  In this article, we study two methods for solving monotone inclusions in real
Hilbert spaces involving the sum of a maximally monotone operator, a
monotone-Lipschitzian operator, a cocoercive operator, and a normal cone to a
vector subspace. Our algorithms split and exploits the intrinsic properties of
each operator involved in the inclusion. We derive our methods by combining
partial inverse techniques with the forward-reflected-backward algorithm and
with the shadow-Douglas--Rachford algorithm, respectively. Our methods inherit
the advantages of those methods, requiring only one activation of the
Lipschitzian operator, one activation of the cocoercive operator, two
projections onto the closed vector subspace, and one calculation of the
resolvent of the maximally monotone operator. Additionally, to allow larger
step-sizes in one of the proposed methods, we revisit FSDR by extending its
convergence for larger step-sizes. Furthermore, we provide methods for solving
monotone inclusions involving a sum of maximally monotone operators and for
solving a system of primal-dual inclusions involving a mixture of sums, linear
compositions, parallel sums, Lipschitzian operators, cocoercive operators, and
normal cones. We apply our methods to constrained composite convex optimization
problems as a specific example. Finally, in order to compare our methods with
existing methods in the literature, we provide numerical experiments on
constrained total variation least-squares optimization problems and computed
tomography inverse problems. We obtain promising numerical results.
