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Demonstratio Mathematica

Editor-in-Chief: Vetro, Calogero

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Volume 51, Issue 1


Approximation solvability for a system of implicit nonlinear variational inclusions with Η-monotone operators

Jong Kyu Kim / Muhammad Iqbal Bhat
Published Online: 2018-11-08 | DOI: https://doi.org/10.1515/dema-2018-0020


In this paper, we introduce and study a new system of variational inclusions which is called a system of nonlinear implicit variational inclusion problems with A-monotone and H-monotone operators in semi-inner product spaces. We define the resolvent operator associated with A-monotone and H-monotone operators and prove its Lipschitz continuity. Using resolvent operator technique, we prove the existence and uniqueness of solution for this new system of variational inclusions. Moreover, we suggest an iterative algorithm for approximating the solution of this system and discuss the convergence analysis of the sequences generated by the iterative algorithm under some suitable conditions.

Keywords: system of nonlinear implicit variational inclusion problem; A- and H-monotone operators; semi-inner product space; resolvent operator; iterative algorithm and convergence analysis

MSC 2010: 49J40; 49J53; 47H06


  • [1] Hassouni A., Moudafi A., A perturbed algorithm for variational inclusions, J. Math. Anal. Appl., 1994, 185(3), 706-712Google Scholar

  • [2] Adly S., Perturbed algorithms and sensitivity analysis for a general class of variational inclusions, J.Math. Anal. Appl., 1996, 201(2), 609-630Google Scholar

  • [3] Huang N. J., Generalized nonlinear variational inclusions with noncompact valued mappings, Appl. Math. Lett., 1996, 9(3), 25-29CrossrefGoogle Scholar

  • [4] Ding X. P., Perturbed proximal point algorithm for generalized quasi-variational inclusions, J.Math. Anal. Appl., 1997, 210(1), 88-101Google Scholar

  • [5] Kazmi K. R., Mann and Ishikawa type perturbed iterative algorithms for generalized quasivariational inclusions, J. Math. Anal. Appl., 1997, 209(2), 572-584Google Scholar

  • [6] Kazmi K. R., Bhat M. I., Iterative algorithm for a system of nonlinear variational-like inclusions, Comput. Math. Appl., 2004, 48(12), 1929-1935CrossrefGoogle Scholar

  • [7] Kazmi K. R., Bhat, M. I., Convergence and stability of iterative algorithms for generalized set-valued variational-like inclusions in Banach spaces, Appl. Math. Comput., 2005, 166(1), 164-180Google Scholar

  • [8] Bhat M. I., Zahoor B., Generalized variational-like inclusion problem involving (H(·, ·) − _)-monotone operators in Banach spaces, J. Nonlinear Anal. Optim., 2017, 8(1), 7-19Google Scholar

  • [9] Bhat M. I., Zahoor B., (H(·, ·), _)-monotone operatorswith an application to a system of set-valued variational-like inclusions in Banach spaces, Nonlinear Funct. Anal. Appl., 2017, 22(3), 673-692Google Scholar

  • [10] Ceng L. C., Wen C. F., Yao Y., Iteration approaches to hierarchial variational inequalities for infinite nonexpansive mappings and finding zero points of m-accretive operators, J. Nonlinear Var. Anal., 2017, 1(2), 213-235Google Scholar

  • [11] Chang S. S., Kim J. K., Kim K. H., On the existence and iterative approximation problems of solutions for set-valued variational inclusions in Banach spaces, J. Math. Anal. Appl., 2002, 268(1), 89-108Google Scholar

  • [12] Fang Y. P., Huang N. J., H-monotone operator and resolvent operator technique for variational inclusions, Appl. Math. Comput., 2003, 145(2-3), 795-803Google Scholar

  • [13] Fang Y. P., Huang N. J., Thompson H. B., A new system of variational inclusions with (H, _) monotone operators in Hilbert spaces, Comput. Math. Appl., 2005, 49(2-3), 365-374CrossrefGoogle Scholar

  • [14] He X. F., Lao J., He Z., Iterative methods for solving variational inclusions in Banach spaces, J. Comput. Appl. Math., 2007, 203(1), 80-86Web of ScienceGoogle Scholar

  • [15] Huang N. J., Fang Y. P., Generalized m-accretive mappings in Banach spaces, J. Sichuan University, 2001, 38(4), 591-592Google Scholar

  • [16] Huang N. J., Fang Y. P., Cho Y. J., Perturbed three-step approximation processes with errors for a class of general implicit variational inclusions, J. Nonlinear Convex Anal., 2003, 4(2), 301-308Google Scholar

  • [17] Kalia R. N., Verma R. U., H-monotone nonlinear variational inclusion systems, Nonlinear Funct. Anal. Appl., 2006, 11(2), 195-200Google Scholar

  • [18] Kim J. K., Kim D. S., A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces, J. Convex Anal., 2004, 11(1), 235-243Google Scholar

  • [19] Kim J. K., Buong Ng., New explicit iteration method for variational inequalities on the set of common fixed points for a finite family of nonexpansive mappings, J. Inequal. Appl., 2013, 2013: 419Web of ScienceCrossrefGoogle Scholar

  • [20] Kim J. K., Kim K. S., New systems of generalized mixed variational inequalities with nonlinear mappings in Hilbert spaces, J. Comput. Anal. Appl., 2010, 12(3), 601-612Google Scholar

  • [21] Lan H. Y., Kim J. H., Cho Y. J., On a new system of nonlinear A-monotone multivalued variational inclusions, J. Math. Anal. Appl., 2007, 327(1), 481-493CrossrefGoogle Scholar

  • [22] Lee C. H., Ansari Q. H., Yao J. C., A perturbed algorithm for strongly nonlinear variational like inclusions, Bull. Aust. Math. Soc., 2000, 62(3), 417-426Google Scholar

  • [23] Li X., Huang N. J., Graph convergence for the H(·, ·)-accretive operators in Banach spaces with an application, Appl. Math. Comput., 2011, 217(22), 9053-9061Google Scholar

  • [24] Luo X. P., Huang N. J., Generalized H − _-accretive operators in Banach spaces with an application to variational inclusions, Appl. Math. Mech., (English Edition), 2010, 31(4), 501-510CrossrefWeb of ScienceGoogle Scholar

  • [25] Alimohammady M., Balooce J., Cho Y. J., Roohi M., Iterative algorithms for new class of extended general nonconvex setvalued variational inequalities, Nonlinear Anal. TMA, 2010, 73, 3907-3923Google Scholar

  • [26] Sahu N. K., Nahak C., Nanda S., Graph convergence and approximation solvability of a class of implicit variational inclusion problems in Banach spaces, J. Indian Math. Soc., 2014, 81(1-2), 155-172Google Scholar

  • [27] Sahu N. K., Mohapatra R. N., Nahak C., Nanda S., Approximation solvability of a class of A-monotone implicit variational inclusion problems in semi-inner product spaces, Appl. Math. Comput., 2014, 236, 109-117Web of ScienceGoogle Scholar

  • [28] Shan S. Q., Xiao Y. B., Huang N. J., A new system of generalized implicit set-valued variational inclusions in Banach spaces, Nonlinear Funct. Anal. Appl., 2017, 22(5), 1091-1105Google Scholar

  • [29] Verma R. U., General nonlinear variational inclusion problems involving A-monotone mappings, Appl. Math. Lett., 2006, 19(9), 960-963CrossrefGoogle Scholar

  • [30] Verma R. U., A generalization to variational convergence of operators, Adv. Nonlinear Var. Inequal., 2008, 11, 97-101Google Scholar

  • [31] Verma R. U., On general over-relaxed proximal point algorithm and applications, Optimization, 2011, 60(4), 531-536Google Scholar

  • [32] Verma R. U., General class of implicit variational inclusions and graph convergence on A-maximal relaxed monotonicity, J. Optim. Theory Appl., 2012, 155(1), 196-214Google Scholar

  • [33] Xu B., Iterative schemes for generalized implicit quasi variational inclusions, Nonlinear Funct. Anal. Appl., 2002, 7(2), 199- 211Google Scholar

  • [34] Zou Y. Z., Huang N. J., A new system of variational inclusions involving H(·; ·)-accretive operator in Banach spaces, Appl. Math. Comput., 2009, 212(1), 135-144Web of ScienceGoogle Scholar

  • [35] Akram M., Chen J.-W., Dilshad M., Generalized Yosida approximation operators with an application to a system of Yosida inclusions, J. Nonlinear Funct. Anal., 2018, Article id 17Google Scholar

  • [36] Aubin J. P., Frankowska H., Set-Valued Analysis, Birkhäuser, Cambridge, MA, 1990Google Scholar

  • [37] Bynum W. L., Weak parallelogram laws for Banach spaces, Canad. Math. Bull., 1976, 19(3), 269-275CrossrefGoogle Scholar

  • [38] Giles J. R., Classes of semi-inner product spaces, Trans. Amer. Math. Soc., 1967, 129, 436-446Google Scholar

  • [39] Lumer G., Semi inner product spaces, Trans. Am. Math. Soc., 1961, 100, 29-43Google Scholar

  • [40] Nadler S. B., Multivalued contraction mappings, Pacific J. Math., 1969, 30(2), 475-488Google Scholar

  • [41] Rockafellar R. T., Wets R. J. B., Variational Analysis, Springer, Berlin, 1998Google Scholar

  • [42] Xu H. K., Inequalities in Banach spaces with applications, Nonlinear Anal. TMA, 1991, 16(12), 1127-1138.Google Scholar

About the article

Received: 2018-04-06

Accepted: 2018-08-21

Published Online: 2018-11-08

Published in Print: 2018-10-01

Citation Information: Demonstratio Mathematica, Volume 51, Issue 1, Pages 241–254, ISSN (Online) 2391-4661, DOI: https://doi.org/10.1515/dema-2018-0020.

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© by Jong Kyu Kim, Muhammad Iqbal Bhat, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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