We are concerned with the problem of solving variational inequalities which are defined on the set of fixed points of a multivalued nonexpansive mapping in a reflexive Banach space. Both implicit and explicit approaches are studied. Strong convergence of the implicit method is proved if the space satisfies Opial’s condition and has a duality map weakly continuous at zero, and the strong convergence of the explicit method is proved if the space has a weakly continuous duality map. An essential assumption on the multivalued nonexpansive mapping is that the mapping be single valued on its nonempty set of fixed points.

中文翻译：

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关于求解Banach空间中多值映射的不动点集上定义的变分不等式

我们关注解决在自反Banach空间中的多值非膨胀映射的不动点集合上定义的变分不等式的问题。研究了隐式和显式方法。如果空间满足Opial条件并且具有在零处弱连续的对偶图，则证明隐式方法的强收敛性；如果空间具有弱连续对偶图，则证明隐式方法的强收敛性。多值非膨胀映射的基本假设是，映射在其非空固定点集上是单值的。