A general composite iterative algorithm for nonexpansive mappings in Hilbert spaces

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

来源：Computers & Mathematics with Applications, 2011, 61(9): 2447-2455.

DOI：10.1016/j.camwa.2011.02.025

• Nonexpansive mappings
• Composite iterative method
• Variational inequality
• Fixed point
• Projection
• Viscosity approximation

Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient alpha is an element of (0, 1), F : H -> H is a k-Lipschitzian and eta-strongly monotone operator with k > 0, eta > 0, and A : H -> H is a strongly positive bounded linear operator with coefficient (gamma) over bar is an element of (1, 2). Let 0 < mu < 2 eta/k(2), 0 < gamma < mu(eta - mu k(2)/2)/alpha = tau/alpha. It is shown that the sequence {x(n)} generated by the following general composite iterative method:
{y(n) = (I - alpha(n)mu F)Tx(n) + alpha(n)gamma f(x(n)), x(n+1) = (I - beta(n)A)Tx(n) + beta(n)y(n), for all n >= 0,
where {alpha(n)} subset of [0, 1] and {beta(n)} subset of (0.1], converges strongly to a fixed point (x) over bar is an element of Fix(T), which solves the variational inequality <(I - A)(x) over bar, x - (x) over bar > <= 0, for all x is an element of Fix(T).

• 出版日期2011-5
• 单位上海大学; 上海师范大学