Extended Newton-type Method for Nonsmooth Generalized Equation under n,α-point-based Approximation - M. Z. Khaton; M. H. Rashid (International Journal of Mathematics and Mathematical Sciences)
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⇉2Y 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∈fx+Fx 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 ∇2f, is K,α-Hölder. Moreover, we have constructed an n,α-point-based approximation for the normal maps fC+F when f has an n,α-point-based approximation. Finally, a numerical experiment is provided to validate the theoretical result of this study.