Abstract and Applied Analysis
 Abstr. Appl. Anal.
 Volume 2014, Special Issue (2014), Article ID 659870, 11 pages.
Stable Perturbed Iterative Algorithms for Solving New General Systems of Nonlinear Generalized Variational Inclusion in Banach Spaces
Tingjian Xiong and Hengyou 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, Tingjian; Lan, Hengyou. 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
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 quasicontractive mapping in quniformly smooth Banach Spaces,” Journal of Mathematical Analysis and Applications, vol. 272, no. 2, pp. 435–447, 2002.Mathematical Reviews (MathSciNet): MR1930850
Digital Object Identifier: doi:10.1016/S0022247X(02)001506  R. P. Agarwal, J. W. Chen, Y. J. Cho, and Z. P. Wan, “Stability analysis for parametric generalized vector quasivariationallike inequality problems,” Journal of Inequalities and Applications, vol. 2012, article 57, 2012.Mathematical Reviews (MathSciNet): MR2915351
 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.Mathematical Reviews (MathSciNet): MR1483163
 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.Mathematical Reviews (MathSciNet): MR1426206
 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.Mathematical Reviews (MathSciNet): MR1449511
Zentralblatt MATH: 0902.49010
Digital Object Identifier: doi:10.1006/jmaa.1997.5370  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. 23, pp. 365–374, 2005.Mathematical Reviews (MathSciNet): MR2123413
 Y. P. Fang and N. J. Huang, “\emphHmonotone operator and resolvent operator technique for variational inclusion,” Applied Mathematics and Computation, vol. 145, no. 23, pp. 795–803, 2003.Mathematical Reviews (MathSciNet): MR2009301
Digital Object Identifier: doi:10.1016/S00963003(03)002753  F. Gürsoy, V. Karakaya, and B. E. Rhoades, “Some convergence and stability results for the Kirk multistep and KirkSP fixed point iterative algorithms,” Abstract and Applied Analysis, vol. 2014, Article ID 806537, 12 pages, 2014.Mathematical Reviews (MathSciNet): MR3166657
 N. J. Huang and Y. P. Fang, “Generalized maccretive 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 maccretive 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 quasivariational inclusions in $q$uniformly smooth Banach spaces,” International Mathematical Forum, vol. 7, no. 3, pp. 2465–2480, 2012.Mathematical Reviews (MathSciNet): MR2967441
 M. M. Jin, “Iterative algorithm for a new system of nonlinear setvalued variational inclusions involving $(H,\eta )$monotone mappings,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 3, article 114, 2006.Mathematical Reviews (MathSciNet): MR2221353
 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.Mathematical Reviews (MathSciNet): MR2458172
 M. M. Jin, “Perturbed iterative algorithms for generalized nonlinear setvalued quasivariational inclusions involving generalized maccretive mappings,” Journal of Inequalities and Applications, vol. 2007, Article ID 29863, 12 pages, 2007.
 M. M. Jin, “Perturbed algorithm and stability for strongly nonlinear quasivariational inclusion involving $H$accretive operators,” Mathematical Inequalities & Applications, vol. 9, no. 4, pp. 771–779, 2006.Mathematical Reviews (MathSciNet): MR2268184
 K. R. Kazmi and M. I. Bhat, “Iterative algorithm for a system of nonlinear variationallike inclusions,” Computers & Mathematics with Applications, vol. 48, no. 12, pp. 1929–1935, 2004.Mathematical Reviews (MathSciNet): MR2116967
 K. R. Kazmi and M. I. Bhat, “Convergence and stability of iterative algorithms of generalized setvalued variationallike inclusions in Banach spaces,” Applied Mathematics and Computation, vol. 166, no. 1, pp. 164–180, 2005.Zentralblatt MATH: 1079.65073
Mathematical Reviews (MathSciNet): MR2145883
Digital Object Identifier: doi:10.1016/j.amc.2004.04.057  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.Mathematical Reviews (MathSciNet): MR2743801
 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.Mathematical Reviews (MathSciNet): MR2111365
 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 setvalued mappings in ordered Hilbert space,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1384–1388, 2012.Mathematical Reviews (MathSciNet): MR2947412
Digital Object Identifier: doi:10.1016/j.aml.2011.12.007  H. G. Li, D. Qiu, J. M. Zheng, and M. M. Jin, “Perturbed Ishikawahybrid quasiproximal 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.Mathematical Reviews (MathSciNet): MR3212959
 H. G. Li, D. Qiu, and Y. Zou, “Characterizations of weakANODD setvalued 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.Mathematical Reviews (MathSciNet): MR3213083
 H. G. Li, A. J. Xu, and M. M. Jin, “A Hybrid proximal point threestep algorithm for nonlinear setvalued quasivariational inclusions system involving $\left(A,\eta \right)$accretive mappings,” Fixed Point Theory and Applications, vol. 2010, Article ID 635382, 2010.Mathematical Reviews (MathSciNet): MR2661188
 H. G. Li, A. J. Xu, and M. M. Jin, “An Ishikawahybrid proximal point algorithm for nonlinear setvalued inclusions problem based on $(A,\eta )$accretive framework,” Fixed Point Theory and Applications, vol. 2010, Article ID 501293, 12 pages, 2010.Mathematical Reviews (MathSciNet): MR2661199
 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.Mathematical Reviews (MathSciNet): MR2999892
 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.Mathematical Reviews (MathSciNet): MR3085738
Digital Object Identifier: doi:10.1186/1029242X2012133  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.Mathematical Reviews (MathSciNet): MR2722560
 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.Mathematical Reviews (MathSciNet): MR2682954
 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.Mathematical Reviews (MathSciNet): MR2915713
 H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991.Mathematical Reviews (MathSciNet): MR1111623
 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. \endinputMathematical Reviews (MathSciNet): MR1353071
Zentralblatt MATH: 0872.47031
Digital Object Identifier: doi:10.1006/jmaa.1995.1289
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