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Viscosity approximation method for accretive operator in Banach space. (English) Zbl 1196.47045
The authors study the strong convergence of a sequence $$\{ x_{n}\} _{n=1}^{\infty }$$ in a closed convex subset $$C$$ of a uniformly smooth Banach space $$X$$ to a fixed point $$F(A)$$ of an $$m$$-accretive operator $$A$$ with a zero. Their sequence is generated by the scheme
$x_{n+1} = \alpha _{n}f(x_{n}) + (1- \alpha _{n})J_{r_{n}}x_{n}, \quad n \geq 0,$
where $$\alpha _{n}, r_{n}$$ are sequences satisfying some conditions, $$J_{r}$$ denotes the resolvent $$(I + rA)^{-1}$$ for $$r > 0$$, and $$f: C \to C$$ is a fixed contractive map. Their results extend and improve those of H.-K. Xu [J. Math. Anal. Appl. 314, No. 2, 631–643 (2006; Zbl 1086.47060)].

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems 47H06 Nonlinear accretive operators, dissipative operators, etc.
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##### References:
 [1] Browder, F.E., Convergence of approximations to points of non-expansive maps in Banach spaces, Arch. ration. mech. anal., 24, 82-90, (1967) · Zbl 0148.13601 [2] Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060 [3] Xu, H.K., Strong convergence of an iterative method for nonexpansive and accretive operators, J. math. anal. appl., 314, 631-643, (2006) · Zbl 1086.47060 [4] Takahashi, W., Nonlinear functional analysis — fixed point theory and its applications, (2000), Yokohama Publishers inc Yokohama, (in Japanese) · Zbl 0997.47002 [5] Xu, H.K., Iterative algorithms for nonlinear operators, J. London math. soc., 66, 240-256, (2002) · Zbl 1013.47032 [6] Xu, H.K., An iterative approach to quadratic optimization, J. optim. theory appl., 116, 659-678, (2003) · Zbl 1043.90063 [7] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, (1976), Noordhoff
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