Well-posedness for systems of time-dependent hemivariational inequalities in Banach spaces

Volume 10, Issue 8, pp 4318--4336 http://dx.doi.org/10.22436/jnsa.010.08.26

Publication Date: August 24, 2017 Submission Date: July 10, 2017

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Authors

Lu-Chuan Ceng - Department of Mathematics, Shanghai Normal University, Shanghai 200234, China. Yeong-Cheng Liou - Department of Healthcare Administration and Medical Informatics, Center for Big Data Analytics and Intelligent Healthcare, and Research Center of Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 807, Taiwan. - Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 807, Taiwan. Jen-Chih Yao - Center for General Education, China Medical University, Taichung 40402, Taiwan. Yonghong Yao - Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China.

Abstract

In this paper, we generalize the concept of \(\alpha\)-well-posedness to a system of time-dependent hemivariational inequalities without Volterra integral terms in Banach spaces. We establish some metric characterizations of \(\alpha\)-well-posedness and prove some equivalence results of strong \(\alpha\)-well-posedness (resp., in the generalized sense) between a system of time-dependent hemivariational inequalities and its derived system of inclusion problems.

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Keywords

• System of time-dependent hemivariational inequalities
• \(\alpha\)-well-posedness
• monotonicity
• Clarke’s generalized gradient
• regularity.

MSC

•  49K40
•  47J20
•  49J52

References

• [1] S. Carl, V. K. Le, D. Motreanu, Nonsmooth variational problems and their inequalities , Comparison principles and applications, Springer Monographs in Mathematics, Springer, New York (2007)
  • View Article
  • MathSciNet
  • Google Scholar


• [2] L.-C. Ceng, H. Gupta, C.-F. Wen , Well-posedness by perturbations of variational-hemivariational inequalities with perturbations, Filomat, 26 (2012), 881–895.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] L.-C. Ceng, N. Hadjisavvas, S. Schaible, J. C. Yao, Well-posedness for mixed quasivariational-like inequalities, J. Optim. Theory Appl., 139 (2008), 109–125.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] L.-C. Ceng, Y. C. Lin, Metric characterizations of \(\alpha\)-well-posedness for a system of mixed quasivariational-like inequalities in Banach spaces, J. Appl. Math., 2012 (2012), 22 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] L.-C. Ceng, C.-F. Wen, Well-posedness by perturbations of generalized mixed variational inequalities in Banach spaces, J. Appl. Math., 2012 (2012), 38 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] L.-C. Ceng, N.-C. Wong, J.-C. Yao, Well-posedness for a class of strongly mixed variational-hemivariational inequalities with perturbations, J. Appl. Math., 2012 (2012), 21 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] L. C. Ceng, J. C. Yao , Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems, Nonlinear Anal., 69 (2008), 4585–4603.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] J.-W. Chen, Y. J. Cho, X.-Q. Ou, Levitin-Polyak well-posedness for set-valued optimization problems with constraints, Filomat, 28 (2014), 1345–1352.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] F. H. Clarke, Optimization and nonsmooth analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1990)
  • Google Scholar


• [10] N. Costea, V. Rădulescu , Hartman-Stampacchia results for stably pseudomonotone operators and non-linear hemivariational inequalities, Appl. Anal., 89 (2010), 175–188.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] Y.-P. Fang, R. Hu, N.-J. Huang , Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints, Comput. Math. Appl., 55 (2008), 89–100.
  • View Article
  • Google Scholar


• [12] F. Giannessi, A. A. Khan , Regularization of non-coercive quasi variational inequalities, Control Cybernet., 29 (2000), 91–110.
  • MathSciNet
  • Google Scholar
  • MATH


• [13] D. Goeleven, D. Mentagui, Well-posed hemivariational inequalities, Numer. Funct. Anal. Optim., 16 (1995), 909–921.
  • View Article
  • MathSciNet
  • Google Scholar


• [14] R. Hu, Y.-P. Huang, N.-J. Huang, M.-M. Wong , Well-posedness of systems of equilibrium problems, Taiwanese J. Math., 14 (2010), 2435–2446.
  • View Article
  • Google Scholar
  • MATH


• [15] X. X. Huang, X. Q. Yang, Generalized Levitin-Polyak well-posedness in constrained optimization, SIAM J. Optim., 17 (2006), 243–258.
  • View Article
  • MathSciNet
  • Google Scholar


• [16] K. Kuratowski , Topology, Vol. II , New edition, revised and augmented, Translated from the French by A. Kirkor, Academic Press, New York-London; Panstwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw (1968)
  • Google Scholar


• [17] B. Lemaire,, Well-posedness, conditioning and regularization of minimization, inclusion and fixed-point problems, Proceedings of the 4th International Conference on Mathematical Methods in Operations Research and 6th Workshop on Well-posedness and Stability of Optimization Problems, Sozopol, (1997), Pliska Stud. Math. Bulgar., 12 (1998), 71–84.
  • MathSciNet
  • Google Scholar


• [18] E. S. Levitin, B. T. Polyak , Convergence of minimizing sequences in problems on the relative extremum, (Russian) Dokl. Akad. Nauk SSSR, 168 (1966), 997–1000.
  • MathSciNet
  • Google Scholar


• [19] X.-B. Li, F.-Q. Xia , Levitin-Polyak well-posedness of a generalized mixed variational inequality in Banach spaces, Nonlinear Anal., 75 (2012), 2139–2153.
  • View Article
  • MathSciNet
  • Google Scholar


• [20] M. B. Lignola , Well-posedness and L-well-posedness for quasivariational inequalities, J. Optim. Theory Appl., 128 (2006), 119–138.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] L.-J. Lin, C.-S. Chuang , Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint, Nonlinear Anal., 70 (2009), 3609–3617.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] Z.-H. Liu , Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities, Inverse Problems, 21 (2005), 13–20.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] R. Lucchetti, F. Patrone , A characterization of Tyhonov well-posedness for minimum problems, with applications to variational inequalities, Numer. Funct. Anal. Optim., 3 (1981), 461–476.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] S. Migórski, A. Ochal, M. Sofonea, Nonlinear inclusions and hemivariational inequalities, Models and analysis of contact problems, Advances in Mechanics and Mathematics, Springer, New York (2013)
  • View Article
  • MathSciNet


• [25] D. Motreanu, P. D. Panagiotopoulos , Minimax theorems and qualitative properties of the solutions of hemivariational inequalities, Nonconvex Optimization and its Applications, Kluwer Academic Publishers, Dordrecht (1999)
  • View Article
  • MathSciNet


• [26] Z. Naniewicz, P. D. Panagiotopoulos , Mathematical theory of hemivariational inequalities and applications, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York (1995)
  • MathSciNet
  • Google Scholar


• [27] P. D. Panagiotopoulos, Nonconvex energy functions, Hemivariational inequalities and substationarity principles, Acta Mech., 48 (1983), 111–130.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] P. D. Panagiotopoulos, Hemivariational inequalities , Applications in mechanics and engineering, Springer-Verlag, Berlin (1993)
  • View Article
  • Google Scholar


• [29] P. D. Panagiotopoulos, M. Fundo, V. Rădulescu , Existence theorems of Hartman-Stampacchia type for hemivariational inequalities and applications, J. Global Optim., 15 (1999), 41–54.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] D. Repovš, C. Varga , A Nash type solution for hemivariational inequality systems, Nonlinear Anal., 74 (2011), 5585– 5590.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] A. N. Tikhonov, On the stability of the functional optimization problem , USSR J. Comput. Math. Math. Phys., 6 (1966), 28–33.
  • View Article
  • Google Scholar


• [32] Y.-M. Wang, Y.-B. Xiao, X. Wang, Y. J. Cho, Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. Appl., 9 (2016), 1178–1192.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [33] Y.-B. Xiao, N.-J. Huang , Browder-Tikhonov regularization for a class of evolution second order hemivariational inequalities, J. Global Optim., 45 (2009), 371–388.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] Y.-B. Xiao, N.-J. Huang, Well-posedness for a class of variational-hemivariational inequalities with perturbations, J. Optim. Theory Appl., 151 (2011), 33–51.
  • View Article
  • MathSciNet
  • Google Scholar


• [35] Y.-B. Xiao, N.-J. Huang, J. Lu, A system of time-dependent hemivariational inequalities with Volterra integral terms, J. Optim. Theory Appl., 165 (2015), 837–853.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [36] Y.-B. Xiao, N.-J. Huang, M.-M. Wong, Well-posedness of hemivariational inequalities and inclusion problems, Taiwanese J. Math., 15 (2011), 1261–1276.
  • View Article
  • MathSciNet
  • Google Scholar


• [37] Y.-H. Yao, M. A. Noor, Y.-C. Liou, S. M. Kang, Iterative algorithms for general multivalued variational inequalities, Abstr. Appl. Anal., 2012 (2012), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [38] Y.-H. Yao, N. Shahzad, An algorithmic approach to the split variational inequality and fixed point problem, J. Nonlinear Convex Anal., 18 (2017), 977–991.
  • MathSciNet
  • Google Scholar
  • MATH


• [39] H. Zegeye, N. Shahzad, Y.-H. Yao, Minimum-norm solution of variational inequality and fixed point problem in Banach spaces, Optimization, 64 (2015), 453–471.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [40] E. Zeidler, Nonlinear functional analysis and its applications, Translated from the German by the author and Leo F. Boron, Springer-Verlag, New York (1990)
  • View Article
  • MathSciNet
  • Google Scholar


• [41] Y.-L. Zhang, Y.-R. He, On stably quasimonotone hemivariational inequalities, Nonlinear Anal., 74 (2011), 3324–3332.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [42] T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257–266.
  • View Article
  • Google Scholar