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Convergence to common fixed point of nonexpansive semigroups. (English) Zbl 1204.47076

Summary: Let \(E\) be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, \(C\) be a closed convex subset of \(E\), \(\mathcal S=\{T(s):s\geq 0\}\) be a nonexpansive semigroup on \(C\) such that the set of common fixed points of \(\{T(s):s\geq 0\}\) is nonempty. Let \(f: C\to C\) be a contraction, \(\{\alpha_n\}\), \(\{\beta_n\}\), \(\{t_n\}\) be real sequences such that \(0<\alpha_n, \beta_n\leq 1\), \(\lim_{n\to\infty}\alpha_n=0\), \(\lim_{n\to\infty}\beta_n=0\) and \(\lim_{n\to\infty}t_n=\infty\), \(y_0\in C\). In this paper, we show that the two iterative sequences defined as follows:
\[ \begin{aligned} x_n&=\alpha_nf(x_n)+(1-\alpha_n)\frac 1{t_n}\int_0^{t_n} T(s)x_n\,ds,\\ y_{n-1}&=\beta_nf(y_n)+(1-\beta_n)\frac 1{t_n}\int_0^{t_n} T(s)y_n\,ds, \end{aligned} \]
converge strongly to a common fixed point of \(\{T(s):s\geq 0\}\) which solves some variational inequality when \(\{\alpha_n\}\), \(\{\beta_n\}\) satisfy some appropriate conditions.

MSC:

47H20 Semigroups of nonlinear operators
47J25 Iterative procedures involving nonlinear operators
65J15 Numerical solutions to equations with nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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References:

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