  The goal of this paper is to present two algorithms for solving systems of
inclusion problems, with all component of the systems being a sum of two
maximal monotone operators. The algorithms are variants of the forward-backward
splitting method and one being a hybrid with the alternating projection method.
They consist of approximating the solution sets involved in the problem by
separating halfspaces which are a well-studied strategy. The schemes contain
two part, the first one is an explicit Armijo-type search in the spirit of the
extragradient-like methods for variational inequalities. The second part is the
projection step, being this the main difference between the algorithms. While
the first algorithm computes the projection onto the intersection of the
separating halfspaces, the second choose one component of the system and
project onto the separating halfspace of this case. In the iterative process,
the forward-backward operator is computed once per inclusion problem,
representing a relevant computational saving if compared with similar
algorithms in the literature. The convergence analysis of the proposed methods
is given assuming monotonicity in all operators, without Lipschitz continuity
assumption. We also present some numerical experiments.
