×

Convergence to common fixed point for generalized asymptotically nonexpansive semigroup in Banach spaces. (English) Zbl 1177.47076

Let \(K\) be a nonempty closed convex subset of a reflexive and strictly convex Banach space \(E\) with a uniformly Gâteaux differentiable norm, \({\mathcal F}=\{T(h) :h\geq 0\}\) be a generalized asymptotically nonexpansive self-mapping semigroup of \(K\), and \(f:K\to K\) be a fixed contractive mapping with contractive coefficient \(\beta\in(0,1)\). The authors prove that the following implicit and modified implicit viscosity iterative schemes \(\{x_n\}\) defined by \(x_n=\alpha_nf(x_n)+(1-\alpha_n)T(t_n)x_n\) and \(x_n=\alpha_ny_n+(1-\alpha_n)T(t_n)x_n\), \(y_n=\beta_n f(x_{n-1})+(1-\beta_n)x_{n-1}\), strongly converge to \(p\in F\) as \(n\to\infty\), and \(p\) is the unique solution to the following variational inequality: \(\langle f(p)-p,j(y-p)\rangle\leq 0\) for all \(y\in F\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H20 Semigroups of nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] F. E. Browder, “Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, no. 1, pp. 82-90, 1967. · Zbl 0148.13601 · doi:10.1007/BF00251595
[2] H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279-291, 2004. · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059
[3] Y. Song and S. Xu, “Strong convergence theorems for nonexpansive semigroup in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 152-161, 2008. · Zbl 1138.47040 · doi:10.1016/j.jmaa.2007.05.021
[4] Y. Song and R. Chen, “Convergence theorems of iterative algorithms for continuous pseudocontractive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 2, pp. 486-497, 2007. · Zbl 1126.47054 · doi:10.1016/j.na.2006.06.009
[5] C. E. Chidume and E. U. Ofoedu, “Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 128-141, 2007. · Zbl 1127.47051 · doi:10.1016/j.jmaa.2006.09.023
[6] S. Reich, “On the asymptotic behavior of nonlinear semigroups and the range of accretive operators,” Journal of Mathematical Analysis and Applications, vol. 79, no. 1, pp. 113-126, 1981. · Zbl 0457.47053 · doi:10.1016/0022-247X(81)90013-5
[7] W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Applications, Yokohama, Yokohama, Japan, 2000. · Zbl 0997.47002
[8] W. Takahashi and Y. Ueda, “On Reich’s strong convergence theorems for resolvents of accretive operators,” Journal of Mathematical Analysis and Applications, vol. 104, no. 2, pp. 546-553, 1984. · Zbl 0599.47084 · doi:10.1016/0022-247X(84)90019-2
[9] K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 8, pp. 2350-2360, 2007. · Zbl 1130.47045 · doi:10.1016/j.na.2006.08.032
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.