Extended splitting methods for systems of three-operator monotone inclusions with continuous operators

In this article, we propose two new splitting methods for solving systems of three-operator monotone inclusions in real Hilbert spaces, where the first operator is continuous monotone, the second is maximal monotone and the third is maximal monotone and is linearly composed. These methods primarily involve evaluating the first operator and computing resolvents with respect to the other two operators. For one method corresponding to Lipschitz continuous operator, we give back-tracking techniques to determine step lengths. Moreover, we propose a dual-first version of this method. For the other method, which corresponds to a uniformly continuous operator, we develop innovative back-tracking techniques, incorporating additional conditions to determine step lengths. The weak convergence of either method is proven using characteristic operator techniques. Notably, either method fully decouples the third operator from its linear composition operator. Numerical results demonstrate the effectiveness of our proposed splitting methods, together with their special cases and variants, in solving test problems.