• Previous Article
    Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities
  • NACO Home
  • This Issue
  • Next Article
    Another note on some quadrature based three-step iterative methods for non-linear equations
2013, 3(3): 557-565. doi: 10.3934/naco.2013.3.557

A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces

1. 

Department of Mathematics, Kyungsung University, Busan 608-736

Received  July 2012 Revised  April 2013 Published  July 2013

In this paper, we introduce a countably infinite iterative scheme and consider a sufficient and necessary condition for the existence of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces.
Citation: Byung-Soo Lee. A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 557-565. doi: 10.3934/naco.2013.3.557
References:
[1]

K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Analysis, 67 (2007), 2350-2360. doi: 10.1016/j.na.2006.08.032.

[2]

S. S. Chang, L. Yang and X. R. Wang, Stronger convergence theorem for an infinite family of uniformly quasi-Lipschitzian mappings in convex metric spaces, Appl. Math. Comput., 217 (2010), 277-282. doi: 10.1016/j.amc.2010.05.058.

[3]

S. B. Diaz and F. B. Metcalf, On the structure of the set of sequential limit points of successive approximations, Bull. Amer. Math. Soc., 73 (1967), 516-519. doi: 10.1090/S0002-9904-1967-11725-7.

[4]

H. Fukhar-ud-din and S. H. Khan, Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl., 328 (2007), 821-829. doi: 10.1016/j.jmaa.2006.05.068.

[5]

K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174. doi: 10.1090/S0002-9939-1972-0298500-3.

[6]

A. R. Khan and M. A. Ahmed, Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications, Com. Math. Appl., 59 (2010), 2990-2995. doi: 10.1016/j.camwa.2010.02.017.

[7]

A. R. Khan, A. A. Domlo and H. Fukhar-ud-din, Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 341 (2008), 1-11. doi: 10.1016/j.jmaa.2007.06.051.

[8]

B. S. Lee, Strong convergence theorems with a Noor-type iterative scheme in convex metrix spaces, Com. Math. Appl., 61 (2011), 3218-3225. doi: 10.1016/j.camwa.2011.04.017.

[9]

Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 259 (2001), 1-7. doi: 10.1006/jmaa.2000.6980.

[10]

W. Nilsrakoo and S. Saejung, Weak and strong convergence theorems for countable Lipschitzian mapping and its applications, Nonlinear Analysis, 69 (2008), 2695-2708. doi: 10.1016/j.na.2007.08.044.

[11]

W. Nilsrakoo and S. Saejung, Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications, J. Math. Anal. Appl., 356 (2009), 154-167. doi: 10.1016/j.jmaa.2009.03.002.

[12]

Y. Song and Y. Zheng, Strong convergence of iteration algorithms for a countable family of nonexpansive mappings, Nonlinear Analysis, 71 (2009), 3072-3082. doi: 10.1016/j.na.2009.01.219.

[13]

W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces, Nonlinear Analysis, 70 (2009), 719-734. doi: 10.1016/j.na.2008.01.005.

[14]

S. Temir and O. Gul, Convergence theorem for I-asymptotically quasi-nonexpansive mapping in Hilbert space, J. Math. Anal. Appl., 329 (2007), 759-765. doi: 10.1016/j.jmaa.2006.07.004.

[15]

Y. X. Tian, Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings, Comput. Math. Appl., 49 (2005), 1905-1912. doi: 10.1016/j.camwa.2004.05.017.

show all references

References:
[1]

K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Analysis, 67 (2007), 2350-2360. doi: 10.1016/j.na.2006.08.032.

[2]

S. S. Chang, L. Yang and X. R. Wang, Stronger convergence theorem for an infinite family of uniformly quasi-Lipschitzian mappings in convex metric spaces, Appl. Math. Comput., 217 (2010), 277-282. doi: 10.1016/j.amc.2010.05.058.

[3]

S. B. Diaz and F. B. Metcalf, On the structure of the set of sequential limit points of successive approximations, Bull. Amer. Math. Soc., 73 (1967), 516-519. doi: 10.1090/S0002-9904-1967-11725-7.

[4]

H. Fukhar-ud-din and S. H. Khan, Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl., 328 (2007), 821-829. doi: 10.1016/j.jmaa.2006.05.068.

[5]

K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174. doi: 10.1090/S0002-9939-1972-0298500-3.

[6]

A. R. Khan and M. A. Ahmed, Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications, Com. Math. Appl., 59 (2010), 2990-2995. doi: 10.1016/j.camwa.2010.02.017.

[7]

A. R. Khan, A. A. Domlo and H. Fukhar-ud-din, Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 341 (2008), 1-11. doi: 10.1016/j.jmaa.2007.06.051.

[8]

B. S. Lee, Strong convergence theorems with a Noor-type iterative scheme in convex metrix spaces, Com. Math. Appl., 61 (2011), 3218-3225. doi: 10.1016/j.camwa.2011.04.017.

[9]

Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 259 (2001), 1-7. doi: 10.1006/jmaa.2000.6980.

[10]

W. Nilsrakoo and S. Saejung, Weak and strong convergence theorems for countable Lipschitzian mapping and its applications, Nonlinear Analysis, 69 (2008), 2695-2708. doi: 10.1016/j.na.2007.08.044.

[11]

W. Nilsrakoo and S. Saejung, Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications, J. Math. Anal. Appl., 356 (2009), 154-167. doi: 10.1016/j.jmaa.2009.03.002.

[12]

Y. Song and Y. Zheng, Strong convergence of iteration algorithms for a countable family of nonexpansive mappings, Nonlinear Analysis, 71 (2009), 3072-3082. doi: 10.1016/j.na.2009.01.219.

[13]

W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces, Nonlinear Analysis, 70 (2009), 719-734. doi: 10.1016/j.na.2008.01.005.

[14]

S. Temir and O. Gul, Convergence theorem for I-asymptotically quasi-nonexpansive mapping in Hilbert space, J. Math. Anal. Appl., 329 (2007), 759-765. doi: 10.1016/j.jmaa.2006.07.004.

[15]

Y. X. Tian, Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings, Comput. Math. Appl., 49 (2005), 1905-1912. doi: 10.1016/j.camwa.2004.05.017.

[1]

Emeka Chigaemezu Godwin, Adeolu Taiwo, Oluwatosin Temitope Mewomo. Iterative method for solving split common fixed point problem of asymptotically demicontractive mappings in Hilbert spaces. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022005

[2]

Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2161-2180. doi: 10.3934/jimo.2020063

[3]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709

[4]

Se-Hyun Ku. Expansive flows on uniform spaces. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1585-1598. doi: 10.3934/dcds.2021165

[5]

Habib ur Rehman, Poom Kumam, Yusuf I. Suleiman, Widaya Kumam. An adaptive block iterative process for a class of multiple sets split variational inequality problems and common fixed point problems in Hilbert spaces. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022007

[6]

Adeolu Taiwo, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2733-2759. doi: 10.3934/jimo.2020092

[7]

Fuzhong Cong, Hongtian Li. Quasi-effective stability for a nearly integrable volume-preserving mapping. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1959-1970. doi: 10.3934/dcdsb.2015.20.1959

[8]

Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165

[9]

Alfonso Artigue. Expansive flows of surfaces. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 505-525. doi: 10.3934/dcds.2013.33.505

[10]

Jorge Groisman. Expansive homeomorphisms of the plane. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 213-239. doi: 10.3934/dcds.2011.29.213

[11]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399

[12]

Alfonso Artigue. Lipschitz perturbations of expansive systems. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1829-1841. doi: 10.3934/dcds.2015.35.1829

[13]

Keonhee Lee, Arnoldo Rojas. Eventually expansive semiflows. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022102

[14]

Shaotao Hu, Yuanheng Wang, Bing Tan, Fenghui Wang. Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of nonexpansive mappings in Hilbert spaces. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022060

[15]

Thierry Daudé, Damien Gobin, François Nicoleau. Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds. Inverse Problems and Imaging, 2016, 10 (3) : 659-688. doi: 10.3934/ipi.2016016

[16]

Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621

[17]

Lijia Yan. Some properties of a class of $(F,E)$-$G$ generalized convex functions. Numerical Algebra, Control and Optimization, 2013, 3 (4) : 615-625. doi: 10.3934/naco.2013.3.615

[18]

Mohammad Eslamian, Ahmad Kamandi. A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021210

[19]

John Banks. Topological mapping properties defined by digraphs. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 83-92. doi: 10.3934/dcds.1999.5.83

[20]

Mads Kyed. On a mapping property of the Oseen operator with rotation. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1315-1322. doi: 10.3934/dcdss.2013.6.1315

 Impact Factor: 

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]