Some equivalence results for well-posedness of generalized hemivariational inequalities with clarkes generalized directional derivative
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Authors
Lu-Chuan Ceng
- Department of Mathematics, Shanghai Normal University, Shanghai 200234, China.
Yeong-Cheng Liou
- Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan.
Ching-Feng Wen
- Center for Fundamental Science and Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 80708, Taiwan.
Abstract
In this paper, we are devoted to exploring conditions of well-posedness for generalized hemivariational
inequalities with Clarke's generalized directional derivative in re
exive Banach spaces. By using some
equivalent formulations of the generalized hemivariational inequality with Clarke's generalized directional
derivative under different monotonicity assumptions, we establish two kinds of conditions under which the
strong \(\alpha\)-well-posedness and the weak \(\alpha\)-well-posedness for the generalized hemivariational inequality with
Clarke's generalized directional derivative are equivalent to the existence and uniqueness of its solution,
respectively.
Share and Cite
ISRP Style
Lu-Chuan Ceng, Yeong-Cheng Liou, Ching-Feng Wen, Some equivalence results for well-posedness of generalized hemivariational inequalities with clarkes generalized directional derivative, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 5, 2798--2812
AMA Style
Ceng Lu-Chuan, Liou Yeong-Cheng, Wen Ching-Feng, Some equivalence results for well-posedness of generalized hemivariational inequalities with clarkes generalized directional derivative. J. Nonlinear Sci. Appl. (2016); 9(5):2798--2812
Chicago/Turabian Style
Ceng, Lu-Chuan, Liou, Yeong-Cheng, Wen, Ching-Feng. "Some equivalence results for well-posedness of generalized hemivariational inequalities with clarkes generalized directional derivative." Journal of Nonlinear Sciences and Applications, 9, no. 5 (2016): 2798--2812
Keywords
- Generalized hemivariational inequality
- Clarke's generalized directional derivative
- contraction
- well-posedness
- relaxed monotonicity.
MSC
References
-
[1]
S. Carl, V. K. Le, D. Motreanu, Nonsmooth variational problems and their inequalities: comparison principles and applications, Springer, New York (2007)
-
[2]
S. Carl, D. Motreanu, General comparison principle for quasilinear elliptic inclusions, Nonlinear Anal., 70 (2009), 1105-1112.
-
[3]
L.-C. Ceng, H. Gupta, C.-F. Wen, Well-posedness by perturbations of variational-hemivariational inequalities with perturbations, Filomat, 26 (2012), 881-895.
-
[4]
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.
-
[5]
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.
-
[6]
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.
-
[7]
F. H. Clarke, Optimization and nonsmooth analysis, SIAM, Philadelphia (1990)
-
[8]
Y.-P. Fang, N.-J. Huang, J.-C. Yao, Well-posedness of mixed variational inequalities, inclusion problems and fixed-point problems, J. Global Optim., 41 (2008), 117-133.
-
[9]
Y.-P. Fang, N.-J. Huang, J.-C. Yao, Well-posedness by perturbations of mixed variational inequalities in Banach spaces, European J. Oper. Res., 201 (2010), 682-692.
-
[10]
F. Giannessi, A. A. Khan, Regularization of non-coercive quasi variational inequalities, Control Cybernet., 29 (2000), 91-110.
-
[11]
D. Goeleven, D. Mentagui , Well-posed hemivariational inequalities, Numer. Funct. Anal. Optim., 16 (1995), 909-921.
-
[12]
D. Goeleven, D. Motreanu , Variational and hemivariational inequalities, theory, methods and applications, Volume II: Unilateral Problems, Springer, Dordrecht (2003)
-
[13]
R. Hu, Y.-P. Fang, Levitin-Polyak well-posedness by perturbations of inverse variational inequalities, Optim. Lett., 7 (2013), 343-359.
-
[14]
X. X. Huang, X. Q. Yang, Generalized Levitin-Polyak well-posedness in constrained optimization, SIAM J. Optim., 17 (2006), 243-258.
-
[15]
X. X. Huang, X. Q. Yang, D. L. Zhu , Levitin-Polyak well-posedness of variational inequality problems with functional constraints, J. Glob. Optim., 44 (2009), 159-174.
-
[16]
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.
-
[17]
M. B. Lignola, J. Morgan, Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution, J. Glob. Optim., 16 (2000), 57-67.
-
[18]
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.
-
[19]
Z. Liu, D. Motreanu, A class of variational-hemivariational inequalities of elliptic type, Nonlinearity, 23 (2010), 1741-1752.
-
[20]
Z. Liu, J. Zou, Strong convergence results for hemivariational inequalities, Sci. China Ser., 49 (2006), 893-901.
-
[21]
R. Lucchetti, F. Patrone, A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities, Numer. Funct. Anal. Optim., 3 (1981), 461-476.
-
[22]
S. Migorski, A. Ochal, M. Sofonea, Nonlinear inclusions and hemivariational inequalities: models and analysis of contact problems, Springer, New York (2013)
-
[23]
D. Motreanu, P. D. Panagiotopoulos, Minimax theorems and qualitative properties of the solutions of hemivariational inequalities, Kluwer Academic Publishers, Dordrecht (1999)
-
[24]
S. B. Nadler, Multivalued contraction mappings, Pacific J. Math., 30 (1969), 475-488.
-
[25]
Z. Naniewicz, P. D. Panagiotopoulos , Mathematical theory of hemivariational inequalities and applications, Marcel Dekker, New York (1995)
-
[26]
P. D. Panagiotopoulos , Nonconvex energy functions, Hemivariational inequalities and substationarity principles, Acta Mech., 48 (1983), 111-130.
-
[27]
J.-W. Peng, S.-Y. Wu, The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems, Optim. Lett., 4 (2010), 501-512.
-
[28]
A. N. Tykhonov, On the stability of the functional optimization problem, USSR J. Comput. Math. Phys., 6 (1966), 28-33.
-
[29]
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.
-
[30]
Y.-B. Xiao, N.-J. Huang, M.-M. Wong, Well-posedness of hemivariational inequalities and inclusion problems, Taiwanese J. Math., 15 (2011), 1261-1276.
-
[31]
Y.-B. Xiao, X. Yang, N.-J. Huang, Some equivalence results for well-posedness of hemivariational inequalities, J. Global Optim., 61 (2015), 789-802.
-
[32]
E. Zeidler, Nonlinear functional analysis and its applications: Vol. II, Springer, Berlin (1990)
-
[33]
J. Zeng, S. J. Li, W. Y. Zhang, X. W. Xue, Hadamard well-posedness for a set-valued optimization problem, Optim. Lett., 7 (2013), 559-573.
-
[34]
T. Zolezzi , Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257-266.