## Hybrid proximal-type and hybrid shrinking projection algorithms for equilibrium problems, maximal monotone operators, and relatively nonexpansive mappings.(English)Zbl 1208.49036

Summary: We introduce two hybrid proximal-type algorithms and two hybrid shrinking projection algorithms by using the hybrid proximal-type method and the hybrid shrinking projection method, respectively, for finding a common element of the set of solutions of an equilibrium problem, the set of fixed points of a relatively nonexpansive mapping, and the set of solutions to the equation $$0 \in Tx$$ for a maximal monotone operator $$T$$ defined on a uniformly smooth and uniformly convex Banach space. The strong convergence of the sequences generated by the proposed algorithms is established. Our results improve and generalize several known results in the literature.

### MSC:

 49M30 Other numerical methods in calculus of variations (MSC2010) 49J40 Variational inequalities 47J20 Variational and other types of inequalities involving nonlinear operators (general) 47H05 Monotone operators and generalizations 65K10 Numerical optimization and variational techniques
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