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In this paper, a conceptual algorithm modifying theforward-backward-half-forward (FBHF) splitting method for solving threeoperator monotone inclusion problems is investigated. The FBHF splitting methodadjusts and improves Tseng's forward-backward-forward (FBF) splitting methodwhen the inclusion problem has a third-part operator that is cocoercive. TheFBHF method recovers the FBF iteration (when this aforementioned part is zero),and it also works without using the widely used Lipschitz continuityassumption. The conceptual algorithm proposed in this paper also has thoseadvantages, and it derives two variants (called Method 1 and Method 2) bychoosing different projection (forward) steps. Both proposed methods also workefficiently without assuming the Lipschitz continuity and without directlyusing the cocoercive constant. Moreover, they have the following desiredfeatures: (i) very general iterations are derived for both methods, recoveringthe FBF and the FBHF iterations and allowing possibly larger stepsizes if theprojection steps are over-relaxing; and (ii) strong convergence to the bestapproximation solution of the problem is proved for Method 2. To the best ofour knowledge, this is the first time that an FBF-type method convergesstrongly for finding the best approximation solution of the three operatormonotone inclusion.

On the Weak and Strong Convergence of a Conceptual Algorithm for Solving Three Operator Monotone Inclusions