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**Composite iterative algorithms for variational inequality and fixed point problems in real smooth and uniformly convex Banach spaces.**
*(English)*
Zbl 1271.47054

Summary: We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich’s extragradient method and the viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

49J40 | Variational inequalities |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

### Keywords:

composite implicit and explicit iterative algorithms; strong convergence; Korpelevich’s extragradient method; viscosity approximation method; real smooth uniformly convex Banach space
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\textit{L.-C. Ceng} and \textit{C.-F. Wen}, J. Appl. Math. 2013, Article ID 761864, 21 p. (2013; Zbl 1271.47054)

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### References:

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