On relaxed inertial projection and contraction algorithms for solving monotone inclusion problems

We present three novel algorithms based on the forward-backward splitting technique for the solution of monotone inclusion problems in real Hilbert spaces. The proposed algorithms work adaptively in the absence of the Lipschitz constant of the single-valued operator involved thanks to the fact that there is a non-monotonic step size criterion used. The weak and strong convergence and the R-linear convergence of the developed algorithms are investigated under some appropriate assumptions. Finally, our algorithms are put into practice to address the restoration problem in the signal and image fields, and they are compared to some pertinent algorithms in the literature.