In this paper, we study the variational inclusion problem which consists of finding zeros of the sum of a single and multivalued mappings in real Hilbert spaces. Motivated by the viscosity approximation, projection and contraction and inertial forward–backward splitting methods, we introduce two new forward–backward splitting methods for solving this variational inclusion. We present weak and strong convergence theorems for the proposed methods under suitable conditions. Our work generalize and extend some related results in the literature. Several numerical examples illustrate the potential applicability of the methods and comparisons with related methods emphasize it further.

在本文中，我们研究了变分包含问题，该问题包含在实希尔伯特空间中找到单值和多值映射之和的零。受粘度逼近，投影和收缩以及惯性前向后拆分方法的启发，我们引入了两种新的前向后拆分方法来解决这种变分包含问题。我们提出了在适当条件下所提出方法的弱收敛定理和强收敛定理。我们的工作概括并扩展了文献中的一些相关结果。几个数值例子说明了该方法的潜在适用性，并且与相关方法的比较进一步强调了该方法。