  The Douglas-Rachford algorithm is a classical and powerful splitting method
for minimizing the sum of two convex functions and, more generally, finding a
zero of the sum of two maximally monotone operators. Although this algorithm is
well understood when the involved operators are monotone or strongly monotone,
the convergence theory for weakly monotone settings is far from being complete.
In this paper, we propose an adaptive Douglas-Rachford splitting algorithm for
the sum of two operators, one of which is strongly monotone while the other one
is weakly monotone. With appropriately chosen parameters, the algorithm
converges globally to a fixed point from which we derive a solution of the
problem. When one operator is Lipschitz continuous, we prove global linear
convergence, which sharpens recent known results.
