On Josephy-Halley method for generalized equations

We extend the classical third-order Halley iteration to the setting of generalized equations of the form $$0 \in f(x) + F(x),$$ where $f\colon X\longrightarrow Y$ is twice continuously Fr\'echet-differentiable on Banach spaces and $F\colon X\tto Y$ is a set-valued mapping with closed graph. Building on predictor-corrector framework, our scheme first solves a partially linearized inclusion to produce a predictor $u_{k+1}$, then incorporates second-order information in a Halley-type corrector step to obtain $x_{k+1}$. Under metric regularity of the linearization at a reference solution and H\"older continuity of $f''$, we prove that the iterates converge locally with order $2+p$ (cubically when $p=1$). Moreover, by constructing a suitable scalar majorant function we derive semilocal Kantorovich-type conditions guaranteeing well-definedness and R-cubic convergence from an explicit neighbourhood of the initial guess. Numerical experiments-including one- and two-dimensional test problems confirm the theoretical convergence rates and illustrate the efficiency of the Josephy-Halley method compared to its Josephy-Newton counterpart.