We introduce a Halpern-type sequence and give a necessary and sufficient condition for a strong convergence of this sequence. In particular, we obtain two strong convergence theorems for approximation of a fixed point of nonexpansive mappings or of quasi-nonexpansive ones. We also apply our result for various iterative methods in variational inequality problem. For the Lipschitz continuous mappings, we deal with the extragradient method of Korpelevič, the subgradient extragradient method of Censor et al. and the extragradient of Tseng where the step size rule is priorly or posteriorly chosen. For the non-Lipschitz continuous mappings, we use our results to deduce the convergence results of Shehu and Iyiola and of Thong and Gibali. Our approach allows us to conclude many new results with some new assumptions.

我们引入了一个 Halpern 型序列，并给出了该序列强收敛的充分必要条件。特别是，我们获得了两个强收敛定理，用于逼近非扩张映射或准非扩张映射的不动点。我们还将我们的结果应用于变分不等式问题中的各种迭代方法。对于 Lipschitz 连续映射，我们处理 Korpelevič 的外梯度方法，Censor 等人的次梯度外梯度方法。以及 Tseng 的外梯度，其中步长规则是先选择或后选择的。对于非 Lipschitz 连续映射，我们使用我们的结果推导出 Shehu 和 Iyiola 以及 Thong 和 Gibali 的收敛结果。我们的方法使我们能够通过一些新假设得出许多新结果。