In this article, we consider and study a system of generalized set-valued variational inequalities involving relaxed cocoercive mappings in Hilbert spaces. Using the projection method and Banach contraction principle, we prove the existence of a solution for the considered problem. Further, we propose an iterative algorithm and discuss its convergence. Moreover, we establish equivalence between the system of variational inequalities and altering points problem. Some parallel iterative algorithms are proposed, and the strong convergence of the sequences generated by these iterative algorithms is discussed. Finally, a numerical example is constructed to illustrate the convergence analysis of the proposed parallel iterative algorithms.

中文翻译：

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广义集值变分不等式系统的一些迭代算法的收敛性

在本文中，我们考虑并研究了一个希尔伯特空间中涉及松弛矫顽映射的广义集值变分不等式系统。使用投影方法和Banach压缩原理，我们证明了所考虑问题的一种解决方案。此外，我们提出了一种迭代算法并讨论了其收敛性。此外，我们建立了变分不等式和变分问题之间的等价性。提出了一些并行迭代算法，并讨论了这些迭代算法生成的序列的强收敛性。最后，通过算例说明了所提出的并行迭代算法的收敛性。