Chen, Rudong; Song, Yunyan Convergence to common fixed point of nonexpansive semigroups. (English) Zbl 1204.47076 J. Comput. Appl. Math. 200, No. 2, 566-575 (2007). 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. Cited in 44 Documents 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. Keywords:nonexpansive semigroup; uniformly convex Banach space; uniformly Gâteaux differentiable; common fixed point PDF BibTeX XML Cite \textit{R. Chen} and \textit{Y. Song}, J. Comput. Appl. Math. 200, No. 2, 566--575 (2007; Zbl 1204.47076) Full Text: DOI OpenURL References: [1] Browder, F.E., Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. rational mech. anal., 24, 82-90, (1967) · Zbl 0148.13601 [2] Bruck, R.E., Nonexpansive retracts of Banach spaces, Bull. amer. math. soc., 76, 384-386, (1970) · Zbl 0224.47034 [3] R. Chen, P.-K. Lin, Y. Song, An approximation method for strictly pseudocontractive mappings of Browder-Petryshyn type, Nonlinear Anal., in press. [4] R. Chen, Y. Song, H. Zhou, Viscosity approximation methods for continuous pseudocontractive mappings, Acta Math. Sinica, in press. · Zbl 1202.47070 [5] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. math. anal. appl., 279, 372-379, (2003) · Zbl 1035.47048 [6] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. math. anal. appl., 75, 287-292, (1980) · Zbl 0437.47047 [7] Shimizu, T.; Takahashi, W., Strong convergence theorems for asymptotically nonexpansive mappings, Nonlinear anal., 26, 265-272, (1996) · Zbl 0861.47030 [8] Shioji, N.; Takahashi, W., Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. amer. math. soc., 125, 3641-3645, (1997) · Zbl 0888.47034 [9] Shioji, N.; Takahashi, W., Strong convergence of averaged approximants for asymptotically nonexpansive mappings in Banach spaces, J. approx. theory, 97, 53-64, (1999) · Zbl 0932.47042 [10] Shioji, N.; Takahashi, W., Strong convergence theorems for continuous semigroups in Banach spaces, Math. japon., 1, 57-66, (1999) · Zbl 0940.47047 [11] Takahashi, W., Nonlinear functional analysis – fixed point theory and its applications, (2000), Yokohama Publishers Inc Yokohama, (in Japanese) · Zbl 0997.47002 [12] Takahashi, W.; Ueda, Y., On Reich’s strong convergence theorems for resolvents of accretive operators, J. math. anal. appl., 104, 546-553, (1984) · Zbl 0599.47084 [13] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math., 58, 486-491, (1992) · Zbl 0797.47036 [14] Xu, H.K., An iterative approach to quadratic optimization, J. optim. theory appl., 116, 659-678, (2003) · Zbl 1043.90063 [15] Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.