Strong convergence of an iterative algorithm for accretive operators and nonexpansive mappings

Volume 9, Issue 5, pp 2394--2409 http://dx.doi.org/10.22436/jnsa.009.05.40

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1475 Downloads
• 2189 Views

Authors

Jong Soo Jung - Department of Mathematics, Dong-A University, Busan 49315, Korea.

Abstract

In this paper, an iterative algorithm for finding a common point of the set of zeros of an accretive operator and the set of fixed points of a nonexpansive mapping is considered in a uniformly convex Banach space having a weakly continuous duality mapping. Under suitable control conditions, strong convergence of the sequence generated by proposed algorithm to a common point of two sets is established. The main theorems develop and complement the recent results announced by researchers in this area.

Share and Cite

Keywords

• Iterative algorithm
• accretive operator
• resolvent
• zeros
• nonexpansive mappings
• fixed points
• variational inequality
• weakly continuous duality mapping
• uniformly convex
• contractive mapping
• weakly contractive mapping.

MSC

•  47H06
•  47H09
•  47H10
•  47J25
•  49M05
•  65J15

References

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


• [2] Y. I. Alber, S. Guerre-Delabriere, L. Zelenko, Principle of weakly contractive maps in metric spaces, Comm. Appl. Nonlinear Anal., 5 (1998), 45-68.


• [3] Y. I. Alber, S. Reich, J. C. Yao , Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces, Abstr. Appl. Anal., 4 (2003), 193-216.


• [4] T. D. Benavides, G. L. Acedo, H. K. Xu, Iterative solutions for zeros of accretive operators, Math. Nachr., 248-249 (2003), 62-71.


• [5] H. Bréziz, P. L. Lions, Products infinis de resolvents, Israel J. Math., 29 (1978), 329-345.


• [6] R. E. Jr. Bruck, A strongly convergent iterative solution of \(0 \in U(x)\) for a maximal monotone operator U in Hilbert space, J. Math. Anal. Appl., 48 (1974), 114-126.


• [7] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht (1990)


• [8] J. S. Jung, Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces, Nonlinear Anal., 64 (2006), 2536-2552.


• [9] J. S. Jung, Convergence of composite iterative methods for finding zeros of accretive operators, Nonlinear Anal., 71 (2009), 1736-1746.


• [10] J. S. Jung, Some results on Rockafellar-type iterative algorithms for zeros of accretive operators , J. Inequal. Appl., 2013 (2013), 19 pages.


• [11] J. S. Jung, Iterative algorithms for zeros of accretive operators and fixed points of nonexpansive mappings in Banach spaces, J. Comput. Anal. Appl., 20 (2016), 688-706.


• [12] J. S. Jung, W. Takahashi , Dual convergence theorems for the infinite products of resolvents in Banach spaces, Kodai Math. J., 14 (1991), 358-365.


• [13] J. S. Jung, W. Takahashi , On the asymptotic behavior of infinite products of resolvents in Banach spaces, Nonlinear Anal., 20 (1993), 469-479.


• [14] L. S. Liu, Iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194 (1995), 114-125.


• [15] S. Reich, On infinite products of resolvents, Atti. Accad. naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 63 (1997), 338-340.


• [16] B. E. Rhodes, Some theorems on weakly contractive maps, Nonlinear Anal., 47 (2001), 2683-2693.


• [17] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.


• [18] Y. Song, New iterative algorithms for zeros of accretive operators, J. Korean Math. Soc., 46 (2009), 83-97.


• [19] Y. Song, J. I. Kang, Y. J. Cho, On iterations methods for zeros of accretive operators in Banach spaces, Appl. Math. Comput., 216 (2010), 1007-1017.


• [20] Y. Song, C. Yang , A note on a paper ''A regularization method for the proximal point algorithm'', J. Global Optim., 43 (2009), 115-125.


• [21] H. K. Xu , Inequality in Banach spaces with applications , Nonlinear Anal., 16 (1991), 1127-1138.


• [22] H. K. Xu , Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.


• [23] H. K. Xu, A regularization method for the proximal point algorithm, J. Global Optim., 36 (2006), 115-125.


• [24] Q. Zhang, Y. Song, Halpern type proximal point algorithm of accretive operators, Nonlinear Anal., 75 (2012), 1859-1868.