Banach spaces”, SpringerPlus 4, Art. 297, 9 p. (2015; doi:10.1186/s40064-015-1044-1)]. Finally, applications to convex minimization problems are given.
MSC:
| 47J05 | Equations involving nonlinear operators (general) |
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H05 | Monotone operators and generalizations |
| 47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |
Keywords:
iterative algorithm; strong convergence; strongly monotone mappings; Lipschitz; convex minimization problemReferences:
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