GENERAL GAUSS-TYPE PROXIMAL POINT METHOD AND ITS CONVERGENCE ANALYSIS 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.
Let X and Y be Banach spaces and Ω be an open subset of X. In the present paper, a general Gauss-type Proximal Point Algorithm (GGPPA) is introduced for approximating the solution of a generalized equation 0∈f(x)+F(x), where f:X→ Y is a differentiable function on Ω and F:X⇉2^Y is a set valued mapping with closed graph. Semilocal and local convergences of the GGPPA are presented by considering a sequence of Lipschitz continuous functions gk:X→ Y such that gk (0)=0 around the origin with positive Lipschitz constants λk and the metric regularity condition of F. Finally, we give a numerical example to verify the convergence results of the GGPPA.
Keywords: Generalized equations, set-valued mapping, Metrically regular mapping, Lipschitz-like mapping, semi-local convergence