In this paper, we introduce new approximate projection and proximal algorithms for solving multivalued variational inequalities involving pseudomonotone and Lipschitz continuous multivalued cost mappings in a real Hilbert space. The first proposed algorithm combines the approximate projection method with the Halpern iteration technique. The second one is an extension of the Halpern projection method to variational inequalities by using proximal operators. The strongly convergent theorems are established under standard assumptions imposed on cost mappings. Finally we introduce a new and interesting example to the multivalued cost mapping, and show its pseudomontone and Lipschitz continuous properties. We also present some numerical experiments to illustrate the behavior of the proposed algorithms.

在本文中，我们介绍了新的近似投影和近邻算法，用于解决实际希尔伯特空间中涉及伪单调和Lipschitz连续多值成本映射的多值变分不等式。首先提出的算法将近似投影方法与Halpern迭代技术相结合。第二个是通过使用近端算子将Halpern投影方法扩展到变分不等式。强收敛定理是在对成本映射施加的标准假设下建立的。最后，我们为多值成本映射引入了一个有趣的新示例，并展示了其伪montone和Lipschitz连续属性。我们还提出了一些数值实验来说明所提出算法的行为。