Annals of Functional Analysis

A general iterative algorithm for nonexpansive mappings in Banach spaces

Bashir Ali, Yekini Shehu, and ‎Godwin C‎. ‎Ugwunnadi

Full-text: Open access


Let $E$ be a real $q$-uniformly smooth Banach space whose duality map is weakly sequentially continuous. Let $T:E\to E$ be a nonexpansive mapping with $F(T)\neq\emptyset.$ Let $A:E\to E$ be an $\eta$-strongly accretive map which is also $\kappa$-Lipschitzian. Let $f:E\to E$ be a contraction map with coefficient $0\lt \alpha\lt1.$ Let a sequence $\{y_{n}\}$ be defined iteratively by $y_{0}\in E,~~ y_{n+1}=\alpha_n\gamma f(y_n)+(I-\alpha_n\mu A)Ty_n,n\geq0,$ where $\{\alpha_n\},~~\gamma$ and $\mu$ satisfy some appropriate conditions. Then, we prove that $\{y_{n}\}$ converges strongly to the unique solution $x^{*} \in F(T)$ of the variational inequality $\langle(\gamma f-\mu A)x^{*},j(y-x^{*})\rangle\leq0,~\forall~y\in F(T).$ Convergence of the correspondent implicit scheme is also proved without the assumption that $E$ has weakly sequentially continuous duality map. Our results are applicable in $l_{p}$ spaces, $1\lt p \lt \infty$.

Article information

Ann. Funct. Anal., Volume 2, Number 2 (2011), 10- 21.

First available in Project Euclid: 12 May 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.
Secondary: 47H10‎ ‎47J20

$\eta-$strongly accretive maps ‎$\kappa-$Lipschitzian ‎maps ‎nonexpansive maps ‎$q-$uniformly smooth Banach spaces


Ali, Bashir; ‎Ugwunnadi, ‎Godwin C‎.; Shehu, Yekini. A general iterative algorithm for nonexpansive mappings in Banach spaces. Ann. Funct. Anal. 2 (2011), no. 2, 10-- 21. doi:10.15352/afa/1399900190.

Export citation


  • W.L. Bynum, Normal structure coefficients for Banach spaces, Pacific J. Math. 86 (1980), no. 2, 427–436.
  • C.E. Chidume, J. Li and A. Udomene, Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 133 (2005), no. 2, 473– 480.
  • C.E. Chidume, Geometric properties of Banach spaces and nonlinear iterations, Lecture Notes in Mathematics, 1965. Springer-Verlag London, Ltd., London, 2009.
  • T.C. Lim and H.K. Xu, Fixed point theorms for asymptotically nonexpansive mappings, Nonlinear Anal. 22 (1994), 1345–1355.
  • A. Moudafi, Viscosity approximation methods for fixed-point problems, J. Math. Anal. Appl. 241 (2000), no. 1, 46–55.
  • M. Tian, A general iterative algorithm for nonexpansive mappings in Hilbert spaces, Nonliear Anal. 73 (2010), no. 3, 689–694.
  • G. Marino and H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006), no. 1, 43–52.
  • J.S. Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005), no. 2, 509–520.
  • Y. Su, M. Shang and D. Wang, Strong convergence of monotone $CQ$ algorithm for relatively nonexpansive mappings, Banach J. Math. Anal. 2 (2008), no. 1, 1–10.
  • Y. Shehu, Iterative methods for fixed points and equilibrium problems, Ann. Funct. Anal. 1 (2010), no. 2, 121–132.
  • H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2) 66 (2002), no. 1, 240–256.
  • H.-K. Xu, Viscosity approximation methods for nonexpansive mapping, J. Math. Anal. Appl. 298 (2004), no. 1, 279–291.
  • H.-K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116 (2003), 659–678.
  • H.-K. Xu and T.H. Kim Convergence of hybrid steepest-decent methods for variational inequalities, J. Optim. Theory Appl. 119 (2003), no. 3, 185–201.
  • H.-K. Xu, Inequality in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127–1138.
  • Z.B. Xu and G.F. Roach, Characteristic inequalities of uniformly smooth Banach spaces, J. Math. Anal. Appl. 157 (1991), no. 1, 189–210.
  • I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Inherently parallel algorithms in feasibility and optimization and their applications (Haifa, 2000), 473–504, Stud. Comput. Math., 8, North-Holland, Amsterdam, 2001.