Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces

作者：Ceng, Lu Chuan; Guu, Sy Ming^*; Yao, Jen Chih

来源：Computers & Mathematics with Applications, 2009, 58(3): 605-617.

DOI：10.1016/j.camwa.2009.02.035

• Nonexpansive mapping
• Variational inequality
• Fixed point
• Hybrid viscosity-like approximation method
• Strong convergence
• Constrained generalized pseudoinverse

Consider on a real Hilbert space H a nonexpansive mapping T with a fixed point, a contraction f with coefficient 0 < alpha < 1, and two strongly positive linear bounded operators A, B with coefficients (gamma) over bar is an element of (0, 1) and beta > 0, respectively. Let 0 < gamma alpha < beta. We introduce a general iterative algorithm defined by
x(n 1) := (I - lambda(n 1)A)Tx(n) lambda(n 1)[Tx(n) - mu(n 1)(BTx(n) - gamma f(x(n)))], for all n >= 1,
with mu(n) -> mu(n -> infinity), and prove the strong convergence of the iterative algorithm to a fixed point (x) over tilde is an element of Fix(T) =: C which is the unique solution of the variational inequality (for short, VI(A - I mu(B - gamma f), C)): <[A - I mu(B - gamma f)](x) over tilde, x - (x) over tilde > >= 0, for all x is an element of C. On the other hand,assume C is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on H. We devise another iterative algorithm which generates a sequence {x(n)} from an arbitrary initial point x(0) is an element of H. The sequence {x(n)} is proven to converge strongly to an element of C which is the unique solution x* of the VI(A - I mu(B - gamma f), C). Applications to constrained generalized pseudoinverses are included.

• 出版日期2009-8
• 单位中山大学; 上海师范大学