  Sparsity regularization has been largely applied in many fields, such as
signal and image processing and machine learning. In this paper, we mainly
consider nonconvex minimization problems involving three terms, for the
applications such as: sparse signal recovery and low rank matrix recovery. We
employ a three-operator splitting proposed by Davis and Yin (called DYS) to
solve the resulting possibly nonconvex problems and develop the convergence
theory for this three-operator splitting algorithm in the nonconvex case. We
show that if the step size is chosen less than a computable threshold, then the
whole sequence converges to a stationary point. By defining a new decreasing
energy function associated with the DYS method, we establish the global
convergence of the whole sequence and a local convergence rate under an
additional assumption that this energy function is a Kurdyka-$\L$ojasiewicz
function. We also provide sufficient conditions for the boundedness of the
generated sequence. Finally, some numerical experiments are conducted to
compare the DYS algorithm with some classical efficient algorithms for sparse
signal recovery and low rank matrix completion. The numerical results indicate
that DYS method outperforms the exsiting methods for these specific
applications.
