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On Chidume’s open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces. (English) Zbl 0909.47049

If \(K\) is a convex subset of a Banach space \(X\), \(T:K\to 2^K\) a multivalued mapping then the Ishikawa iteration process is defined by \[ x_{n+1} \in (1-\alpha_n)x_n + \alpha_n Ty_n, \;y_n \in (1-\beta_n)x_n + \beta_n Tx_n, \;n \geq 1, \] where \(x_1 \in K\) is arbitrary, \(\alpha_n, \;\beta_n\) are given numbers, \(0\leq \alpha_n\), \(\beta_n < 1\), \(\sum_{n=1}^{\infty}\alpha_n = \infty\), \(\alpha_n \to 0\), \(\beta_n \to 0\). The author proves that if \(T\) is a multivalued strongly pseudocontractive mapping with nonempty closed values and either \(T\) is uniformly continuous or the dual to \(X\) is uniformly convex then this iterative process converges to the unique fixed point of \(T\) in \(K\). Further, the Ishikawa process for the operators of the type \(Sx = f-Tx+x\) is studied. It is proved that if \(f\in X\) is arbitrary, \(T:X \to 2^X\) is a multivalued strongly accretive mapping with nonempty closed values, the range of \(I-T\) is bounded and either \(T\) is uniformly continuous or the dual to \(X\) is uniformly convex then the Ishikawa process for \(S\) converges to the unique solution of the inclusion \(f\in Tx\). These results answer some open questions formulated by C. E. Chidume.
Reviewer: M.Kučera (Praha)

MSC:

47J25 Iterative procedures involving nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H04 Set-valued operators
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