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:

 [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
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