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Strong Convergence of an explicit iteration method in uniformly convex Banach spaces

Year 2018, Volume: 6 Issue: 1, 178 - 187, 15.04.2018

Abstract

We obtain the necessary and sufficient conditions for the convergence of an explicit iterative procedure to a common fixed point of a finite family of non-self asymptotically quasi-nonexpansive type mappings in real Banach spaces. We also prove the strong convergence of this iterative method to a common fixed point of a finite family of non-self asymptotically quasi-nonexpansive in the intermediate sense mappings in uniformly convex Banach spaces. Our results mainly generalize and extend those obtained by Wang [L. Wang, Explicit iteration method for common fixed points of a finite family of nonself asymptotically nonexpansive mappings, Computers \& Mathematics with applications, 53, (2007), 1012 - 1019.]

References

  • [1] S. S. Chang, Y. J. Cho, H. Zhou, Demiclosedness principle and weak convergence theorems for asymptotically nonexpansive mappings, J. Korean Math. Soc. 38 (2001), 1245-1260.
  • [2] C. E. Chidume, Nonexpansive mappings, Generalizations and iterative algorithms, Nonlinear Anal. Appl., vols. 1,2, Kluwer Academic, Dordrecht, 2003, pp. 383-421.
  • [3] C. E. Chidume, N. Shahzaad, strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal., 62(6) (2005), 1149-1156.
  • [4] C. E. Chidume, E. U. Ofoedu, H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl.280 (2003), 364-374.
  • [5] K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174.
  • [6] W. Kaczor, Weak convergence of almost orbits of asymptotically nonexpansive semigroups, J. Math. Anal. Appl., 272 (2002), 565-574.
  • [7] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
  • [8] S. Y. Matsuhita, D. Kuroiwa, Strong convergence of averaging iteration of nonexpansive nonself mappings, J. Math. Anal. Appl., 294 (2004), 206-214.
  • [9] M. O. Osilike, A. Udomene, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling , 32(2000), 1181-1191.
  • [10] M. O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl., 294 (2004), 73-81.
  • [11] S. Plubteing and R. Wangkeeree, Strong convergence theorems for three-step iterations with errors non-Lipschitzian nonself mappings in Banach spaces, Computers & Mathematics with applications, 51(2006), 1093-1102.
  • [12] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274-276.
  • [13] B. E. Rhoades, Fixed point iterations for certain nonlinear mappings, J. Math. Anal. Appl., 183(1994), 118-120.
  • [14] N. Shahzad, Approximating fixed points of nonself nonexpansive mappings in Banach spaces, Nonlinear Anal. 61 (2005), 1031-1039.
  • [15] J. schu, Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43(1991), 153-159.
  • [16] 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.
  • [17] W. Takahashi, T.Tamura, convergence theorem for a pair of nonexpansive mappings, J. Conv. Anal., 5(1998), 45-56.
  • [18] K. K. Tan, H. K. Xu, The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach space, Proc. Amer. Math. Soc. 114 (1992), 399-404.
  • [19] K. K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308.
  • [20] K. K. Tan, H. K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122 (1994), 733-739.
  • [21] Y. X. Tian, S. S. Chang, J. L. Huang, On the approximation problem of common fixed points for a finite family of non-self asymptotically quasinonexpansive type mappings in Banach spaces, Computers & Mathematics with applications, 53(2007), 1847-1853.
  • [22] L. Wang, Strong and weak convergence theorems for nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl., 323(2006), 550-557.
  • [23] L. Wang, Explicit iteration method for common fixed points of a finite family of nonself asymptotically nonexpansive mappings, Computers & Mathematics with applications, 53, (2007), Pages 1012 - 1019.
  • [24] H. K. Xu, X. M. Yin, Strong convergence theorems for nonexpansive nonself mappings, Nonlinear Anal. 24(2)(1995), 223-228.
  • [25] H. K. Xu, R. Ori, An implicit iterative process for nonexpansive mappings, Num. Funct. Anal. Optim. 22(2001), 767-773.
  • [26] H. Y. Zhou, S. S. Chang, Convergence of implicit iteration perocess for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. Funct. Anal. 23 (2002), 911-921.
Year 2018, Volume: 6 Issue: 1, 178 - 187, 15.04.2018

Abstract

References

  • [1] S. S. Chang, Y. J. Cho, H. Zhou, Demiclosedness principle and weak convergence theorems for asymptotically nonexpansive mappings, J. Korean Math. Soc. 38 (2001), 1245-1260.
  • [2] C. E. Chidume, Nonexpansive mappings, Generalizations and iterative algorithms, Nonlinear Anal. Appl., vols. 1,2, Kluwer Academic, Dordrecht, 2003, pp. 383-421.
  • [3] C. E. Chidume, N. Shahzaad, strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal., 62(6) (2005), 1149-1156.
  • [4] C. E. Chidume, E. U. Ofoedu, H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl.280 (2003), 364-374.
  • [5] K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174.
  • [6] W. Kaczor, Weak convergence of almost orbits of asymptotically nonexpansive semigroups, J. Math. Anal. Appl., 272 (2002), 565-574.
  • [7] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
  • [8] S. Y. Matsuhita, D. Kuroiwa, Strong convergence of averaging iteration of nonexpansive nonself mappings, J. Math. Anal. Appl., 294 (2004), 206-214.
  • [9] M. O. Osilike, A. Udomene, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling , 32(2000), 1181-1191.
  • [10] M. O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl., 294 (2004), 73-81.
  • [11] S. Plubteing and R. Wangkeeree, Strong convergence theorems for three-step iterations with errors non-Lipschitzian nonself mappings in Banach spaces, Computers & Mathematics with applications, 51(2006), 1093-1102.
  • [12] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274-276.
  • [13] B. E. Rhoades, Fixed point iterations for certain nonlinear mappings, J. Math. Anal. Appl., 183(1994), 118-120.
  • [14] N. Shahzad, Approximating fixed points of nonself nonexpansive mappings in Banach spaces, Nonlinear Anal. 61 (2005), 1031-1039.
  • [15] J. schu, Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43(1991), 153-159.
  • [16] 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.
  • [17] W. Takahashi, T.Tamura, convergence theorem for a pair of nonexpansive mappings, J. Conv. Anal., 5(1998), 45-56.
  • [18] K. K. Tan, H. K. Xu, The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach space, Proc. Amer. Math. Soc. 114 (1992), 399-404.
  • [19] K. K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308.
  • [20] K. K. Tan, H. K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122 (1994), 733-739.
  • [21] Y. X. Tian, S. S. Chang, J. L. Huang, On the approximation problem of common fixed points for a finite family of non-self asymptotically quasinonexpansive type mappings in Banach spaces, Computers & Mathematics with applications, 53(2007), 1847-1853.
  • [22] L. Wang, Strong and weak convergence theorems for nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl., 323(2006), 550-557.
  • [23] L. Wang, Explicit iteration method for common fixed points of a finite family of nonself asymptotically nonexpansive mappings, Computers & Mathematics with applications, 53, (2007), Pages 1012 - 1019.
  • [24] H. K. Xu, X. M. Yin, Strong convergence theorems for nonexpansive nonself mappings, Nonlinear Anal. 24(2)(1995), 223-228.
  • [25] H. K. Xu, R. Ori, An implicit iterative process for nonexpansive mappings, Num. Funct. Anal. Optim. 22(2001), 767-773.
  • [26] H. Y. Zhou, S. S. Chang, Convergence of implicit iteration perocess for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. Funct. Anal. 23 (2002), 911-921.
There are 26 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Ahmed A. Abdelhakim This is me

R. A. Rashwan

Publication Date April 15, 2018
Submission Date November 7, 2017
Published in Issue Year 2018 Volume: 6 Issue: 1

Cite

APA A. Abdelhakim, A., & A. Rashwan, R. (2018). Strong Convergence of an explicit iteration method in uniformly convex Banach spaces. Konuralp Journal of Mathematics, 6(1), 178-187.
AMA A. Abdelhakim A, A. Rashwan R. Strong Convergence of an explicit iteration method in uniformly convex Banach spaces. Konuralp J. Math. April 2018;6(1):178-187.
Chicago A. Abdelhakim, Ahmed, and R. A. Rashwan. “Strong Convergence of an Explicit Iteration Method in Uniformly Convex Banach Spaces”. Konuralp Journal of Mathematics 6, no. 1 (April 2018): 178-87.
EndNote A. Abdelhakim A, A. Rashwan R (April 1, 2018) Strong Convergence of an explicit iteration method in uniformly convex Banach spaces. Konuralp Journal of Mathematics 6 1 178–187.
IEEE A. A. Abdelhakim and R. A. Rashwan, “Strong Convergence of an explicit iteration method in uniformly convex Banach spaces”, Konuralp J. Math., vol. 6, no. 1, pp. 178–187, 2018.
ISNAD A. Abdelhakim, Ahmed - A. Rashwan, R. “Strong Convergence of an Explicit Iteration Method in Uniformly Convex Banach Spaces”. Konuralp Journal of Mathematics 6/1 (April 2018), 178-187.
JAMA A. Abdelhakim A, A. Rashwan R. Strong Convergence of an explicit iteration method in uniformly convex Banach spaces. Konuralp J. Math. 2018;6:178–187.
MLA A. Abdelhakim, Ahmed and R. A. Rashwan. “Strong Convergence of an Explicit Iteration Method in Uniformly Convex Banach Spaces”. Konuralp Journal of Mathematics, vol. 6, no. 1, 2018, pp. 178-87.
Vancouver A. Abdelhakim A, A. Rashwan R. Strong Convergence of an explicit iteration method in uniformly convex Banach spaces. Konuralp J. Math. 2018;6(1):178-87.
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