STABLE AND ALMOST STABLE ITERATION SCHEMES FOR STRICTLY HEMI-CONTRACTIVE OPERATORS IN ARBITRARY BANACH SPACES

Zeqing Liu, Jeong Sheok Ume · Numerical Functional Analysis and Optimization · 2002

Let K be a nonempty closed convex subset of an arbitrary Banach space X and be a Lipschitz strictly hemi-contractive operator. We establish that a certain Ishikawa iteration scheme with errors both converges strongly to a unique fixed point of T and is almost T-stable on K or T-stable on K. Our results extend, improve and unify the corresponding results in Refs. 1-14 Chang, S.S. 1997. Some Problems and Results in the Study of Nonlinear Analysis. Nonlinear Anal. T. M. A., 30(7): 4197–4208. Chang, S.S., Cho, Y.J., Lee, B.S., Jung, J.S. and Kang, S.M. 1998. Iterative Approximations of Fixed Points and Solutions for Strongly Accretive and Strongly Pseudo-Contractive Mappings in Banach Spaces. J. Math. Anal. Appl., 224: 149–165. Chidume, C.E. 1987. Iterative Approximation of Fixed Points of Lipschitzian Strictly Pseudo-Contractive Mappings. Proc. Amer. Math. Soc., 99(2): 283–288. Chidume, C.E. 1990. An Iterative Process for Nonlinear Lipschitzian Strongly Accretive Mappings in Lp Spaces. J. Math. Anal. Appl., 151: 453–461. Chidume, C.E. 1994. Approximation of Fixed Points of Strongly Pseudo-Contractive Mappings. Proc. Amer. Math. Soc., 120(2): 545–551. Chidume, C.E. 1995. Iterative Solutions of Nonlinear Equations with Strongly Accretive Operators. J. Math. Anal. Appl., 192: 502–518. Chidume, C.E. 1996. Iterative Solutions of Nonlinear Equations in Smooth Banach Spaces. Nonlinear Anal. T. M. A., 26(11): 1823–1834. Chidume, C.E. and Osilike, M.O. 1994. Fixed Point Iterations for Strictly Hemi-Contractive Maps in Uniformly Smooth Banach Spaces. Numer. Funct. Anal. Optimiz., 15: 779–790. Chidume, C.E. and Osilike, M.O. 1995. Ishikawa Iteration Process for Nonlinear Lipschitz Strongly Accretive Mappings. J. Math. Anal. Appl., 192: 727–741. Chidume, C.E. and Osilike, M.O. 1998. Nonlinear Accretive and Pseudo-Contractive Operator Equations in Banach Spaces. Nonlinear Anal. T. M. A., 31: 779–789. Deng, L. 1993. On Chidume's Open Questions. J. Math. Anal. Appl., 174: 441–449. Deng, L. 1993. An Iterative Process for Nonlinear Lipschitzian and Strongly Accretive Mappings in Uniformly Convex and Uniformly Smooth Banach Spaces. Acta Appl. Math. Mech., 32: 183–196. Deng, L. 1994. Iteration Processes for Nonlinear Lipschitzian Strongly Accretive Mappings in Lp Spaces. J. Math. Anal. Appl., 188: 128–140. Deng, L. and Ding, X.P. 1995. Iterative Approximation of Lipschitz Strictly Pseudocontractive Mappings in Uniformly Smooth Banach Spaces. Nonlinear Anal. T. M. A., 24: 981–987. , [21] Liu, L.W. 1997. Approximation of Fixed Points of a Strictly Pseudocontractive Mapping. Proc. Amer. Math. Soc., 125(5): 1363–1366. [Crossref], [Web of Science ®] , [Google Scholar], 23-24 Osilike, M.O. 1996. Stable Iteration Procedures for Strong Pseudocontractions and Nonlinear Operator Equations of the Accretive Type. J. Math. Anal. Appl., 204: 677–692. Osilike, M.O. 1997. Stable Iteration Procedures for Nonlinear Pseudocontractive and Accretive Operator in Arbitrary Banach Spaces. Indian J. Pure Appl. Math., 28: 1017–1029. , [27] Tan, K.K. and Xu, H.K. 1993. Iterative Solutions to Nonlinear Equations of Strongly Accretive Operators in Banach Spaces. J. Math. Anal. Appl., 178: 9–21. [Crossref], [Web of Science ®] , [Google Scholar], [30] Zeng, L.C. 1998. Iterative Approximation of Solutions to Nonlinear Equations of Strongly Accretive Operators in Banach Spaces. Nonlinear Anal. T. M. A., 31: 589–598. [Crossref], [Web of Science ®] , [Google Scholar] and others.

Read the paper · More papers on PaperTik