In this paper, we investigate the problem of finding a fixed point of the nonsmooth function, max f1(x), f2(x),., fn(x). First, we recall the definition of Newton derivative and examine some basic properties. | ISSN 1859-1531 - THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO. 6(127).2018 37 APPLYING SEMISMOOTH NEWTON METHOD TO FIND FIXED POINTS OF NONSMOOTH FUNCTIONS OF ONE VARIABLE Pham Quy Muoi, Phan Quang Nhu Anh, Duong Xuan Hiep, Phan Duc Tuan University of Education – The University of Danang; pqmuoi@; nhuanh83@; dxhiep1994@; pdtuan@ Abstract - In this paper, we investigate the problem of finding a fixed point of the nonsmooth function, max f1 (x ), f2 (x ), , fn (x ) . First, we recall the definition of Newton derivative and examine some basic properties. Then, we investigate the Newton differentiability of function max f1(x ), f2(x ), , fn (x ) . We give the necessary and sufficient conditions for Newton differentiability of this function in two cases: A special case: max f1(x ), f2(x ) and the general case: max f1(x ), f2(x ), , fn (x ) . We emphasize that, the sufficient condition for the special case is much weaker than that of the general case. After that, we apply the semismooth Newton method to find a fixed point of the above function. The local quadratic order convergence of the method is proven. Finally, we present the numerical results for some specific examples. Key words - Newton Derivative; Newton differential; Fixed point; Semismooth Newton method; Nonsmooth function 1. Introduction Thought out the history, fixed point theory has been widely considered by numerous researchers both domestically and internationally. There is a large amount of published research including Banach, Browder and Borel’s fixed point theory, . [1, 4]. Fixed point theory has wide applications in abundant areas such as partial differential equation theory, economics (game theory), . [3, 4]. In numerical programming, the familiar method which has been used is fixed point’s iteration as well as advanced one [2]. As we know, fixed point iteration converges in linear speed. Recently, in optimization theory for .
