Relaxed Inertial Method for Solving Split Monotone Variational Inclusion Problem with Multiple Output Sets Without Co-coerciveness and Lipschitz Continuity

In this paper, we study the concept of split monotone variational inclusion problem with multiple output sets. We propose a new relaxed inertial iterative method with self-adaptive step sizes for approximating the solution of the problem in the framework of Hilbert spaces. Our proposed algorithm does not require the co-coerciveness nor the Lipschitz continuity of the associated single-valued operators. Moreover, some parameters are relaxed to accommodate a larger range of values for the step sizes. Under some mild conditions on the control parameters and without prior knowledge of the operator norms, we obtain strong convergence result for the proposed method. Finally, we apply our result to study certain classes of optimization problems and we present several numerical experiments to demonstrate the implementability of the proposed method. Several of the existing results in the literature could be viewed as special cases of our result in this paper.