Common Fixed Points of Two Generalized Asymptotically Quasi-Nonexpansive Mappings

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Abstract

In this paper, we consider an iterative procedure for approximating common fixed points of two generalized asymptotically quasi-nonexpansive mappings and we prove some strong and weak convergence theorems for such mappings in uniformly con- vex Banach spaces.This will extend the results of Cholamjiak and Suantai, Khan and those generalized therein to the case of generalized asymptotically quasi-nonexpansive mappings and by a faster iterative procedure.

Abstract

In this paper, we consider an iterative procedure for approximating common fixed points of two generalized asymptotically quasi-nonexpansive mappings and we prove some strong and weak convergence theorems for such mappings in uniformly con- vex Banach spaces.This will extend the results of Cholamjiak and Suantai, Khan and those generalized therein to the case of generalized asymptotically quasi-nonexpansive mappings and by a faster iterative procedure.

References

  • 1. Agarwal, R.P.; O’Regan, D.; Sahu, D.R. - Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8 (2007), 61-79.

  • 2. Ceng, L.C.; Cubiotti, P.; Yao, J.C. - Approximation of common fixed points of families of nonexpansive mappings, Taiwanese J. Math., 12 (2008), 487-500.

  • 3. Ceng, L.C.; Cubiotti, P.; Yao, J.C. - An implicit iterative scheme for monotone variational inequalities and fixed point problems, Nonlinear Anal., 69 (2008), 2445-2457.

  • 4. Ceng, L.C.; Schaible, S.; Yao, J.C. - Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonex- pansive mappings, J. Optim. Theory Appl., 139 (2008), 403-418.

  • 5. Cho, Y.J.; Zhou, H.; Guo, G. - Weak and strong convergence theorems for three- step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl., 47 (2004), 707-717.

  • 6. Cholamjiak, W.; Suantai, S. - Weak and strong convergence theorems for a finite family of generalized asymptotically quasi-nonexpansive mappings, Comput. Math. Appl., 60 (2010), 1917-1923.

  • 7. Das, G.; Debata, J.P. - Fixed points of quasinonexpansive mappings, Indian J. Pure Appl. Math., 17 (1986), 1263-1269.

  • 8. Fukhar-ud-din, H.; Khan, S.H. - Convergence of iterates with errors of asymp- totically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl., 328 (2007), 821-829.

  • 9. Goebel, K.; Kirk, W.A. - A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174.

  • 10. Khan, S.H. - A two-step iterative process for two asymptotically quasi-nonexpansive mappings, World Academy of Science, Engineering and Technology, 75 (2011), 379-386.

  • 11. Khan, S.H.; Cho, Y.J.; Abbas, M. - Convergence to common fixed points by a modified iteration process, J. Appl. Math. Comput., 35 (2011), 607-616.

  • 12. Khan, S.H.; Kim, J.K. - Common fixed points of two nonexpansive mappings by a modified faster iteration scheme, Bull. Korean Math. Soc., 47 (2010), 973-985.

  • 13. Khan, S.H.; Takahashi, W. - Approximating common fixed points of two asymp- totically nonexpansive mappings, Sci. Math. Jpn., 53 (2001), 143-148.

  • 14. Nammanee, K.; Noor, M.A.; Suantai, S. - Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 314 (2006), 320-334.

  • 15. Nilsrakoo, W.; Saejung, S. - A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings, Appl. Math. Comput., 181 (2006), 1026-1034.

  • 16. Opial, Z. - Weak convergence of the sequence of successive approximations for non- expansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597.

  • 17. Schu, J. - Weak and strong convergence to fixed points of asymptotically nonexpan- sive mappings, Bull. Austral. Math. Soc., 43 (1991), 153-159.

  • 18. Shahzad, N.; Zegeye, H. - Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps, Appl. Math. Comput., 189 (2007), 1058-1065

  • 19. Takahashi, W.; Kim, G.-E. - Approximating fixed points of nonexpansive mappings in Banach spaces, Math. Japon., 48 (1998), 1-9.

  • 20. Takahashi, W. - Iterative methods for approximation of fixed points and their ap- plications, New trends in mathematical programming (Kyoto, 1998), J. Oper. Res. Soc. Japan, 43 (2000), 87-108.

  • 21. Takahashi, W.; Tamura, T. - Limit theorems of operators by convex combinations of nonexpansive retractions in Banach spaces, J. Approx. Theory, 91 (1997), 386-397.

  • 22. Tan, Kok-Keong; Xu, Hong Kun - Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308.

1. Agarwal, R.P.; O’Regan, D.; Sahu, D.R. - Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8 (2007), 61-79.

2. Ceng, L.C.; Cubiotti, P.; Yao, J.C. - Approximation of common fixed points of families of nonexpansive mappings, Taiwanese J. Math., 12 (2008), 487-500.

3. Ceng, L.C.; Cubiotti, P.; Yao, J.C. - An implicit iterative scheme for monotone variational inequalities and fixed point problems, Nonlinear Anal., 69 (2008), 2445-2457.

4. Ceng, L.C.; Schaible, S.; Yao, J.C. - Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonex- pansive mappings, J. Optim. Theory Appl., 139 (2008), 403-418.

5. Cho, Y.J.; Zhou, H.; Guo, G. - Weak and strong convergence theorems for three- step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl., 47 (2004), 707-717.

6. Cholamjiak, W.; Suantai, S. - Weak and strong convergence theorems for a finite family of generalized asymptotically quasi-nonexpansive mappings, Comput. Math. Appl., 60 (2010), 1917-1923.

7. Das, G.; Debata, J.P. - Fixed points of quasinonexpansive mappings, Indian J. Pure Appl. Math., 17 (1986), 1263-1269.

8. Fukhar-ud-din, H.; Khan, S.H. - Convergence of iterates with errors of asymp- totically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl., 328 (2007), 821-829.

9. Goebel, K.; Kirk, W.A. - A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174.

10. Khan, S.H. - A two-step iterative process for two asymptotically quasi-nonexpansive mappings, World Academy of Science, Engineering and Technology, 75 (2011), 379-386.

11. Khan, S.H.; Cho, Y.J.; Abbas, M. - Convergence to common fixed points by a modified iteration process, J. Appl. Math. Comput., 35 (2011), 607-616.

12. Khan, S.H.; Kim, J.K. - Common fixed points of two nonexpansive mappings by a modified faster iteration scheme, Bull. Korean Math. Soc., 47 (2010), 973-985.

13. Khan, S.H.; Takahashi, W. - Approximating common fixed points of two asymp- totically nonexpansive mappings, Sci. Math. Jpn., 53 (2001), 143-148.

14. Nammanee, K.; Noor, M.A.; Suantai, S. - Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 314 (2006), 320-334.

15. Nilsrakoo, W.; Saejung, S. - A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings, Appl. Math. Comput., 181 (2006), 1026-1034.

16. Opial, Z. - Weak convergence of the sequence of successive approximations for non- expansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597.

17. Schu, J. - Weak and strong convergence to fixed points of asymptotically nonexpan- sive mappings, Bull. Austral. Math. Soc., 43 (1991), 153-159.

18. Shahzad, N.; Zegeye, H. - Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps, Appl. Math. Comput., 189 (2007), 1058-1065

19. Takahashi, W.; Kim, G.-E. - Approximating fixed points of nonexpansive mappings in Banach spaces, Math. Japon., 48 (1998), 1-9.

20. Takahashi, W. - Iterative methods for approximation of fixed points and their ap- plications, New trends in mathematical programming (Kyoto, 1998), J. Oper. Res. Soc. Japan, 43 (2000), 87-108.

21. Takahashi, W.; Tamura, T. - Limit theorems of operators by convex combinations of nonexpansive retractions in Banach spaces, J. Approx. Theory, 91 (1997), 386-397.

22. Tan, Kok-Keong; Xu, Hong Kun - Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308.

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