The shrinking projection method for solving variational inequality problems and fixed point problems in Banach spaces.

*(English)*Zbl 1184.49019Summary: We consider a hybrid projection algorithm based on the shrinking projection method for two families of quasi-\(\varphi\)-nonexpansive mappings. We establish strong convergence theorems for approximating the common element of the set of the common fixed points of such two families and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. As applications, at the end of the paper we first apply our results to consider the problem of finding a zero point of an inverse-strongly monotone operator and we finally utilize our results to study the problem of finding a solution of the complementarity problem. Our results improve and extend the corresponding results announced by recent results.

##### MSC:

49J40 | Variational inequalities |

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

47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

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\textit{R. Wangkeeree} and \textit{R. Wangkeeree}, Abstr. Appl. Anal. 2009, Article ID 624798, 26 p. (2009; Zbl 1184.49019)

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