  In this paper, we propose an algorithm combining the forward-backward
splitting method and the alternative projection method for solving the system
of splitting inclusion problem. We want to find a point in the interception of
a finite number of sets that we don't know, the solution of each component of
the system. The algorithm consists of approximate the sets involved in the
problem by separates halfspaces which are a known strategy. By finding these
halfspaces in each iteration we use only one inclusion problem of the system.
The iterations consist of two parts, the first contains an explicit Armijo-type
search in the spirit of the extragradient-like methods for variational
inequalities. In the iterative process, the operator forward-backward is
computed only one time for each inclusion problem, this represents a great
computational saving because the computational cost of this operator is nothing
cheap. The second part consists of special projection step, projecting in the
separating halfspace. The convergence analysis of the proposed scheme is given
assuming monotonicity all operators, without any Lipschitz continuity
assumption.
