Results in Mathematics

, 74:130 | Cite as

Approximation of Common Fixed Points and the Solution of Image Recovery Problem

  • Javid AliEmail author
  • Faeem Ali


In the present paper, we study one step iterative scheme to approximate common fixed points of two generalized non-expansive mappings in uniformly convex Banach spaces and using the same scheme we prove some weak and strong convergence results for such mappings. Further, we establish some weak and strong convergence results for a finite family of generalized non-expansive mappings to approximate common fixed points using proposed algorithm in uniformly convex Banach spaces. As an application, we apply our main result to approximate the solution of image recovery problem in Banach space setting. To support our results we present some illustrative numerical examples. Our results are new and generalize several relevant results in the literature.


Generalized non-expansive mappings common fixed points one step iterative scheme weak and strong convergence results uniformly convex Banach spaces image recovery 

Mathematics Subject Classification

47H09 47H10 



The authors are grateful to the anonymous referee for his valuable comments which improve the paper. The second author would like to thank Council of Scientific and Industrial Research, Government of India for SRF (09/112(0536)/2016-EMR-I).


  1. 1.
    Bregman, L.M.: The method of successive projection for finding a common point of convex sets. Sov. Math. 6, 688–692 (1965)zbMATHGoogle Scholar
  2. 2.
    Browder, F.E.: Non-expansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54, 1041–1044 (1965)CrossRefGoogle Scholar
  3. 3.
    Chidume, C.E., Ali, B.: Weak and strong convergence theorems for finite families of asymptotically non-expansive mappings in Banach spaces. J. Math. Anal. Appl. 330, 377–387 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Crombez, G.: Image recovery by convex combinations of projections. J. Math. Anal. Appl. 155, 413–419 (1991)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Das, G., Debata, J.P.: Fixed points of quasi-non-expansive mappings. Indian J. Pure Appl. Math. 17, 1263–1269 (1986)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Fukher-ud-din, H., Saleh, K.: One-step iterations for a finite family of generalized non-expansive mappings in CAT(0) spaces. Bull. Malays. Math. Sci. Soc. 41(2), 597–608 (2018)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hardy, G.F., Rogers, T.D.: A generalization of a fixed point theorem of Reich. Can. Math. Bull. 16, 201–206 (1973)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ishikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Khan, S.H.: Convergence of a one step iteration scheme for quasi-asymptotically non-expansive mappings. World Acad. Sci. Eng. Technol. 63, 504–506 (2012)Google Scholar
  10. 10.
    Khan, S.H.: Iterative approximation of common attractive points of further generalized hybrid mappings. Fixed Point Theory Appl. 2018, 8 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Khan, S.H., Fukhar-ud-din, H.: Weak and strong convergence of a scheme with errors for two non-expansive mappings. Nonlinear Anal. 61, 1295–1301 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Khan, M.A.A., Kohlenbach, U.: Quantitative image recovery theorems. Nonlinear Anal. 106, 138–150 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Khan, S.H., Abbas, M., Khan, A.R.: Common fixed points of two non-expansive mappings by a new one step iterative scheme. Iran. J. Sci. Technol. Trans. A Sci. 33(A3), 249–257 (2009)MathSciNetGoogle Scholar
  14. 14.
    Kimura, Y., Nakajo, K.: The problem of image recovery by the metric projections in Banach spaces. In: Abstract and Applied Analysis, vol. 2013. Hindawi Publishing Corporation, New YorkMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kitahara, S., Takahashi, W.: Image recovery by convex combinations of sunny non-expansive retractions. Topol. Methods Nonlinear Anal. 2, 333–342 (1993)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kuhfittig, P.K.F.: Common fixed points of non-expansive mappings by iteration. Pac. J. Math. 97, 137–139 (1981)CrossRefGoogle Scholar
  17. 17.
    Liu, Z., Feng, C., Ume, J.S., Kang, S.M.: Weak and strong convergence for common fixed points of a pair of non-expansive and asymptotically non-expansive mappings. Taiwansese J. Math. 1, 27–42 (2007)zbMATHGoogle Scholar
  18. 18.
    Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Opial, Z.: Weak convergence of the sequence of successive approximations for non-expansive mappings. Bull. Am. Math. Soc. 73, 595–597 (1967)CrossRefGoogle Scholar
  20. 20.
    Plubtieng, S., Ungchittrakool, K.: Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively non-expansive mappings in Banach spaces. Fixed Point Theory Appl. 2008, Article ID 583082 (2008)Google Scholar
  21. 21.
    Precup, R., Rodríguez-López, J.: Multiplicity results for operator systems via fixed point index. Results Math. 74, 25 (2019). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rhoades, B.E.: Finding common fixed points of non-expansive mappings by iteration. Bull. Aust. Math. Soc. 62, 307–310 (2000); Corrigendum, Bull. Aust. Math. Soc. 63: 345–346 (2001)Google Scholar
  23. 23.
    Schu, J.: Weak and strong convergence to fixed points of asymptotically non-expansive mappings. Bull. Aust. Math. Soc. 43(1), 153–159 (1991)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Senter, H.F., Dotson, W.G.: Approximating fixed points of non-expansive mappings. Proc. Am. Math. Soc. 44(2), 375–380 (1974)CrossRefGoogle Scholar
  25. 25.
    Takahashi, W.: Iterative methods for approximation of fixed points and their applications. J. Oper. Res. Soc. Jpn. 43, 87–108 (2000)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Takahashi, W., Shimoji, K.: Convergence theorems for non-expansive mappings and feasibility problems. Math. Comput. Model. 32, 1463–1471 (2000)CrossRefGoogle Scholar
  27. 27.
    Takahashi, W., Tamura, T.: Limit theorems of operators by convex combinations of non-expansive retractions in Banach spaces. J. Approx. Theory 91, 386–397 (1997)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Takahashi, W., Tamura, T.: Convergence theorems for a pair of non-expansive mappings. J. Convex Anal. 5(1), 45–58 (1998)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Uddin, I., Ali, J., Nieto, J.J.: An iteration scheme for a family of multivalued non-expansive mappings in CAT(0) spaces with an application to image recovery. RACSAM 112(2), 373–384 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Uddin, I., Garodia, C., Nieto, J.J.: Mann iteration for monotone non-expansive mappings in ordered CAT(0) space with an application to integral equations. J. Inequal. Appl. 2018(1), 339 (2018)CrossRefGoogle Scholar
  31. 31.
    Woldeamanuel, S.T., Sangago, M.G., Hailu, H.Z.: Approximating a common fixed point of finite family of asymptotically quasi-non-expansive mappings in Banach spaces. Afr. Mat. 27, 949–961 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Xu, Y.: Ishikawa and Mann Iteration process with errors for nonlinear strongly accretive operator equations. J. Math. Anal. Appl. 224, 91–101 (1998)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Yao, Y., Chen, R.: Weak and strong convergence of a modified Mann iteration for asymptotically non-expansive mappings. Nonlinear Funct. Anal. Appl. 12, 307–315 (2007)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Youla, D., Webb, H.: Image restoration by the method of convex projections. IEEE Trans. Medical Image 1(2), 81–101 (1982)CrossRefGoogle Scholar
  35. 35.
    Zhang, S.S.: Generalized mixed equilibrium problem in Banach spaces. Appl. Math. Mech. 30(9), 1105–1112 (2009)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia

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