  In this paper we focus on the convergence analysis of the forward-backward
splitting method for solving nonsmooth optimization problems in Hilbert spaces
when the objective function is the sum of two convex functions. Assuming that
one of the functions is Fr\'echet differentiable and using two new
linesearches, the weak convergence is established without any Lipschitz
continuity assumption on the gradient. Furthermore, we obtain many complexity
results of cost values at the iterates when the stepsizes are bounded below by
a positive constant.
