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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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[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
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[7] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, (1976), Noordhoff
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