×

zbMATH — the first resource for mathematics

On the convergence of iterative processes for generalized strongly asymptotically \(\phi\)-pseudocontractive mappings in Banach spaces. (English) Zbl 1295.47078
Summary: We prove the equivalence and the strong convergence of iterative processes involving generalized strongly asymptotically \(\phi\)-pseudocontractive mappings in uniformly smooth Banach spaces.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] F. E. Browder, “Nonlinear mappings of nonexpansive and accretive type in Banach spaces,” Bulletin of the American Mathematical Society, vol. 73, pp. 875-882, 1967. · Zbl 0176.45302
[2] T. Kato, “Nonlinear semigroups and evolution equations,” Journal of the Mathematical Society of Japan, vol. 19, pp. 508-520, 1967. · Zbl 0163.38303
[3] C. Chidume, Geometric properties of Banach spaces and nonlinear iterations, vol. 1965 of Lecture Notes in Mathematics, Springer, London, UK, 2009. · Zbl 1167.47002
[4] K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171-174, 1972. · Zbl 0256.47045
[5] K. Deimling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, pp. 365-374, 1974. · Zbl 0288.47047
[6] M. O. Osilike, “Iterative solution of nonlinear equations of the \varphi -strongly accretive type,” Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 259-271, 1996. · Zbl 0860.65039
[7] C. H. Xiang, “Fixed point theorem for generalized \varphi -pseudocontractive mappings,” Nonlinear Analysis, vol. 70, no. 6, pp. 2277-2279, 2009. · Zbl 1207.47059
[8] J. Schu, “Iterative construction of fixed points of asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 158, no. 2, pp. 407-413, 1991. · Zbl 0734.47036
[9] E. U. Ofoedu, “Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 722-728, 2006. · Zbl 1109.47061
[10] H. Zhou, “Demiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces,” Nonlinear Analysis, vol. 70, no. 9, pp. 3140-3145, 2009. · Zbl 1207.47052
[11] C. E. Chidume and M. O. Osilike, “Equilibrium points for a system involving m-accretive operators,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 44, no. 1, pp. 187-199, 2001. · Zbl 1001.47032
[12] C. E. Chidume and H. Zegeye, “Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps,” Proceedings of the American Mathematical Society, vol. 132, no. 3, pp. 831-840, 2004. · Zbl 1051.47041
[13] S. S. Chang, “Some results for asymptotically pseudo-contractive mappings and asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 129, no. 3, pp. 845-853, 2001. · Zbl 0968.47017
[14] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985. · Zbl 0559.47040
[15] S. S. Chang, K. K. Tan, H. W. J. Lee, and C. K. Chan, “On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 313, no. 1, pp. 273-283, 2006. · Zbl 1086.47044
[16] F. Gu, “The new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 766-776, 2007. · Zbl 1153.47306
[17] Z. Huang and F. Bu, “The equivalence between the convergence of Ishikawa and Mann iterations with errors for strongly successively pseudocontractive mappings without Lipschitzian assumption,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 586-594, 2007. · Zbl 1101.47038
[18] L. S. Liu, “Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114-125, 1995. · Zbl 0872.47031
[19] B. E. Rhoades and S. M. Soltuz, “The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps,” Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 266-278, 2004. · Zbl 1053.47055
[20] B. E. Rhoades and S. M. Soltuz, “The equivalence between Mann-Ishikawa iterations and multistep iteration,” Nonlinear Analysis, vol. 58, no. 1-2, pp. 219-228, 2004. · Zbl 1064.47070
[21] Z. Huang, “Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively \varphi -pseudocontractive mappings without Lipschitzian assumptions,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 935-947, 2007. · Zbl 1153.47307
[22] Y. Xu, “Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations,” Journal of Mathematical Analysis and Applications, vol. 224, no. 1, pp. 91-101, 1998. · Zbl 0936.47041
[23] P.-E. MaingĂ©, “Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 469-479, 2007. · Zbl 1111.47058
[24] C. E. Chidume and C. O. Chidume, “Convergence theorem for zeros of generalized lipschitz generalized phi-quasi-accretive operators,” Proceedings of the American Mathematical Society, vol. 134, no. 1, pp. 243-251, 2006. · Zbl 1072.47062
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.