On the strong convergence of a general-type Krasnosel'skii–Mann's algorithm depending on the coefficients

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• 作者：Newab Hussain ; Giuseppe Marino ; Luigi Muglia…
• 关键词：Iterative methods ; nonexpansive mappings ; strongly monotone operators ; variational inequality
• 刊名：Journal of Fixed Point Theory and Applications
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：18
• 期：1
• 页码：1-25
• 全文大小：852 KB
• 参考文献：1.Atsushiba S., Takahashi W.: Strong convergence theorems for a finite family of nonexpansive mappings and applications. Indian J. Math. 41, 435–453 (1999)MathSciNet MATH
  2.Baillon J.-B., Haddad G.: Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones. Israel J. Math. 26, 137–150 (1977)CrossRef MathSciNet MATH
  3.F. Cianciaruso, G. Marino, L. Muglia and Y. Yao, On a two-step algorithm for hierarchical fixed point problems and variational inequalities. J. Inequal. Appl. 2009 (2009), Article ID 208692, 13 pages.
  4.K. Deimling, Nonlinear Functional Analysis. Dover Publications, 2010 (originally publlished by Springer, Berlin, 1985).
  5.Halpern B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957–961 (1967)CrossRef MathSciNet MATH
  6.Hundal H. S.: An alternating projection that does not converge in norm. Nonlinear Anal. 57, 35–61 (2004)CrossRef MathSciNet MATH
  7.N. Hussain, G. Marino and A. A. N. Abdou, On Mann’s method with viscosity for nonexpansive and nonspreading mappings in Hilbert spaces. Abstr. Appl. Anal. 2014 (2014), Article ID 152530, 11 pages.
  8.N. Hussain, G. Marino, L. Muglia and B. Alamri, On some Mann’s type iterative algorithms. Fixed Point Theory Appl. 2015 (2015), doi:10.​1186/​s13663-015-0267-8 , 16 pages.
  9.Hussain N., Takahashi W.: Weak and strong convergence theorems for semigroups of mappings without continuity in Hilbert spaces. J. Nonlinear Convex Anal. 14, 769–783 (2013)MathSciNet MATH
  10.Ishikawa S.: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. Amer. Math. Soc. 59, 65–71 (1976)CrossRef MathSciNet MATH
  11.Maingé P.-E., Moudafi A.: Strong convergence of an iterative method for hierarchical fixed-points problems. Pac. J. Optim. 3, 529–538 (2007)MathSciNet MATH
  12.Mann W. R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)CrossRef MathSciNet MATH
  13.G. Marino, F. Cianciaruso and N. Hussain, Ergodic approximations via matrix regularization approach. Fixed Point Theory Appl. 2015 (2015), doi:10.​1186/​s13663-015-0321-6 , 11 pages.
  14.Marino G., Muglia L.: On the auxiliary mappings generated by a family of mappings and solutions of variational inequalities problems. Optim. Lett. 9, 263–282 (2015)CrossRef MathSciNet MATH
  15.Marino G., Muglia L., Yao Y.: The uniform asymptotical regularity of families of mappings and solutions of variational inequality problems. J. Nonlinaer Convex Anal. 15, 477–492 (2014)MathSciNet MATH
  16.Marino G., Xu H.-K.: Explicit hierarchical fixed point approach to variational inequalities. J. Optim. Theory Appl. 149, 61–78 (2011)CrossRef MathSciNet MATH
  17.Moudafi A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000)CrossRef MathSciNet MATH
  18.A. Moudafi and P.-E. Maingé, Towards viscosity approximations of hierarchical fixed-points problems. Fixed Point Theory Appl. 2006 (2006), Art. ID 95453, 10 pages.
  19.Nakajo K., Takahashi W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)CrossRef MathSciNet MATH
  20.Reich S., Xu H.-K.: An iterative approach to a constrained least squares problem. Abstr. Appl. Anal. 8, 503–512 (2003)CrossRef MathSciNet
  21.Shimoji K., Takahashi W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J. Math. 5, 387–404 (2001)MathSciNet MATH
  22.Takahashi W., Toyoda M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)CrossRef MathSciNet MATH
  23.Xu H.-K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. (2) 66, 240–256 (2002)CrossRef MathSciNet MATH
  24.Xu H.-K.: Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl. 150, 360–378 (2011)CrossRef MathSciNet MATH
  25.Xu H.-K., Kim T. H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185–201 (2003)CrossRef MathSciNet MATH
  
• 作者单位：Newab Hussain (1)
  Giuseppe Marino (1) (2)
  Luigi Muglia (2)
  Afrah A. N. Abdou (1)
  
  1. Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia
  2. Dipartimento di Matematica, Universitá della Calabria, 87036, Arcavacata di Rende (CS), Italy
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Mathematical Methods in Physics
  
• 出版者：Birkh盲user Basel
• ISSN：1661-7746

文摘

Let H be a Hilbert space, \({(W_n)_{n \in \mathbb{N}}}\) a suitable family of mappings, S a nonexpansive mapping and D a strongly monotone operator. We are interested in the strong convergence of the general scheme $$x_{n + 1} = \gamma x_{n} + (1 - \gamma)W_{n} (\alpha_{n}S_{x_{n}} + (1 - \alpha_{n})(I - \mu_{n}D)x_{n}),\quad \gamma \in [0, 1),$$in dependence of the coefficients \({(\alpha_{n})_{n \in \mathbb{N}}}\) and \({(\mu_{n})_{n \in \mathbb{N}}}\) . Keywords Iterative methods nonexpansive mappings strongly monotone operators variational inequality Mathematics Subject Classification 47H09 58E35 47H10 65J25 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (25) References1.Atsushiba S., Takahashi W.: Strong convergence theorems for a finite family of nonexpansive mappings and applications. Indian J. Math. 41, 435–453 (1999)MathSciNetMATH2.Baillon J.-B., Haddad G.: Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones. Israel J. Math. 26, 137–150 (1977)CrossRefMathSciNetMATH3.F. Cianciaruso, G. Marino, L. Muglia and Y. Yao, On a two-step algorithm for hierarchical fixed point problems and variational inequalities. J. Inequal. Appl. 2009 (2009), Article ID 208692, 13 pages.4.K. Deimling, Nonlinear Functional Analysis. Dover Publications, 2010 (originally publlished by Springer, Berlin, 1985).5.Halpern B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957–961 (1967)CrossRefMathSciNetMATH6.Hundal H. S.: An alternating projection that does not converge in norm. Nonlinear Anal. 57, 35–61 (2004)CrossRefMathSciNetMATH7.N. Hussain, G. Marino and A. A. N. Abdou, On Mann’s method with viscosity for nonexpansive and nonspreading mappings in Hilbert spaces. Abstr. Appl. Anal. 2014 (2014), Article ID 152530, 11 pages.8.N. Hussain, G. Marino, L. Muglia and B. Alamri, On some Mann’s type iterative algorithms. Fixed Point Theory Appl. 2015 (2015), doi:10.​1186/​s13663-015-0267-8, 16 pages.9.Hussain N., Takahashi W.: Weak and strong convergence theorems for semigroups of mappings without continuity in Hilbert spaces. J. Nonlinear Convex Anal. 14, 769–783 (2013)MathSciNetMATH10.Ishikawa S.: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. Amer. Math. Soc. 59, 65–71 (1976)CrossRefMathSciNetMATH11.Maingé P.-E., Moudafi A.: Strong convergence of an iterative method for hierarchical fixed-points problems. Pac. J. Optim. 3, 529–538 (2007)MathSciNetMATH12.Mann W. R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)CrossRefMathSciNetMATH13.G. Marino, F. Cianciaruso and N. Hussain, Ergodic approximations via matrix regularization approach. Fixed Point Theory Appl. 2015 (2015), doi:10.​1186/​s13663-015-0321-6, 11 pages.14.Marino G., Muglia L.: On the auxiliary mappings generated by a family of mappings and solutions of variational inequalities problems. Optim. Lett. 9, 263–282 (2015)CrossRefMathSciNetMATH15.Marino G., Muglia L., Yao Y.: The uniform asymptotical regularity of families of mappings and solutions of variational inequality problems. J. Nonlinaer Convex Anal. 15, 477–492 (2014)MathSciNetMATH16.Marino G., Xu H.-K.: Explicit hierarchical fixed point approach to variational inequalities. J. Optim. Theory Appl. 149, 61–78 (2011)CrossRefMathSciNetMATH17.Moudafi A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000)CrossRefMathSciNetMATH18.A. Moudafi and P.-E. Maingé, Towards viscosity approximations of hierarchical fixed-points problems. Fixed Point Theory Appl. 2006 (2006), Art. ID 95453, 10 pages.19.Nakajo K., Takahashi W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)CrossRefMathSciNetMATH20.Reich S., Xu H.-K.: An iterative approach to a constrained least squares problem. Abstr. Appl. Anal. 8, 503–512 (2003)CrossRefMathSciNet21.Shimoji K., Takahashi W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J. Math. 5, 387–404 (2001)MathSciNetMATH22.Takahashi W., Toyoda M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)CrossRefMathSciNetMATH23.Xu H.-K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. (2) 66, 240–256 (2002)CrossRefMathSciNetMATH24.Xu H.-K.: Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl. 150, 360–378 (2011)CrossRefMathSciNetMATH25.Xu H.-K., Kim T. H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185–201 (2003)CrossRefMathSciNetMATH About this Article Title On the strong convergence of a general-type Krasnosel’skii–Mann’s algorithm depending on the coefficients Journal Journal of Fixed Point Theory and Applications Volume 18, Issue 1 , pp 1-25 Cover Date2016-03 DOI 10.1007/s11784-015-0261-0 Print ISSN 1661-7738 Online ISSN 1661-7746 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Analysis Mathematical Methods in Physics Keywords 47H09 58E35 47H10 65J25 Iterative methods nonexpansive mappings strongly monotone operators variational inequality Authors Newab Hussain (1) Giuseppe Marino (1) (2) Luigi Muglia (2) Afrah A. N. Abdou (1) Author Affiliations 1. Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia 2. Dipartimento di Matematica, Universitá della Calabria, 87036, Arcavacata di Rende (CS), Italy Continue reading... To view the rest of this content please follow the download PDF link above.