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http://dx.doi.org/10.22771/nfaa.2021.26.04.08

ON OPTIMAL SOLUTIONS OF WELL-POSED PROBLEMS AND VARIATIONAL INEQUALITIES  

Ram, Tirth (Department of Mathematics, University of Jammu)
Kim, Jong Kyu (Department of Mathematics Education, Kyungnam University)
Kour, Ravdeep (Department of Mathematics, University of Jammu)
Publication Information
Nonlinear Functional Analysis and Applications / v.26, no.4, 2021 , pp. 781-792 More about this Journal
Abstract
In this paper, we study well-posed problems and variational inequalities in locally convex Hausdorff topological vector spaces. The necessary and sufficient conditions are obtained for the existence of solutions of variational inequality problems and quasi variational inequalities even when the underlying set K is not convex. In certain cases, solutions obtained are not unique. Moreover, counter examples are also presented for the authenticity of the main results.
Keywords
Variational inequality; well-posed problem; global minimizer; optimal solutions;
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1 Y.M. Wang, Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. App.9 (2016), 1178-1192.   DOI
2 J. Hadamard, Sur les problemes aux derivees partielles et leur signiication physique, Princeton Univ. Bull., 13 (1902), 49-52.
3 N. Kikuchi and J.T. Oden, Contact Problems in Elasticity, SIAM, (1987).
4 J.K. Kim, Sensitivity analysis for general nonlinear nonvex set-valued variational inequalities in Banach spaces, J. of Comput. Anal. Appl., 22(2) (2017), 327-335,
5 M.A. Noor, On a class of variational inequalities, J. Math Anal. Appl., 128 (1987), 138-155.   DOI
6 R. Larsen, Functional Analysis, Marcel Dekker, Inc. New York, (1973).
7 J.K. Kim, P.N. Anh and T.T.H. Anh and N.D. Hien, Projection methods for solving the variational inequalities involving unrelated nonexpansive mappings, J. Nonlinear and Convex Anal., 21(11) (2020), 2517-2537.
8 J.K. Kim and Salahuddin, Local sharp vector variational type inequality and optimization problems, Mathematics,8(10):1844, (2020), 1-10. https://doi.org/10.3390/math8101844   DOI
9 D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
10 R. Luccheti and F. Patrone, A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities, Numer. Funct. Anal.Optim., 3 (1981), 461-476.   DOI
11 R. Luccheti and F. Patrone, Some properties of well posed variational equalities governed by linear operators, Numer. Funct. Anal.Optim., 5 (1982-1983), 349-361.   DOI
12 G. Hartmann and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math., 115 (1966), 271-310.   DOI
13 Q.H. Ansari, E. Kobis and J.C. Yao, Vector Variational Inequalities and Vector Optimization International Publishing AG, Switzerland, (2018).
14 M. Balaj, Stampacchia variational inequality with weak convex mappings, A Journal of Mathematical Programming and Operations Research, 67 (2018), 1571-1577.
15 J.K. Kim and Salahuddin, System of hierarchical nonlinear mixed variational inequalities, Nonlinear Funct. Anal. and Appl., 24(2) (2019), 207-220.
16 X.X. Huang and X.Q. Yang, Generalized Levitin-Polyak well-posedness in constrained optimization, SIAM J. Optim, 17 (2006), 243-258.   DOI
17 J.K. Kim, Salahuddin and W.H. Lim, Solutions of general variational inequality problems in Banach spaces, Linear and Nonlinear Anal., 6(3) (2020), 333-345.
18 E.S. Levitin and B.T. Polyak, Convergence of minimizing sequence in conditional extremum problems, Sov. Math. Dokl. 7 (1966), 764-767.
19 M. Sofonea and Y.B. Xiao, On the well-posedness in the sense of Tykhonov, Journal of Optimization Theory and Applications 183 (2019), 139-157.   DOI
20 A.H. Siddiqi, Q.H. Ansari and M.F. Khan Variational-like inequalities for multivalued maps, Indian Journal of Pure and Applied Mathematics, 30 (1999), 161-166.
21 P.N. Anh, H.T.C. Thach and J.K. Kim, Proximal-like subgradient methods for solving multi-valued variational inequalities, Nonlinear Funct. Anal. Appl, 25(3) (2020), 437-451, https://doi.org/10.22771/nfaa.2020.25.03.03   DOI
22 S.N. Mishra, P.K. Das and S.K. Mishra, On generalized harmonic vector variational inequalities using HC*- condition, Nonlinear Funct. Anal. Appl, 24(3) (2019), 639-649, https://doi.org/10.22771/nfaa.2019.24.03.14   DOI
23 M.A. Noor, Well-posed variational inequalities, J. Appl. Math and Computing Vol., 11 (2003), 165-172.   DOI
24 G. Fichera, Problemi elastostatici con uincoli unilatesali it problemadi signorini con ambigue condizioni al contorno, Atti. Acad. Naz. Lincei, Mem cl. Acta Mat. Nature Sez. La, 7 (1963-1964), 91-140.
25 A.S. Antipin, M. Jacimovic and N. Mijajlovic, Extragradient method for solving quasi-variational inequalities, Optimization, 67(1) (2018), 103-112.   DOI
26 C. Baiocchi and A. Capelo, Variational and Quasi Variational Inequalities, J. Wiley and Sons, New York, London, (1984).
27 J. Crank, Free and Moving Boundary Value Problems, Clarendon press, Oxford, U.K, (1984).
28 A.L. Dontchev and T. Zolezzi, Well Posed Optimization Problems, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, 1993.
29 R. Glowinski, J. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North Holland, New York, (1981).
30 X.X. Huang, Extended and strongly extended well-posedness of set valued optimization problems, Math Methods Oper. Res. 53 (2001), 101-116.   DOI
31 G. Wang, S.S. Chang, Salahuddin and J.A Liu, Generalized vector variational inequalities and applications, PanAmerican Mathematical Journal, 26 (2016), 77-88.
32 R. Hu, Equivalence results of well-posedness for split variational-hemivariational inequalities, North Holland, J. Nonlinear Convex Anal. 20 (2019), 447-459.
33 M. Sofonea and Y.B. Xiao,Tykonov Well-posedness of elliptic variationalhemivariational inequalities, Electron. J. Differ. Equ. 64 (2018), 1-19.
34 A.N. Tykhonov, On the stability of functional optimization problems, USSR, Comput. Math. Math. Phys. 6 (1966), 28-33.   DOI