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Viscosity approximation methods for pseudocontractive mappings in Banach spaces. (English) Zbl 1178.47049
Let $K$ be a closed convex subset of a Banach space $E$ and let $T : K \to E$ be a continuous weakly inward pseudocontractive mapping. Then for $t\in(0,1)$, there exists a sequence $\{y_t\} \subset K$ satisfying $y_t = (1 - t)f(y_t) + tT(y_t)$, where $f \in \Pi_K := \{f : K\to K,\text{ a contraction with a suitable contractive constant}\}$. Suppose further that $F(T) \ne \emptyset$ and $E$ is reflexive and strictly convex which has uniformly Gâteaux differentiable norm. Then it is proved that $\{y_t\}$ converges strongly to a fixed point of $T$ which is also a solution of certain variational inequality. Moreover, an explicit iteration process which converges strongly to a fixed point of $T$ and hence to a solution of certain variational inequality is constructed, provided that $T$ is Lipschitzian.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 65J15 Equations with nonlinear operators (numerical methods)
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##### References:
 [1] Barbu, V.; Precupanu, Th.: Convexity and optimization in Banach spaces. (1978) · Zbl 0379.49010 [2] Browder, F. E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. ration. Mech. anal. 24, 82-90 (1967) · Zbl 0148.13601 [3] Chidume, C. E.; Moore, C.: The solution by iteration of nonlinear equations in uniformly smooth Banach spaces. J. math. Anal. appl. 215, 132-146 (1997) · Zbl 0906.47050 [4] Chidume, C. E.; Mutangadura, S.: An example on the Mann iteration methods for Lipschitzian pseudocontractions. Proc. am. Math. soc. 129, 2359-2363 (2001) · Zbl 0972.47062 [5] Chidume, C. E.; Zegeye, H.: Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps. Proc. am. Math. soc. 132, 831-840 (2004) · Zbl 1051.47041 [6] Chidume, C. E.; Zegeye, H.; Aneke, S. J.: Approximation of fixed points of weakly contractive non-self maps in Banach spaces. J. math. Anal. appl. 270, 189-199 (2002) · Zbl 1005.47053 [7] Cioranescu, I.: Geometry of Banach spaces, duality mapping and nonlinear problems. (1990) · Zbl 0712.47043 [8] Deimling, K.: Zeros of accretive operators. Manuscr. math. 13, 365-374 (1974) · Zbl 0288.47047 [9] Ishikawa, S.: Fixed points by a new iteration method. Proc. am. Math. soc. 44, 147-150 (1974) · Zbl 0286.47036 [10] Ishikawa, S.: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. am. Math. soc. 59, 65-71 (1976) · Zbl 0352.47024 [11] Kato, T.: Nonlinear semi-groups and evolution equations. J. math. Soc. jpn. 19, 508-520 (1967) · Zbl 0163.38303 [12] Lim, T. C.; Xu, H. K.: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear anal. 22, 1345-1355 (1994) · Zbl 0812.47058 [13] Martin, R. H.: Differential equations on closed subsets of a Banach space. Trans. am. Math. soc. 179, 399-414 (1973) · Zbl 0293.34092 [14] Megginson, R. E.: An introduction to Banach space theory. (1998) · Zbl 0910.46008 [15] Morales, C. H.: Strong convergence theorems for pseudocontractive mappings in Banach spaces. Houston J. Math. 16, 549-557 (1990) · Zbl 0728.47037 [16] Morales, C. H.; Jung, J. S.: Convergence of paths for pseudo-contractive mappings in Banach spaces. Proc. am. Math. soc. 128, 3411-3419 (2000) · Zbl 0970.47039 [17] Moudafi, A.: Viscosity approximation methods for fixed point problems. J. math. Anal. appl. 241, 46-55 (2000) · Zbl 0957.47039 [18] Osilike, M. O.: Iterative solution of nonlinear equations of the $\psi$-strongly accretive type. J. math. Anal. appl. 200, 259-271 (1996) · Zbl 0860.65039 [19] Petryshyn, W. V.: Construction of fixed points of demicompact mappings in Hilbert spaces. J. math. Anal. appl. 14, 276-284 (1966) · Zbl 0138.39802 [20] Qihou, L.: The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings. J. math. Anal. appl. 148, 55-62 (1990) · Zbl 0729.47052 [21] Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. math. Anal. appl. 75, 287-292 (1980) · Zbl 0437.47047 [22] Y. Song, R. Chen, Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings, Appl. Math. Comput., in press, doi:10.1016/j.amc.2005.12.013. · Zbl 1139.47050 [23] 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 [24] Takahashi, W.: Nonlinear functional analysis. (2000) · Zbl 0997.47002 [25] Xu, H. K.: Viscosity approximation methods for nonexpansive mappings. J. math. Anal. appl. 298, 279-281 (2004) · Zbl 1061.47060 [26] Zeidler, E.: Nonlinear functional analysis and its applications, part II: Monotone operators. (1985) · Zbl 0583.47051 [27] Zhang, S.: On the convergence problems of Ishikawa and Mann iteration process with errors for $\psi$-pseudo contractive type mappings. Appl. math. Mech. 21, 1-10 (2000)