Extended Newton-type Method for Nonsmooth Generalized Equation under $\left(n,\alpha \right)$-point-based Approximation

Abstract

Let X and Y be Banach spaces and Ω ⊆ X . Let f : Ω ⟶ Y be a single valued function which is nonsmooth. Suppose that F : X⇉ 2 Y is a set-valued mapping which has closed graph. In the present paper, we study the extended Newton-type method for solving the nonsmooth generalized equation 0 ∈ f x + F x and analyze its semilocal and local convergence under the conditions that f + F − 1 is Lipschitz-like and f admits a certain type of approximation which generalizes the concept of point-based approximation so-called n , α -point-based approximation. Applications of n , α -point-based approximation are provided for smooth functions in the cases n = 1 and n = 2 as well as for normal maps. In particular, when 0 < α < 1 and the derivative of f , denoted ∇ f , is ℓ , α -Hölder continuous, we have shown that f admits 1 , α -point-based approximation for n = 1 while f admits 2 , α -point-based approximation for n = 2 , when 0 < α < 1 and the second derivative of f , denoted ∇ 2 f , is K , α -Hölder. Moreover, we have constructed an n , α -point-based approximation for the normal maps f C + F when f has an n , α -point-based approximation. Finally, a numerical experiment is provided to validate the theoretical result of this study.