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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.

On Halpern-type sequences with applications in variational inequality problems