An efficient algorithm based on viscosity tseng-type method for solving split hierarchical monotone variational inclusion problems with multiple output sets,

Very recently, [Wickramasinghe et al., Mann type approximation scheme for solving a new class of split inverse problems in Hilbert spaces Applicable Anal. (2023). DOI: 10.1080/00036811. 2023.2233977] introduced a new class of split inverse problems, known as split hierarchical monotone variational inclusion problems with multiple output sets. The authors proposed a Mann-type iterative method to approximate the solution of the problem in Hilbert spaces. However, to establish their convergence result, the authors required the associated single-valued operators to be co-coercive. This condition is too stringent and limits the scope of application of their results. In this study, we aim to improve their result of [Wickramasinghe et al.] by relaxing the co-coercive condition to monotone. We also introduce a new viscosity Tseng-type iterative method for solving the problem in Hilbert spaces and obtain strong convergence results under more relaxed conditions. Furthermore, our study extends to addressing the split hierarchical variational inequality problem involving multiple output sets, the split hierarchical convex minimization problem with multiple output sets, and the generalized split hierarchical monotone variational inclusion problem. Lastly, we perform numerical experiments in both finite- and infinite-dimensional spaces and provide comparison results with existing methods discussed in the literature.