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 $f\text{\hspace{0.17em}⁣}:X⟶Y$ is twice continuously Fréchet-differentiable on Banach spaces and $F\text{\hspace{0.17em}⁣}:X\text{tto}Y$ 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 Hölder continuity of $f^{\mathrm{\prime }\mathrm{\prime }}$, 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.