The main aim of this work is to use the XOR-operation technique to find the common solutions for a new class of extended system of fuzzy ordered variational inclusions with its corresponding system of fuzzy ordered resolvent equations involving the $\oplus$ operation and fixed point problems, which are slightly different from corresponding problems considered in several recent papers in the literature and are more advantageous. We establish that the system of fuzzy ordered variational inclusions is equivalent to a fixed point problem and a relationship between a system of fuzzy ordered variational inclusions and a system of fuzzy ordered resolvent equations is shown. We prove the existence of a common solution and discuss the convergence of the sequence of iterates generated by the algorithm for a considered problem. The iterative algorithm and results demonstrated in this article have witnessed, a significant improvement for many previously known results of this domain. Some examples are constructed in support of the main results.

This paper investigated a Yosida variational inclusion problem (YVIP) in a real ordered Hilbert space, where logical operations are incorporated through an averaged-operator framework. By reformulating the YVIP and its associated resolvent equation as equivalent fixed-point problems, we designed an iterative scheme that systematically integrates these logical and averaged-operator components. Furthermore, we analyzed the convergence of the proposed algorithms. A comparative study with existing algorithms, supported by numerical experiments, demonstrated the improved computational behavior of the proposed method. To illustrate its practical relevance, a representative MATLAB-based numerical result was also presented.

This paper introduced and considered the Yosida inclusion problem, which is a unified model arising from the monotone inclusion, fixed point, and Cayley inclusion problems. By using the Yosida operator as a regularized and generically differentiable approximation of a maximal monotone operator for solving the Yosida inclusion problem we devised efficient and stable iterative algorithms instead of direct resolvent computation. This approach offers both theoretical and computational advantages for solving complex mathematical operator inclusion problems in Hilbert spaces. We demonstrated that the sequence generated by the proposed inertial $S$-iteration converges strongly. A theoretical example and an application were also presented to illustrate the effectiveness of the proposed algorithms.