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Some Iterative Methods for Fixed Point Problems

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Abstract

In this chapter, we discuss several iterative algorithms. We present and analyze a new unified hybrid steepest-descent-like iterative algorithm for finding a common solution of a generalized mixed equilibrium problem and a common fixed point problem of uncountable family of nonexpansive mappings.

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References

  1. Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-type Mappings with Applications. Springer, New York (2009)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. Ansari, Q.H.: Metric Spaces: Including Fixed Point Theory and Set-Valued Maps. Narosa Publishing House, New Delhi (2010)

    Google Scholar 

  4. Ansari, Q.H. (ed.): Topics in Nonlinear Analysis and Optimization. World Education, Delhi (2012)

    Google Scholar 

  5. Bose, S.C.: Weak convergence to the fixed of an asymptotically nonexpansive map. Proc. Am. Math. Soc. 68, 305–308 (1978)

    Article  MATH  Google Scholar 

  6. Browder, F.E.: Convergence of approximants to fixed point of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration Mech. Anal. 21, 259–269 (1967)

    MathSciNet  Google Scholar 

  7. Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings. J. Math. Anal. Appl. 20, 197–228 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  8. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and imagine reconstruction. Inverse Probl. 20, 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ceng, L.C., Al-Homidan, S., Ansari, Q.H., Yao, J.-C.: An iterative scheme for equilibrium problems and fixed points problems of strict pseudo-contraction mappings. J. Computat. Appl. Math. 223, 967–974 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ceng, L.C., Ansari, Q.H., Schaible, S., Yao, J.C.: Iterative methods for generalized equilibrium problems, systems of general generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert spaces. Fixed Point Theor. 12, 293–308 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Ceng, L.C., Ansari, Q.H., Wen, C.-F.: Implicit relaxed and hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense. Abstr. Appl. Anal. 2013, Article ID 854297 (2013)

    Google Scholar 

  12. Ceng, L.C., Ansari, Q.H., Yao, J.C.: Strong and weak convergence theorems for asymptotically strict pseudocontractive mappings in intermediate sense. J. Nonlinear Convex Anal. 11, 283–308 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Ceng, L.C., Ansari, Q.H., Yao, J.C.: Hybrid pseudoviscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Nonlinear Anal. Hybrid Syst. 4, 743–754 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ceng, L.C., Ansari, Q.H., Yao, J.C.: Hybrid proximal-type and hybrid shrinking projection algorithms for equilibrium problems, maximal monotone operators and relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 31, 763–797 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ceng, L.C., Ansari, Q.H., Yao, J.C.: Mann type iterative methods for finding a common solution of split feasibility and fixed point problems. Positivity 16, 471–495 (2012)

    Article  MathSciNet  Google Scholar 

  16. Ceng, L.C., Yao, J.C.: Strong convergence theorems for variational inequalities and fixed point problems of asymptotically strict pseudocontractive mappings in the intermediate sense. Acta Appl. Math. 115, 167–191 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chang, S.S.: On the approximation problem of fixed points for asymptotically nonexpansive mappings. Indian J. Pure Appl. Math. 32, 1297–1307 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Chidume, C.E.: Global iteration schemes for strongly pseudocontractive maps. Proc. Am. Math. Soc. 126, 2641–2649 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cui, Y.-L., Liu, X.: Notes on Browder’s and Halpern’s methods for nonexpansive mappings. Fixed Point Theor. 10, 89–98 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Deng, L.: Convergence of the Ishikawa iteration process for nonexpansive mappings. J. Math. Anal. Appl. 199, 769–775 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Edelstein, M.: A remark on a theorem of Krasnoselskii. Am. Math. Mon. 73, 509–510 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fukhar-ud-din, H., Khan, A.R.: Approximating common fixed points of asymptotically nonexpansive maps in uniformaly convex Banach spaces. Comput. Math. Appl. 53, 1349–1360 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Goebel, K., Kirk, W.A.: A fixed point theorem of asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171–174 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  24. Halpern, B.: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 73, 957–961 (1967)

    Article  MATH  Google Scholar 

  25. Huang, S.: Hybrid extragradient methods for asymptotically strict pseudocontractions in the intermediate sense and variational inequality problems. Optimization 60, 739–754 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ishikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ishikawa, S.: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. Am. Math. Soc. 59, 65–71 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  28. Khan, A.R., Domlo, A.A., Fukhar-ud-din, H.: Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 341, 1–11 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Khan, A.R., Fukhar ud-din, H., Domlo, A.A.: Approximating fixed points of some maps in uniformly convex metric spaces. Fixed Point Theor. Appl. 2010, Article ID 385986 (2010)

    Google Scholar 

  30. Kim, T.H., Xu, H.K.: Convergence of the modified Mann’s iteration method for asymptotically strict pseudocontractions. Nonlinear Anal. 68, 2828–2836 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kim, T.H., Xu, H.K.: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal. 64, 1140–1152 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Krasnoselskii, M.A.: Two observations about the method of successive approximations. Uspehi Math. Nauk 10, 123–127 (1955)

    MathSciNet  Google Scholar 

  33. Lim, T.C., Xu, H.: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Anal. 22, 1345–1355 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu, Z.Q., Kang, S.M.: Weak and strong convergence for fixed points of asymptotically nonexpansive mappings. Acta Math. Sinica 20, 1009–1018 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–610 (1953)

    Article  MATH  Google Scholar 

  36. Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. Opial, Z.: Weak convergence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  38. Osilike, M.O., Aniagbosor, S.C.: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Math. Comput. Model. 32, 1181–1191 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Picard, E.: Mémoire sur la théorie des equations aux derivée partielles et la metode des approximations successives. J. Math. Pures. Appl. 6, 145–210 (1890)

    MATH  Google Scholar 

  40. Podilchuk, C.I., Mammone, R.J.: Imagine recovery by convex projections using a least-squares constraint. J. Opt. Soc. Am. A 7, 517–521 (1990)

    Article  Google Scholar 

  41. Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theor. 12, 187–204 (2011)

    MathSciNet  MATH  Google Scholar 

  42. Sahu, D.R.: Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces. Comment. Math. Univ. Carol. 46, 653–666 (2005)

    MathSciNet  MATH  Google Scholar 

  43. Sahu, D.R., Wong, N.C., Yao, J.C.: A unified hybrid iterative method for solving variational inequalities involving generalized pseudo-contractive mappings. SIAM J. Contr. Optim. 50, 2335–2354 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Sahu, D.R., Xu, H.K., Yao, J.C.: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Anal. 70, 3502–3511 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sahu, D.R., Beg, I.: Weak and strong convergence for fixed points of nearly asymptotically non-expansive mappings. Int. J. Mod. Math. 3(2), 135–151 (2008)

    MathSciNet  MATH  Google Scholar 

  46. Schaefer, H.: Uber die methode sukzessiver Approximationen. Jber. Deutch. Math. Verein 59, 131–140 (1957)

    MathSciNet  MATH  Google Scholar 

  47. Schu, J.: Iterative construction of fixed points of asymptotically nonexpansive mapping. J. Math. Anal. Appl. 159, 407–413 (1991)

    Article  MathSciNet  Google Scholar 

  48. Schu, J.: Weak and strong convergence of fixed points of asymptotically nonexpansive maps. Bull. Aust. Math. Soc. 43, 153–159 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  49. Takahashi, W.: Viscosity approximation methods for resolvents of accretive operators in Banach spaces. J. Fixed Point Theor. Appl. 1, 135–147 (2007)

    Article  MATH  Google Scholar 

  50. Tan, K.K., Xu, H.K.: Fixed point iteration process for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 122, 733–739 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wong, N.C., Sahu, D.R., Yao, J.C.: Solving variational inequalities involving nonexpansive type mappings. Nonlinear Anal. 69, 4732–4753 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  53. Xu, H.K.: Strong convergence of an iterative method for nonexpansive and accretive operators. J. Math. Anal. Appl. 314, 631–643 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  54. Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  55. Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

    Article  MATH  Google Scholar 

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Ansari, Q.H., Sahu, D.R. (2014). Some Iterative Methods for Fixed Point Problems. In: Almezel, S., Ansari, Q., Khamsi, M. (eds) Topics in Fixed Point Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-01586-6_8

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