# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 47J25 Iterative procedures (nonlinear operator equations) 47H06 Accretive operators, dissipative operators, etc. (nonlinear) 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text:
##### 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