On Split Monotone Variational Inclusion Problem with Multiple Output Sets with Fixed Point Constraints

Abstract In this paper, we introduce and study the concept of split monotone variational inclusion problem with multiple output sets (SMVIPMOS). We propose a new iterative scheme, which employs the viscosity approximation technique for approximating the solution of the SMVIPMOS with fixed point constraints of a nonexpansive mapping in real Hilbert spaces. The proposed method utilises the inertial technique for accelerating the speed of convergence and a self-adaptive step size so that its implementation does not require prior knowledge of the operator norm. Under mild conditions, we obtain a strong convergence result for the proposed algorithm and obtain a consequent result, which complements several existing results in the literature. Moreover, we apply our result to study the notions of split variational inequality problem with multiple output sets with fixed point constraints and split convex minimisation problem with multiple output sets with fixed point constraints in Hilbert spaces. Finally, we present some numerical experiments to demonstrate the implementability of our proposed method.