Viscosity Iteration Technique involving an inertial extrapolation for Monotone Inclusion Problems Using Forward-Backward Splitting Methods

Abstract In this paper, we introduce and study a modified forward-backward splitting method for finding a zero in the sum of two monotone operators in real Hilbert spaces. Our proposed method only requires one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator per iteration. This is an improvement over many others in literature with strongly convergent splitting methods with two forwards and a backward iteration. Furthermore, we also incorporate inertial term in our scheme to speed up the rate of convergence. We obtain a strong convergence result when the set-valued operator is maximal monotone and the single-valued operator is Lipschitz continuous monotone which is weaker assumption than being inverse strongly monotone or cocoercive. 2010 Mathematics Subject Classification: 47H09; 47H10; 49J20; 49J40