  In this paper, a conceptual algorithm modifying the
forward-backward-half-forward (FBHF) splitting method for solving three
operator monotone inclusion problems is investigated. The FBHF splitting method
adjusts and improves Tseng's forward-backward-forward (FBF) splitting method
when the inclusion problem has a third-part operator that is cocoercive. The
FBHF method recovers the FBF iteration (when this aforementioned part is zero),
and it also works without using the widely used Lipschitz continuity
assumption. The conceptual algorithm proposed in this paper also has those
advantages, and it derives two variants (called Method 1 and Method 2) by
choosing different projection (forward) steps. Both proposed methods also work
efficiently without assuming the Lipschitz continuity and without directly
using the cocoercive constant. Moreover, they have the following desired
features: (i) very general iterations are derived for both methods, recovering
the FBF and the FBHF iterations and allowing possibly larger stepsizes if the
projection steps are over-relaxing; and (ii) strong convergence to the best
approximation solution of the problem is proved for Method 2. To the best of
our knowledge, this is the first time that an FBF-type method converges
strongly for finding the best approximation solution of the three operator
monotone inclusion.
