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Nonlinear accretive and pseudo-contractive operator equations in Banach spaces. (English) Zbl 0901.47037

Gegenstand der vorliegenden Arbeit ist das folgende Theorem (Theorem 1):
“Suppose \(E\) is an arbitrary real Banach space and \(K\) is a closed convex subset \(E\). Let \(T: K\to K\) be a Lipschitz strong pseudo-contraction mapping. Let \((\alpha_n)\) and \((\beta_n)\) be real sequences satisfying the following conditions:
(i) \(0\leq\beta_n\), \(\alpha_n< 1\) \((n= 0,1,2,\dots)\),
(ii) \(\displaystyle{\lim_{n\to\infty}} \alpha_n= 0\); \(\displaystyle{\lim_{n\to\infty}} \beta_n =0\),
(iii) \(\displaystyle{\sum^\infty_{n= 0}} \alpha_n= +\infty\).
Then the sequence \((x_n)\) generated from an arbitrary \(x_0\in K\) by (the Ishikawa iteration process) \[ y_n= (1-\beta_n)x_n+ \beta_n Tx_n,\quad x_{n+ 1}= (1-\alpha_n)x_n+ \alpha_n Ty_n \] \((n= 0,1,2,\dots)\) converges strongly to the (unique) fixed point of \(T\).”
Dieses Theorem verallgemeinert frühere Ergebnisse anderer Autoren, speziell ein Ergebnis von L. Liu [Proc. Am. Math. Soc. 125, No. 5, 1363-1366 (1997; Zbl 0870.47039)]. Dabei heißt eine Abbildung \(T\) aus \(E\) nach \(E\) stark pseudo-kontraktiv (strong pseudo-contraction), wenn es für alle \(x\), \(y\) aus dem Definitionsbereich von \(T\) ein Element \(j(x- y)\in J(x- y)\) gibt (\(J(\cdot)\) bezeichne die normierte Dualitätsabbildung von \(E\) nach \(2^{E^*}\)) sowie ein \(t> 1\) existiert, so daß die Ungleichung \(\langle Tx- Ty,j(x- y)\rangle\leq{1\over t}\| x-y\|^2\) gilt (\(\langle.,.\rangle\) ist die zugrundeliegende Dualität). Außer Folgerungen für spezielle Abbildungsklassen bzw. Iterationsverfahren werden auch Abschätzungen für die Konvergenzgeschwindigkeit der Iterationsfolge angegeben. Der wesentliche Fortschritt der vorliegenden Arbeit gegenüber früheren einschlägigen Veröffentlichungen [L. Deng and X. P. Ding, Nonlinear Analysis, Theory Methods Appl. 24, No. 7, 981-987 (1995; Zbl 0827.47041) and the authors, “Fixpoint iterations for strictly hemi-contractive maps in uniformly smooth Banach spaces”, Numer. Funct. Anal. Optimization 15, No. 7-8, 779-790 (1994; Zbl 0810.47057)] ist darin zu sehen, daß (z.B. in Theorem 1) keine Voraussetzungen über die geometrischen Eigenschaften des zugrundeliegenden Banachraumes getroffen werden müssen.

MSC:

47H06 Nonlinear accretive operators, dissipative operators, etc.
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
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