Projection-constrained solvability for multivalued operator inclusions

In this paper, we introduce the notion of finite-approximate solvability for multivalued operator inclusions in Hilbert spaces. We study projection-constrained inclusions of the form h∈Au+F(u), where A is a bounded linear operator and F is a compact convex-valued multivalued perturbation, and develop a nonlinear resolvent framework based on the regularized operators Tα=α(I−π)+AA∗. Using multivalued fixed-point techniques, we establish existence of nonlinear resolvent selections and derive sufficient and necessary residual conditions for finite-approximate solvability. We also show that finite-rank projection geometry, compactness, convexity, and orthogonality assumptions are essential for the solvability mechanism, and develop Galerkin-type approximation schemes for recovering solvability asymptotically. The obtained results extend finite-approximate solvability theory from single-valued operator equations to nonlinear multivalued inclusions.