A new modified forward–backward–forward algorithm for solving inclusion problems

The forward–backward–forward (FBF) splitting method is a popular iterative procedure for finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. In this paper, we introduce a modification of the forward–backward splitting method with an adaptive step size rule for inclusion problems in real Hilbert spaces. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish weak convergence of the proposed algorithm. Moreover, if the single-valued operator is cocoercivity, then the proposed algorithm strongly converges to the unique solution of the problem with an R-linear rate. Finally, we give several numerical experiments to illustrate the convergence of the proposed algorithm and also to compare them with others. © 2022, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.