Iterative algorithms with perturbations for Lipschitz pseudocontractive mappings in Banach spaces

Yao, Zhangsong¹ ; Zhu, Li-Jun² ; Kang, Shin Min³ ; Liou, Yeong-Cheng⁴

¹ School of Mathematics & Information Technology, Nanjing Xiaozhuang University, Nanjing 211171, China
² School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, China
³ Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Korea
⁴ Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan;Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Journal of nonlinear sciences and its applications, Tome 8 (2015) no. 6, p. 935-943.

Résumé

In this paper, we present an iterative algorithm with perturbations for Lipschitz pseudocontractive mappings in Banach spaces. Consequently, we give the convergence analysis of the suggested algorithm. Our result improves the corresponding results in the literature.

Détail

DOI : 10.22436/jnsa.008.06.04

Classification : 47H05, 47H10, 47H17
Keywords: Strong convergence, pseudocontractive mapping, fixed point, Banach space.

Bibliographie

[1] Bruck, R. E. A strongly convergent iterative method for the solution of \(0 \in Ux\) for a maximal monotone operator U in Hilbert space, J. Math. Anal. Appl., Volume 48 (1974), pp. 114-126

[2] Chang, S. S. On Chidume's open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces, J. Math. Anal. Appl., Volume 216 (1997), pp. 94-111

[3] Chidume, C. E. Approximation of fixed points of strongly pseudocontractive mappings, Proc. Amer. Math. Soc., Volume 120 (1994), pp. 545-551

[4] Chidume, C. E. Global iteration schemes for strongly pseudocontractive maps, Proc. Amer. Math. Soc., Volume 126 (1998), pp. 2641-2649

[5] Chidume, C. E.; Moore, C. Fixed point iteration for pseudocontractive maps, Proc. Amer. Math. Soc., Volume 127 (1999), pp. 1163-1170

[6] Chidume, C. E.; Mutangadura, S. A. An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc., Volume 129 (2001), pp. 2359-2363

[7] Chidume, C. E.; Udomene, A. Strong convergence theorems for uniformly continuous pseudocontractive maps, J. Math. Anal. Appl., Volume 323 (2006), pp. 88-99

[8] Chidume, C. E.; Zegeye, H. Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps, Proc. Amer. Math. Soc., Volume 132 (2004), pp. 831-840

[9] Deimling, K. Zeros of accretive operators, Manuscripta Math., Volume 13 (1974), pp. 365-374

[10] Deng, L. Iteration processes for nonlinear Lipschitzian strongly accretive mappings in Lp spaces, J. Math. Anal. Appl., Volume 188 (1994), pp. 128-140

[11] Deng, L.; Ding, X. P. Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth Banach spaces, Nonlinear Anal., Volume 24 (1995), pp. 981-987

[12] Ding, X. P. Iterative process with errors to locally strictly pseudocontractive maps in Banach spaces, Comput. Math. Appl., Volume 32 (1996), pp. 91-97

[13] Goebel, K.; Wirk, W. A. Topics in metric fixed point theory, Cambridge University Press, Cambridge, 1990

[14] Huang, N. J.; Gao, C. J.; Huang, X. P. New iteration procedures with errors for multivalued \(\phi\)-strongly pseudocontractive and \(\phi\)-strongly accretive mappings, Comput. Math. Appl., Volume 43 (2002), pp. 1381-1390

[15] Huang, N. J.; Cho, Y. J.; Lee, B. S.; Jung, J. S. Convergence of iterative processes with errors for set-valued pseudocontractive and accretive type mappings in Banach spaces, Comput. Math. Appl., Volume 40 (2000), pp. 1127-1139

[16] S. Ishikawa Fixed point by a new iteration method , Proc. Amer. Math. Soc., Volume 4 (1974), pp. 147-150

[17] T. Kato Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, Volume 19 (1967), pp. 508-520

[18] Mann, W. R. Mean value methods in iteration, Proc. Amer. Math. Soc., Volume 4 (1953), pp. 506-510

[19] Martin, R. H. Differential equations on closed subsets of Banach space, Trans. Amer. Math. Soc., Volume 179 (1973), pp. 399-414

[20] Morales, C. H.; Jung, J. S. Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. Amer. Math. Soc., Volume 128 (2000), pp. 3411-3419

[21] Reich, S. An iterative procedure for constructing zero of accretive sets in Banach spaces, Nonlinear Anal., Volume 2 (1978), pp. 85-92

[22] Reich, S. Constructive techniques for accretive and monotone operators, Applied Nonlinear Analysis, Academic Press, New York, 1979

[23] Reich, S. Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., Volume 75 (1980), pp. 287-292

[24] Schu, J. Approximating fixed points of Lipschitzian pseudocontractive mappings, Houston J. Math., Volume 19 (1993), pp. 107-115

[25] Xu, H. K. Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., Volume 298 (2004), pp. 279-291

[26] Xu, Y. Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl., Volume 224 (1998), pp. 91-101

[27] Yao, Y.; Liou, Y. C.; Chen, R. Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces, Nonlinear Anal., Volume 67 (2007), pp. 3311-3317

[28] Yao, Y.; Liou, Y. C.; Kang, S. M. Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings, Appl. Math. Comput., Volume 208 (2009), pp. 211-218

[29] Zegeye, H.; Prempeh, E. Strong convergence of approximants to fixed points of Lipschitzian pseudocontractive maps, Comput. Math. Appl., Volume 44 (2002), pp. 339-346

[30] Zegeye, H.; Shahzad, N.; Mekonen, T. Viscosity approximation methods for pseudocontractive mappings in Banach spaces, Appl. Math. Comput., Volume 185 (2007), pp. 538-546

[31] Zeng, L. C. Iterative approximation of solutions to nonlinear equations of strongly accretive operators in Banach spaces, Nonlinear Anal., Volume 31 (1998), pp. 589-598

[32] Zhou, H. Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces, Nonlinear Anal., Volume 68 (2008), pp. 2977-2983