Modified forward–backward splitting method for variational inclusions

In this paper we propose an explicit algorithm for solving a variational inclusion problem of the sum of two operators, the one is maximally monotone and the other is monotone and Lipschitz continuous. The algorithm uses the variable stepsizes which are updated over each iteration by some cheap comptutations. These stepsizes are found without the prior knowledge of the Lipschitz constant of operator as well as without using lineseach procedure. The algorithm thus can be implemented easily. The convergence and the convergence rate of the algorithm are established under mild conditions. Several preliminary numerical results are provided to demonstrate the theoretical results and also to compare the new algorithm with some existing ones.