Iterative methods for solving monotone variational inclusions without prior knowledge of the Lipschitz constant of the single-valued operator

In this work, we investigate a contraction-type method for solving monotone variational inclusion problems in real Hilbert spaces. We obtain strong convergence theorems for two algorithms with a self-adaptive step size for solving monotone variational inclusions. The advantage of our algorithms is that we do not require a cocoercivity assumption nor do we need to know the Lipschitz-type constant of the single-valued operator. Moreover, a convergence rate is derived in the case where one of the operators is maximally and strongly monotone, and the other is monotone and Lipschitz continuous. The performance of our proposed methods is illustrated by numerical experiments regarding signal recovery. Our results improve and extend some known results, and our experiments show that our proposed algorithms are efficient and outperform other algorithms which are available in the literature.