Projection and contraction methods for constrained convex minimization problem and the zero points of maximal monotone operator

In this paper, we introduce a new iterative scheme for the constrained convex minimization problem and the set of zero points of the maximal monotone operator problem, based on the projection and contraction methods. The core idea is to build the corresponding iterative algorithms by constructing reasonable error metric function and profitable direction to assure that the distance form the iteration points generated by the algorithms to a point of the solution set is strictly monotone decreasing. Under suitable conditions, new convergence theorems are obtained, which are useful in nonlinear analysis and optimization. The main advantages of the method presented are its simplicity, robustness, and ability to handle large problems with any start point. As an application, we apply our algorithm to solve the equilibrium problem, the constrained convex minimization problem and the split feasibility problem, the split equality problem in Hilbert spaces.