Chang, Shihsen On Chidume’s open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces. (English) Zbl 0909.47049 J. Math. Anal. Appl. 216, No. 1, 94-111 (1997). 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) Cited in 1 ReviewCited in 96 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H06 Nonlinear accretive operators, dissipative operators, etc. 47H04 Set-valued operators Keywords:Ishikawa iteration process; multivalued strongly accretive mappings; multivalued strongly pseudocontractive mapping with nonempty closed values PDFBibTeX XMLCite \textit{S. Chang}, J. Math. Anal. Appl. 216, No. 1, 94--111 (1997; Zbl 0909.47049) Full Text: DOI References: [1] Browder, F. E., Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. Amer. Math. Soc., 73, 875-882 (1967) · Zbl 0176.45302 [2] Chang, S. S., Variational Inequality and Complementarity Problem Theory with Applications (1991), Shanghai Sci. Technol: Shanghai Sci. Technol Shanghai [3] Chidume, C. E., An iterative process for nonlinear Lipschitzian strongly accretive mappings in\(L_p\)spaces, J. Math. Anal. Appl., 151, 453-461 (1990) · Zbl 0724.65058 [4] Chidume, C. E., Iterative approximation of a fixed point of Lipschitzian strictly pseudo-contractive mappings, Proc. Amer. Math. Soc., 99, 283-288 (1987) · Zbl 0646.47037 [5] Chidume, C. E., Approximation of fixed points of strongly pseudocontractive mappings, Proc. Amer. Math. Soc., 120, 545-551 (1994) · Zbl 0802.47058 [6] Chidume, C. E., Iterative solution of nonlinear equations with strongly accretive operators, J. Math. Anal. Appl., 192, 501-518 (1995) · Zbl 0868.47040 [7] Deimling, K., Zeros of accretive operators, Manuscripta Math., 13, 283-288 (1974) [8] Deng, L., On Chidume’s open questions, J. Math. Anal. Appl., 174, 441-449 (1993) · Zbl 0784.47051 [9] Deng, L., An iterative process for nonlinear Lipschitzian and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces, Acta. Appl. Math., 32, 183-196 (1993) · Zbl 0801.47040 [10] Ishikawa, S., Fixed points and iteration of a nonexpansive mappings in a Banach space, Proc. Amer. Math. Soc., 73, 65-71 (1976) · Zbl 0352.47024 [11] Kato, T., Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 18/19, 508-520 (1967) · Zbl 0163.38303 [12] Kirk, W. A., A fixed point theorem for local pseudo-contraction in uniformly convex spaces, Manuscripta Math., 30, 89-102 (1979) · Zbl 0422.47032 [13] Morales, C., Surjectivity theorems for multi-valued mappings of accretive type, Comment. Math. Univ. Carolin., 26 (1985) [14] Mukerjee, R. N., Construction of fixed points of strictly pseudocontractive mappings in generalized Hilbert spaces and related applications, Indian J. Pure Appl. Math., 15, 276-284 (1966) [15] Nevalinna, O.; Reich, S., Strong convergence of contraction semi-groups and of iterative methods for accretive operators in Banach spaces, Israel J. Math., 32, 44-58 (1979) · Zbl 0427.47049 [16] Reich, S., Constructing zeros of accretive operators, II, Appl. Anal., 9, 159-163 (1979) · Zbl 0424.47034 [17] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 85, 287-292 (1980) · Zbl 0437.47047 [18] Reich, S., An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal., 2, 85-92 (1978) · Zbl 0375.47032 [19] Tan, K. K.; Xu, H. K., Iterative solutions to nonlinear equations of strongly accretive operators in Banach space, J. Math. Anal. Appl., 178, 9-21 (1993) · Zbl 0834.47048 [20] You, Z. Y.; Gong, H. Y.; Xu, Z. B., Nonlinear Analysis (1991), Xi’an Jiaotong Univ. Press This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.