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Implicit relaxed and hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense. (English) Zbl 1273.47107
Summary: We first introduce an implicit relaxed method with regularization for finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping $S$ in the intermediate sense and the set of solutions of the minimization problem (MP) for a convex and continuously Fréchet differentiable functional in the setting of Hilbert spaces. The implicit relaxed method with regularization is based on three well-known methods: the extragradient method, the viscosity approximation method, and the gradient projection algorithm with regularization. We derive a weak convergence theorem for two sequences generated by this method. On the other hand, we also prove a new strong convergence theorem by an implicit hybrid method with regularization for the MP and the mapping $S$. The implicit hybrid method with regularization is based on four well-known methods: the CQ method, the extragradient method, the viscosity approximation method, and the gradient projection algorithm with regularization.
##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 65J20 Improperly posed problems; regularization (numerical methods in abstract spaces)
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##### References:
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