  In this paper, we propose variants of forward-backward splitting method for
solving the system of splitting inclusion problem. We propose a conceptual
algorithm containing three variants, each having a different projection steps.
The algorithm consists in two parts, the first and main 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
represent a great computational saving if we compare with Tseng's algorithm,
because the computational cost of this operator is very high. The second part
of the scheme consists in special projection steps. The convergence analysis of
the proposed scheme is given assuming monotonicity on both operators, without
assuming Lipschitz continuity on the forward operators.

Optimization and Control