Tseng type methods for solving inclusion problems and its applications

In this paper, we introduce two modifications of the forward–backward splitting method with a new step size rule for inclusion problems in real Hilbert spaces. The modifications are based on Mann and viscosity-ideas. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish strong convergence of the proposed algorithms. We present two numerical examples, the first in infinite dimensional spaces, which illustrates mainly the strong convergence property of the algorithm. For the second example, we illustrate the performances of our scheme, compared with the classical forward–backward splitting method for the problem of recovering a sparse noisy signal. Our result extend some related works in the literature and the primary experiments might also suggest their potential applicability.