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1	Convergence Analysis for Inertial Krasnoselskii-Mann Type Iterative Algorithms Huang, Wei-Shiou 16 February 2011 (has links) We consider the problem of finding a common fixed point of an infinite family ${T_n}$ of nonlinear self-mappings of a closed convex subset $C$ of a real Hilbert space $H$. Namely, we want to find a point $x$ with the property (assuming such common fixed points exist): [ xin igcap_{n=1}^infty ext{Fix}(T_n). ] We will use the Krasnoselskii-Mann (KM) Type inertial iterative algorithms of the form $$ x_{n+1} = ((1-alpha_n)I+alpha_nT_n)y_n,quad y_n = x_n + eta_n(x_n-x_{n-1}).eqno()$$ We discuss the convergence properties of the sequence ${x_n}$ generated by this algorithm (). In particular, we prove that ${x_n}$ converges weakly to a common fixed point of the family ${T_n}$ under certain conditions imposed on the sequences ${alpha_n}$ and ${eta_n}$. demiclosedness principle KM Type iterative algorithms inertial iteration fixed point Weak convergence nonexpansive mapping
2	Iterative Methods for Minimization Problems over Fixed Point Sets Chen, Yen-Ling 02 June 2011 (has links) In this paper we study through iterative methods the minimization problem min_{x∈C} £K(x) (P) where the set C of constraints is the set of fixed points of a nonexpansive mapping T in a real Hilbert space H, and the objective function £K:H¡÷R is supposed to be continuously Gateaux dierentiable. The gradient projection method for solving problem (P) involves with the projection P_{C}. When C = Fix(T), we provide a so-called hybrid iterative method for solving (P) and the method involves with the mapping T only. Two special cases are included: (1) £K(x)=(1/2)\|\|x-u\|\|^2 and (2) £K(x)=<Ax,x> - <x,b>. The first case corresponds to finding a fixed point of T which is closest to u from the fixed point set Fix(T). Both cases have received a lot of investigations recently. Halpern's algorithm demiclosedness principle hybrid method quadratic optimization strongly monotone monotone mapping projection iterative method fixed point nonexpansive mapping Minimization
3	Projection Methods for Variational Inequalities Governed by Inverse Strongly Monotone Operators Lin, Yen-Ru 26 June 2010 (has links) Consider the variational inequality (VI) x* ∈C, ‹Fx, x - x ›≥0, x∈C () where C is a nonempty closed convex subset of a real Hilbert space H and F : C¡÷ H is a monotone operator form C into H. It is known that if F is strongly monotone and Lipschitzian, then VI () is equivalently turned into a fixed point problem of a contraction; hence Banach's contraction principle applies. However, in the case where F is inverse strongly monotone, VI () is equivalently transformed into a fixed point problem of a nonexpansive mapping. The purpose of this paper is to present some results which apply iterative methods for nonexpansive mappings to solve VI (). We introduce Mann's algorithm and Halpern's algorithm and prove that the sequences generated by these algorithms converge weakly and respectively, strongly to a solution of VI (*), under appropriate conditions imposed on the parameter sequences in the algorithms. fixed point Variational inequality Mann's algorithm projection monotone mapping demiclosedness principle Halpern's algorithm nonexpansive mapping inverse strongly monotone strongly monotone weak convergence strongly convergence

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