  This note is devoted to the splitting algorithm proposed by Davis and Yin in
2017 for computing a zero of the sum of three maximally monotone operators,
with one of them being cocoercive. We provide a direct proof that guarantees
its convergence when the stepsizes are smaller than four times the cocoercivity
constant, thus doubling the size of the interval established by Davis and Yin.
As a by-product, the same conclusion applies to the forward-backward splitting
algorithm. Further, we use the notion of "strengthening" of a set-valued
operator to derive a new splitting algorithm for computing the resolvent of the
sum. Last but not least, we provide some numerical experiments illustrating the
importance of appropriately choosing the stepsize and relaxation parameters of
the algorithms.
