Strong Convergence of a Multi-Step Implicit Iterative Scheme With Errors for Common Fixed Points of Uniformly L-Lipschitzian Total Asymptotically Strict Pseudocontractive Mappings

Yıl 2020, Cilt: 3 Sayı: 3, 100 - 116, 30.09.2020

Austine Ofem

Öz

In this paper, we introduce a three-step implicit iteration process with errors and prove a strong convergence theorem of the new iterative scheme for a finite family of uniformly L-Lipschitzian total asymptotically strict pseudo contractive mappings in Banach spaces. The results in the paper extend, generalize, and unify well-known results in the existing literature.  In this paper, we compare a three-step implicit iteration process with the existing results in the literature.

 

Anahtar Kelimeler

Fixed point, Banach space, Lipschitzian,, total asymptotically strict pseudocontractive mapping, implicit iteration scheme, strong convergence.

Kaynakça

• [1] A. Bnouhachem, M. A. Noor and T. M. Rassias, Three-steps iterative algorithms for mixed variational inequalities, Appl. Math. Comput., 183 (2006), 436-446.
• [2] S.S.Chang, Some results for asymptotically pseudocontractive mappings and asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 129 (2001), 845-853.
• [3] S. S. Chang, L. Wang, Y. K. Tang, and L. Yang, The split common fixed point problem for total asymptotically strictly pseudocontractive mappings, Journal of Applied Mathematics, 2012 (2012) , Article ID 385638, 13 pages .
• [4] C. E. Chidume and C .O. Chidume, Convergence theorems for fixed points of uniformly continuous generalized Φ-hemi- contractive mappings, J. Math. Anal. Appl., 303 (2005), 545-554.
• [5] C. E. Chidume and C. O. Chidume, Convergence theorem for zeros of generalized Lipschitz generalized phi-quasi-accretive operators, Proc. Amer. Math. Soc., 134 (2006), 243-251.
• [6] R. C. Chen, Y. S. Song, H. Zhou, Convergence theorems for implicit iteration process for a finite family continuous pseudo- contractive mappings, J. Math. Anal. Appl., 314 (2006), 701-706.
• [7] F. Cianciaruso, G. Marino and X. Wang, Weak and strong convergence of the Ishikawa iterative process for a finite family of asymptotically nonexpansive mappings, Applied Mathematics and Computation, 216 (2010), 3558-3567.
• [8] Y. J. Cho, H. Y. Zhou and G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl., 47 (2004), 707-717.
• 9] R. Glowinski and P. Le-Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM, Philadelphia, 1989.
• [10] S. Haubruge, V. H. Nguyen and J. J. Strodiot, Convergence analysis and applications of the Glowinski-Le-Tallec splitting method for finding a zero of the sum of two maximal monotone oper- ators, J. Optim. Theory Appl., 97 (1998), 645-673.
• [11] F.Gu, Convergence theorems for ϕ-pseudocontractive type mappings in normed linear spaces, Northeast Math. J., 17(3) (2001), 340-346.
• [12] 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, 2(329) (2007), 766-776.
• [13] F. Gu, Implicit and explicit iterative process with errors for a common fixed points of a finite family of strictly pseudocon- tractive mappings, An. St. Univ. Ovidius Constanta, 18(1) (2010), 139-154.
• [14] T. L. Hicks and J. R. Kubicek, On the mann iteration process in hilbert spaces, Journal of Mathematical Analysis and Applications, 59 (1977), 498-504.
• [15] D. I. Igbokwe, Approximation of fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces, Journal of Inequality in Pure and Applied Mathematics, 3(1) (2002), 1-11.
• [16] D. Igbokwe and O. Ini, A modified averaging composite Implicit Iteration process for common fixed points of a finite family of k− strictly asymptotically pseudocontractive mappings, Advances in Pure Mathematics, 1 (2011), 204-209.
• [17] S. Ishikawa, Fixed points by a new iteration method, Proceeding of the America Mathematical society, 4(1974) 157-150.
• [18] G. A. Okeke and J. O. Olareju, Modifed Noor iterations with errors for nonlinear equations in Banach spaces, J. Nonlinear Sci. Appl., 7 (2014), 180- 187.
• [19] U. S. Jim, Z. Ongodiebi and F. A. Efiong, A new modified averaging implicit iteration process with errors for common fixed points of a finite family of asymptotically ϕ-demicontractive maps in arbitrary real Banach spaces, International Journal of Pure and Applied Mathematics, 78(3)(2012), 309-321.
• [20] W. R. Mann, Mean value methods in iteration, Proceedings of American Mathematical Science, 4 (2003), 506-510.
• [21] M. A. Noor, T. M. Kassias and Z. Huang, Three-step iterations for nonlinear accretive operator equations, J. Math. Anal. Appl., 274 (2001), 59-68.
• [22] M. O. Osilike, Implicit iteration process for common fixed point of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl., 294 (2004), 73-81.
• [23] M. O. Osilike, Iterative Approximation of fixed points asymptotically demicontractive mappings, Indian J. of Pure Appl. Maths., 29(12)(1998), 1291-1300.
• [24] M. O. Osilike and B. G. Akuchu, Common fixed points of finite family of asymptotically pseudocontractive mappings. Fixed Point Theory and Application, 2004 2004, 81-88.
• [25] M. O. Osilike, S. C. Aniagbosor and B. G. Akuchu, Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces, Pan American Mathematical Journal, 12(2) (2002), 77-88.
• [26] M. O. Osilike and F. U. Isiogugu, Fixed points of asymptotically ϕ-demicontractive mappings in arbitrary Banach spaces, Pan-American Mathematical Journal, 15 (3)(2005), 59-69.
• [27] M.O. Osilike, A. Udomene, D.I. Igbokwe, B.G. Akuchu, Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps, J. Math. Anal. Appl., 326 (2007), 1334-1345.
• [28] L. Qihou, Convergence Theorems of the Sequence of Iterates for Asymptotically Demicontarcive and Hemicontractive Mappings, Nonlinear Analysis: Theory, Methods and Applications, 26 26, (1996), 1835-1842.
• [29] G. S. Saluja, Convergence of the explicit iteration method for strictly asymptotically pseudocontractive mappings in the intermediate sense, Novi Sad J. Math., 44 (1)(2014), 75-90.
• [30] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austr. Math. Soc. , 43(1991), 153-159.
• [31] H. F. Senter and W. G. Dotson: Approximating ?xed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44(1974), 375-380.
• [32] J. Schu, Iterative construction of fixed point of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158(1991), 407-413.
• [33] N. Shahzad and A. Udomene: Approximating common fixed points of two asymptotically quasinonexpansive mappings in Banach spaces, Fixed Point Theory Appl., (2006), article ID 18909, 10 pages.
• [34] Y. Su and S. Li, Composite implicit process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl., 320 (2006), 882-891.
• [35] Z. H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 286 (2003), 351-358.
• [36] S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 311 (2005), 506-517.
• [37] B. S. Thakur, Weak and strong convergence of composite implicit iteration process, Appl. Math. Comput., 190 (2007), 965-973.
• [38] Y. Wang and C. Wang, convergence of a new modified Ishikawa type iteration for common Fixed points of total asymp- totically strict pseudocontractive semigroups, Abstract and Applied Analysis, 2013, Article ID 319241, 7 pages.
• [39] X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proceedings of the American Mathematical Society, 133 (2)(1991), 727-731.
• [40] H. K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mapping, Num. Fun. Anal. Optim., 22(2001),767-773.
• [41] Y. Yao, Convergence of three-step iterations for asymptotically nonexpansive mappings, Appl. Math. Comput., 187 (2007), 883-892.
• [42] L. P. Yang, Convergence of the new composite implicit iteration process with random errors, Nonlinear Anal., 69(10) (2008), 3591-3600.
• [43] L. Yang and F. H. Zhao, Large strong convergence theorems for total asymptotically strict pseudocontractive semigroup in banach spaces, Fixed Point Theory and Applications, 2012, 2012:24
• [44] L. Yang1, S. Chang and F. H. Zhao, Strong convergence theorems for a finite family of total asymptotically strict pseudo- contractive semigroups in Banach spaces, Fixed Point Theory and Applications 2013, 2013:178