Does Newton's method for set-valued maps converges uniformly in mild differentiability context?

In this article, we study the existence of Newton-type sequence for solving the equation y ϵ f(𝓍) + F(𝓍) where y is a small parameter, f is a function whose Fréchet derivative satisfies a Holder condition of the form II∇f(𝓍_k) -∇f(x₂)∥ ≤ K ∥lx₁ - x₂∥^d and F is a set-valued map between two Banach spaces X and Y. We prove that the Newton-type method y ∈ f(𝓍_k) +∇f(𝓍_k)(𝓍_k+1 - 𝓍_k ) +F(𝓍_k+1), is locally convergent to a solution of  y ϵ f(𝓍) + F(𝓍) if the set valued map (f(x* )+ ∇f(x* )(∙-𝓍*)+F(∙))^-1 is Aubin continuous at (0,𝓍*) where 𝓍* is a solution of 0 ϵ f(𝓍) + F(𝓍). Moreover, we show that this convergence is superlinear uniformly in the parameter y and quadratic when d = 1.