Fixed-Point Formulation and Parallel Averaged Operator Algorithms for Generalized Variational Inclusion Problems

In this paper, we study a variational inclusion problem in a real ordered Hilbert space via an averaged-operator-based approach. The proposed framework combines Yosida and Cayley approximation operators with the logical XOR and XNOR operations, resulting in a unified iterative procedure for obtaining solutions. Using the relevant properties of averaged operators and the associated Yosida and Cayley operators, we derive conditions that guarantee the existence of solutions and the convergence of the resulting iterative process. The performance of the proposed schemes is further assessed against several existing iterative methods in terms of convergence characteristics and computational cost. Finally, we present numerical results obtained by MATLAB to corroborate the theoretical analysis and to demonstrate the effectiveness of the proposed methodology.