On a unified Yosida inclusion problem and its computational implications

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.