The conjugate gradient method for split variational inclusion and constrained convex minimization problems

• A new viscosity method of split variational inclusion problem is proposed.
• The method is based on conjugate gradient method and averaged mapping approach.
• The strong convergence of the sequences generated by the method is proved.
• The results are the generalization of the previously known results in this area.

In this paper, we introduce and study a new viscosity approximation method based on the conjugate gradient method and an averaged mapping approach for finding a common element of the set of solutions of a constrained convex minimization problem and the set of solutions of a split variational inclusion problem. Under suitable conditions, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution of the split variational inclusion problem and the set of solutions of the constrained convex minimization problem. The results presented in this paper are the supplement, extension and generalization of the previously known results in this area. Finally, preliminary numerical results indicate the feasibility and efficiency of the proposed methods.