General Version of Gauss-type Proximal Point Method and Its Uniform Convergence Analysis for Metrically Regular Mappings
M. A. Alom
Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh and Department of Mathematics, Khulna University of Engineering and Technology, Khulna-9203, Bangladesh.
M. H. Rashid *
Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh.
K. K. Dey
Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh.
*Author to whom correspondence should be addressed.
Abstract
We study the uniform convergence of the general version of Gauss-type proximal point algorithm (GG-PPA), introduced by Alom et al. [1], for solving the parametric generalized equations y ∈ T(x), where T : X 
2Y is a set-valued mapping with locally closed graph, y is a parameter, and X and Y are Banach spaces. In particular, we establish the uniform convergence of the GG-PPA by considering a sequence of Lipschitz continuous functions gk : X → Y with gk(0) = 0 and positive Lipschitz constants λk in the sense that it is stable under small perturbations when T is metrically regular at a given point. In addition, we give a numerical example to justify theuniform convergence of the GG-PPA.
Keywords: Set-valued mappings, metrically regular mappings, lipschitz-like mappings, semi-local convergence, uniform convergence