On the Convergence of Gauss-type Proximal Point Method for Smooth Generalized Equations
Md. Asraful Alom
Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh and Department of Mathematics, Khulna University of Engineering and Technology, Khulna-9203, Bangladesh.
Mohammed Harunor Rashid *
Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh.
*Author to whom correspondence should be addressed.
Abstract
Let X and Y be Banach spaces and Ω be an open subset of X. Let f : X → Y be a Frechet differentiable function on Ω and F : X 
2Y be a set valued mapping with closed graph. We deal with smooth generalized equations which is de ned by the sum of Frechet di erentiable function and a set valued mapping. Under some sufficient conditions, a Gauss-type proximal point algorithm (G-PPA) is introduced and studied for solving generalized equations of the form 0 ∈ 2 f(x) + F(x). Indeed, when F is metrically regular we analyze semi-local and local convergence of the G-PPA. Furthermore, we give a numerical example to justify the convergence results of the G-PPA.
Keywords: Generalized equations, set-valued mapping, metrically regular mapping, lipschitz-like mapping, semi-local convergence.