On the Josephy–Halley method for generalized equations

We extend the classical third-order Halley iteration to the setting of generalized equations of the form 0∈f(x)+F(x),where f:X⟶Y is twice continuously Fréchet differentiable between Banach spaces and F:X⇉Y is a set-valued mapping with a closed graph. Building on a predictor–corrector framework, our scheme first solves a partially linearized inclusion to produce a predictor uk+1, then incorporates second-order information in a Halley–type corrector step to obtain xk+1. Under metric regularity of the linearization at a reference solution and Hölder continuity of f″ with exponent p∈(0,1], we prove that the iterates converge locally with the order 2 + p (cubically when p = 1). Moreover, by constructing a suitable scalar majorant function, we derive Kantorovich-type conditions guaranteeing well-definedness and R-cubic convergence from an explicit neighbourhood of the initial guess. Numerical experiments across one-, two-, and many-dimensional problems validate the theoretical convergence rates and demonstrate the efficiency of the Josephy–Halley method relative to its Josephy–Newton counterpart.