In this work, we propose a new version of inertial relaxed CQ algorithms for solving the split feasibility problems in the frameworks of Hilbert spaces. We then prove its strong convergence by using a viscosity approximation method under some weakened assumptions. To be more precisely, the computation on the norm of operators and the metric projections is relaxed. Finally, we provide numerical experiments to illustrate the convergence behavior and to show the effectiveness of the sequences constructed by the inertial technique.

Variational inequality theory is an effective tool for engineering, economics, transport and mathematical optimization. Some of the approaches used to resolve variational inequalities usually involve iterative techniques. In this article, we introduce a new modified viscosity-type extragradient method to solve monotone variational inequalities problems in real Hilbert space. The result of the strong convergence of the method is well established without the information of the operator’s Lipschitz constant. There are proper mathematical studies relating our newly designed method to the currently state of the art on several practical test problems.