Solving monotone inclusions involving the sum of three maximally monotone operators and a cocoercive operator with applications

In this paper, we propose two four-operator splitting algorithms for approaching the set of zeros of the sum of three maximally monotone operators and a cocoercive operator. Our methods do not rely on the traditional product space technique. The sequences of the proposed algorithms are proven to be convergent under mild conditions. In applications of interest to us, we employ the proposed algorithms to solve a composite convex minimization problem, for which we also provide convergence analysis. Numerical experiments are performed on the image inpainting problem to demonstrate the efficiency of the proposed algorithms.