We study variational inequalities and fixed point problems in real Hilbert spaces. A new algorithm is proposed for finding a common element of the solution set of a pseudo-monotone variational inequality and the fixed point set of a demicontractive mapping. The advantage of our algorithm is that it does not require prior information regarding the Lipschitz constant of the variational inequality operator and that it only computes one projection onto the feasible set per iteration. In addition, we do not need the sequential weak continuity of the variational inequality operator in order to establish our strong convergence theorem. Next, we also obtain an R-linear convergence rate for a related relaxed inertial gradient method under strong pseudo-monotonicity and Lipschitz continuity assumptions on the variational inequality operator. Finally, we present several numerical examples which illustrate the performance and the effectiveness of our algorithm.

我们研究真实希尔伯特空间中的变分不等式和不动点问题。提出了一种寻找伪单调变分不等式的解集和解收缩映射的不动点集的公共元素的新算法。我们算法的优点是它不需要关于变分不等式算子的 Lipschitz 常数的先验信息，并且每次迭代只计算一个到可行集的投影。此外，我们不需要变分不等式算子的序列弱连续性来建立我们的强收敛定理。接下来，我们也得到了一个R- 在变分不等式算子上的强伪单调性和 Lipschitz 连续性假设下，相关松弛惯性梯度方法的线性收敛率。最后，我们给出了几个数值例子来说明我们算法的性能和有效性。