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Iterative solution of nonlinear equations of accretive and pseudocontractive types. (English) Zbl 1054.47056
The authors prove the following theorem without the assumption of boundedness of $\{x_n\}$. Theorem 1: Let $E$ be a real uniformly smooth Banach space. Let $A: D(A)= E\to 2^E$ be an accretive operator that satisfies the range condition and $A^{-1}(0)\ne\emptyset$. Suppose that $\{\lambda_n\}$ and $\{\theta_n\}$ are real sequences in $(0,1]$ satisfying the following conditions: (i) $\lim_{n\to\infty}\theta_n= 0$; (ii) $\sum^\infty_{n=1} \lambda_n\theta_n= \infty$, $\lim_{n\to\infty} {b(\lambda_n)\over \theta_n}= 0;$\par(iii) $\lim_{n\to\infty} {\theta_{n-1}/\theta_n- 1\over \lambda_n\theta_n}= 0$. Let $z\in E$ arbitrary and the sequence $\{x_n\}$ be generated from $x_0\in E$ by $x_{n+1}= x_n- \lambda_n(x_n- z))$, $u_n\in Ax_n$, for all positive integers $n$. Suppose that $\{u_n\}$ is bounded. Then there exists $d> 0$ such that, whenever $\lambda_n\le d$ and $b(\lambda_n)/\theta_n\le d^2$ for all $n\ge 0$, $\{x_n\}$ converges strongly to a zero of $A$. As a consequence of this result, they also prove the following theorem. Theorem 2: Let $K$ be a closed convex subset of a real uniformly smooth Banach space $E$. Let $T: K\to K$ be a bounded continuous pseudocontractive map with $F(T)\ne\emptyset$. Let $\{\lambda_n\}$ and $\{\theta_n\}$ be real squences satisfying\par\hskip17mm (i) $\lim_{n\to\infty}\theta_n= 0$;\par\hskip17mm (ii) $\sum^\infty_{n=1}\lambda_n\theta_n= \infty$, $\lambda_n(1+ \theta_n)\le 1$, $\lim_{n\to\infty} {b(\lambda_n)\over \theta_n}= 0$;\par\hskip17mm (iii) $\lim_{n\to\infty} {\theta_{n-1}/\theta_n- 1\over \lambda_n\theta_n}= 0$. Let $z\in K$ arbitrary and the sequence $\{x_n\}$ be generated from $x_0\in K$ by $x_{n+1}= x_n- \lambda_n((I- T)x_n+ \theta_n(x_n- z))$ for all positive integers $n$. Then there exists $d> 0$ such that, whenever $\lambda_n\le d$ and $b(\lambda_n)/\theta_n\le d^2$ for all $n\ge 0$, $\{x_n\}$ converges strongly to a fixed point of $T$. This theorem extends the class of Lipschitzian pseudocontractive maps to the class of bounded continuous pseudocontractive maps, which generalizes results of {\it C. E. Chidume} [Proc. Am. Math. Soc. 129, 2245--2251 (2001; Zbl 0979.47038)].

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H06Accretive operators, dissipative operators, etc. (nonlinear)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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Full Text: DOI
References:
[1] Jr., R. E. Bruck: A strongly convergent iterative method for the solution of $0\inUx $for a maximal monotone operator U in Hilbert space. J. math. Anal. appl. 48, 114-126 (1974)
[2] Browder, F. E.: Nonlinear monotone and accretive operators in Banach space. Proc. nat. Acad. sci. USA 61, 388-393 (1968) · Zbl 0167.15205
[3] Chidume, C. E.: Iterative approximation of fixed points of Lipschitz pseudocontractive maps. Proc. amer. Math. soc. 129, 2245-2251 (2001) · Zbl 0979.47038
[4] Chidume, C. E.; Moore, C.: Fixed point iteration for pseudocontractive maps. Proc. amer. Math. soc. 127, 1163-1170 (1999) · Zbl 0913.47052
[5] Chidume, C. E.; Mutangadura, S.: An example on the Mann iteration methods for Lipschitzian pseudocontractions. Proc. amer. Math. soc. 129, 2359-2363 (2001) · Zbl 0972.47062
[6] Cioranescu, I.: Geometry of Banach spaces, duality mapping and nonlinear problems. (1990) · Zbl 0712.47043
[7] Ishikawa, S.: Fixed points by a new iteration method. Proc. amer. Math. soc. 44, 147-150 (1974) · Zbl 0286.47036
[8] Mann, W. R.: Mean value methods in iteration. Proc. amer. Math. soc. 4, 506-510 (1953) · Zbl 0050.11603
[9] Jr., R. H. Martin: A global existence theorem for autonomous differential equations in Banach spaces. Proc. amer. Math. soc. 26, 307-314 (1970)
[10] Liu, Q.: On naimpally and singh’s open question. J. math. Anal. appl. 124, 157-164 (1987) · Zbl 0625.47044
[11] Liu, Q.: The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings. J. math. Anal. appl. 148, 55-62 (1990) · Zbl 0729.47052
[12] Reich, S.: Iterative methods for accretive sets. Nonlinear equations in abstract spaces, 317-326 (1978)
[13] Reich, S.: An iterative procedure for constructing zeros of accretive sets in Banach spaces. Nonlinear anal. 2, 85-92 (1978) · Zbl 0375.47032
[14] Reich, S.: Constructive techniques for accretive and monotone operators. Applied nonlinear analysis, 335-345 (1979)
[15] Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. math. Anal. appl. 75, 287-292 (1980) · Zbl 0437.47047
[16] Schu, J.: Approximating fixed points of Lipschitzian pseudocontractive mappings. Houston J. Math. 19, 107-115 (1993) · Zbl 0804.47057
[17] Xu, H. K.: Inequalities in Banach spaces with applications. Nonlinear anal. 16, 1127-1138 (1991) · Zbl 0757.46033
[18] Xu, Z. B.; Roach, G. F.: Characteristic inequalities for uniformly convex and uniformly smooth Banach spaces. J. math. Anal. appl. 157, 189-210 (1991) · Zbl 0757.46034
[19] Zeidler, E.: Nonlinear functional analysis and its applications, part II: Monotone operators. (1985) · Zbl 0583.47051