Strong Convergence Theorems for Common Points of a Finite Family of Accretive Operators

  • Jeong, Jae Ug (Department of Mathematics, Dongeui University) ;
  • Kim, Soo Hwan (Department of Mathematics, Dongeui University)
  • Received : 2016.03.18
  • Accepted : 2019.08.05
  • Published : 2019.09.23


In this paper, we propose a new iterative algorithm generated by a finite family of accretive operators in a q-uniformly smooth Banach space. We prove the strong convergence of the proposed iterative algorithm. The results presented in this paper are interesting extensions and improvements of known results of Qin et al. [Fixed Point Theory Appl. 2014(2014): 166], Kim and Xu [Nonlinear Anal. 61(2005), 51-60] and Benavides et al. [Math. Nachr. 248(2003), 62-71].


accretive operator;nonexpansive mapping;zero point;resolvent operator


  1. G. L. Acedo and H. K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 67(2007), 2258-2271.
  2. K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, On a strongly nonexpansive sequence in Hilbert spaces, J. Nonlinear Convex Anal., 8(2007), 471-489.
  3. V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Noord-hoffInternational Publishing, Leiden, 1976.
  4. T. D. Benavides, G. L. Acedo and H. K. Xu, Iterative solutions for zeros of accretive operators, Math. Nachr., 248/249(2003), 62-71.
  5. E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63(1994), 123-145.
  6. F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20(1967), 197-228.
  7. G. H. G. Chen and R. T. Rockafellar, Convergence rates in forward-backward splitting, SIAM J. Optim., 7(1997), 421-444.
  8. I. Ciorancscu, Geometry of Banach spaces, duality mappings and nonlinear problems, Kluwer Academic Publishers Group, Dordrecht, 1990.
  9. P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6(2005), 117-136.
  10. P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4(2005), 1168-1200.
  11. O. Guler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29(1991), 403-419.
  12. J. S. Jung, Some results on Rockafellar-type iterative algorithms for zeros of accretive operators, J. Inequal. Appl. (2013), 2013: 255, 19 pp.
  13. T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19(1967), 508-520.
  14. J. K. Kim, Convergence of Ishikawa iterative sequences for accretive Lipschitzian mappings in Banach spaces, Taiwanese J. Math., 10(2006), 553-561.
  15. J. K. Kim and Salahuddin, Extragradient methods for generalized mixed equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Funct. Anal. and Appl. 22(2017), 693-709.
  16. T. H. Kim and H. K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal., 61(2005), 51-60.
  17. L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194(1995), 114-125.
  18. D. S. Mitrinovic, Analytic inequalities, Springer-Verlag, New York, 1970.
  19. A. Moudafi and M. Thera, Finding a zero of the sum of two maximal monotone operators, J. Optim. Theory Appl., 94(1997), 425-448.
  20. X. Qin, S. Y. Cho and L. Wang, Iterative algorithms with errors for zero points of m-accretive operators, Fixed Point Theory Appl., (2013), 2013:148, 17 pp.
  21. X. Qin, S. Y. Cho and L. Wang, Convergence of splitting algorithms for the sum of two accretive operators with applications, Fixed Point Theory Appl., (2014), 2014:166, 12 pp.
  22. X. Qin and Y. Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl., 329(2007), 415-424.
  23. R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149(1970), 75-88.
  24. R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14(1976), 877-898.
  25. T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305(2005), 227-239.
  26. S. Takahashi, W. Takahashi and M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., 147(2010), 27-41.
  27. P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38(2000), 431-446.
  28. H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16(1991), 1127-1138.
  29. S. Yang, Zero theorems of accretive operators in reflexive Banach spaces, J. Nonlinear Funct. Anal., (2013), 2013:2, 12 pp.