Abstract and Applied Analysis

Stable Perturbed Iterative Algorithms for Solving New General Systems of Nonlinear Generalized Variational Inclusion in Banach Spaces

Ting-jian Xiong and Heng-you Lan

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Abstract

We introduce and study a new general system of nonlinear variational inclusions involving generalized m -accretive mappings in Banach space. By using the resolvent operator technique associated with generalized m -accretive mappings due to Huang and Fang, we prove the existence theorem of the solution for this variational inclusion system in uniformly smooth Banach space, and discuss convergence and stability of a class of new perturbed iterative algorithms for solving the inclusion system in Banach spaces. Our results presented in this paper may be viewed as an refinement and improvement of the previously known results.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 659870, 11 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425048203

Digital Object Identifier
doi:10.1155/2014/659870

Mathematical Reviews number (MathSciNet)
MR3272207

Zentralblatt MATH identifier
07022843

Citation

Xiong, Ting-jian; Lan, Heng-you. Stable Perturbed Iterative Algorithms for Solving New General Systems of Nonlinear Generalized Variational Inclusion in Banach Spaces. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 659870, 11 pages. doi:10.1155/2014/659870. https://projecteuclid.org/euclid.aaa/1425048203


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References

  • R. P. Agarwal, Y. J. Cho, J. Li, and N. J. Huang, “Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mapping in q-uniformly smooth Banach Spaces,” Journal of Mathematical Analysis and Applications, vol. 272, no. 2, pp. 435–447, 2002.
  • R. P. Agarwal, J. W. Chen, Y. J. Cho, and Z. P. Wan, “Stability analysis for parametric generalized vector quasi-variational-like inequality problems,” Journal of Inequalities and Applications, vol. 2012, article 57, 2012.
  • M. Alimohammady and M. Roohi, “A system of generalized variational inclusion problems involving ($A,\eta $)-monotone mappings,” Filomat, vol. 23, no. 1, pp. 13–20, 2009.
  • X. P. Ding, “General algorithm of solutions for nonlinear variational inequalities in Banach space,” Computers & Mathematics with Applications, vol. 34, no. 9, pp. 131–137, 1997.
  • X. P. Ding, “Iterative process with errors to locally strictly pseudocontractive maps in Banach spaces,” Computers & Mathematics with Applications, vol. 32, no. 10, pp. 91–97, 1996.
  • X. P. Ding, “Perturbed proximal point algorithms for generalized quasivariational inclusions,” Journal of Mathematical Analysis and Applications, vol. 210, no. 1, pp. 88–101, 1997.
  • Y. P. Fang, N. J. Huang, and H. B. Thompson, “A new system of variational inclusions with $(H,\eta )$-monotone operators in Hilbert spaces,” Computers & Mathematics with Applications, vol. 49, no. 2-3, pp. 365–374, 2005.
  • Y. P. Fang and N. J. Huang, “\emphH-monotone operator and resolvent operator technique for variational inclusion,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 795–803, 2003.
  • F. Gürsoy, V. Karakaya, and B. E. Rhoades, “Some convergence and stability results for the Kirk multistep and Kirk-SP fixed point iterative algorithms,” Abstract and Applied Analysis, vol. 2014, Article ID 806537, 12 pages, 2014.
  • N. J. Huang and Y. P. Fang, “Generalized m-accretive mappings in Banach spaces,” Journal of Sichuan University, vol. 38, no. 4, pp. 591–592, 2001.
  • N. J. Huang, Y. P. Fang, and C. X. Deng, “Nonlinear variational inclusions involving generalized m-accretive mappings,” in Preceedings of the Bellman Continum: International Workshop on Uncertain Systems and Soft Computing, pp. 323–327, Beijing, China, 2002.
  • J. U. Jeong, “Sensitivity analysis for a system of extended generalized nonlinear quasi-variational inclusions in $q$-uniformly smooth Banach spaces,” International Mathematical Forum, vol. 7, no. 3, pp. 2465–2480, 2012.
  • M. M. Jin, “Iterative algorithm for a new system of nonlinear set-valued variational inclusions involving $(H,\eta )$-monotone mappings,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 3, article 114, 2006.
  • M. M. Jin, “A new system of general nonlinear variational inclusions involving (A, $\eta $)-accretive mappings in Banach Spaces,” Mathematical Inequalities and Applications, vol. 11, no. 4, pp. 783–794, 2008.
  • M. M. Jin, “Perturbed iterative algorithms for generalized nonlinear set-valued quasivariational inclusions involving generalized m-accretive mappings,” Journal of Inequalities and Applications, vol. 2007, Article ID 29863, 12 pages, 2007.
  • M. M. Jin, “Perturbed algorithm and stability for strongly nonlinear quasi-variational inclusion involving $H$-accretive operators,” Mathematical Inequalities & Applications, vol. 9, no. 4, pp. 771–779, 2006.
  • K. R. Kazmi and M. I. Bhat, “Iterative algorithm for a system of nonlinear variational-like inclusions,” Computers & Mathematics with Applications, vol. 48, no. 12, pp. 1929–1935, 2004.
  • K. R. Kazmi and M. I. Bhat, “Convergence and stability of iterative algorithms of generalized set-valued variational-like inclusions in Banach spaces,” Applied Mathematics and Computation, vol. 166, no. 1, pp. 164–180, 2005.
  • H. Y. Lan, “Stability of perturbed iterative algorithm for solving a system of generalized nonlinear equations,” Nonlinear Functional Analysis and Applications, vol. 14, no. 1, pp. 1–11, 2009.
  • H. Y. Lan and Q. K. Liu, “Iterative approximation for a system of nonlinear variational inclusions involving generalized $m$-accretive mappings,” Nonlinear Analysis Forum, vol. 9, no. 1, pp. 33–42, 2004.
  • H. Y. Lan and J. K. Kim, “Stable perturbed iteration procedures for solving new strongly nonlinear operator inclusions in Banach spaces,” Nonlinear Functional Analysis and Application, vol. 18, no. 3, pp. 433–444, 2013.
  • H. G. Li, “A nonlinear inclusion problem involving $(\alpha ,\lambda )$ -NODM set-valued mappings in ordered Hilbert space,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1384–1388, 2012.
  • H. G. Li, D. Qiu, J. M. Zheng, and M. M. Jin, “Perturbed Ishikawa-hybrid quasi-proximal point algorithm with accretive mappings for a fuzzy system,” Fixed Point Theory and Applications, vol. 2013, article 281, 2013.
  • H. G. Li, D. Qiu, and M. M. Jin, “GNM ordered variational inequality system with ordered Lipschitz continuous mappings in an ordered Banach space,” Journal of Inequalities and Applications, vol. 2013, article 514, 2013.
  • H. G. Li, D. Qiu, and Y. Zou, “Characterizations of weak-ANODD set-valued mappings with applications to an approximate solution of GNMOQV inclusions involving $\oplus $ operator in ordered Banach spaces,” Fixed Point Theory and Applications, vol. 2013, article 241, 2013.
  • H. G. Li, A. J. Xu, and M. M. Jin, “A Hybrid proximal point three-step algorithm for nonlinear set-valued quasi-variational inclusions system involving $\left(A,\eta \right)$-accretive mappings,” Fixed Point Theory and Applications, vol. 2010, Article ID 635382, 2010.
  • H. G. Li, A. J. Xu, and M. M. Jin, “An Ishikawa-hybrid proximal point algorithm for nonlinear set-valued inclusions problem based on $(A,\eta )$-accretive framework,” Fixed Point Theory and Applications, vol. 2010, Article ID 501293, 12 pages, 2010.
  • Z. Q. Liu, J. S. Ume, and S. M. Kang, “On a system of nonlinear variational inclusions with H$_{h,n}$-monotone operators,” Abstract and Applied Analysis, vol. 2012, Article ID 643828, 21 pages, 2012.
  • J. W. Peng and L. J. Zhao, “General system of $A$-monotone nonlinear variational inclusions problems with applications,” Journal of Inequalities and Applications, vol. 2009, Article ID 364615, 13 pages, 2009.
  • P. Sunthrayuth and P. Kumam, “Iterative algorithms approach to a general system of nonlinear variational inequalities with perturbed mappings and fixed point problems for nonexpansive semigroups,” Journal of Inequalities and Applications, vol. 2012, article 133, 2012.
  • S. Saewan and P. Kumam, “Existence and algorithm for solving the system of mixed variational inequalities in Banach spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 413468, 15 pages, 2012.
  • M. J. Shang, X. F. Su, and X. L. Qin, “An iterative method for a variational inequality and a fixed point problem for nonexpansive mappings,” Acta Mathematica Scientia A, vol. 30, no. 4, pp. 1126–1137, 2010.
  • Y. C. Xu, X. F. He, Z. B. Hou, and Z. He, “Generalized projection methods for Noor variational inequalities in Banach spaces,” Acta Mathematica Scientia Shuxue Wuli Xuebao: Chinese Edition, vol. 30, no. 3, pp. 808–817, 2010.
  • J. H. Zhu, S. S. Chang, and M. Liu, “Algorithms for a system of general variational inequalities in Banach spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 580158, 18 pages, 2012.
  • H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991.
  • L. S. Liu, “Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114–125, 1995. \endinput