Fan-hemicontinuity for the gradient of the norm in several reflexive Banach spaces
DOI:
https://doi.org/10.12775/TMNA.2024.061Keywords
Fan-hemicontinuity, topological pseudomonotonicity, type $S_+$ operators, solutions set, variational inequalityAbstract
The present study approaches variational inequalities governed by Fan-hemicontinuous operators. The Fan-hemicontinuity was recently proved (\cite{bogdan:hemicontinuity}) for the gradient of the norm defined on its scalarly-positive, closed convex subdomain of a Hilbert space. The aim of the present study is to establish whether the positive result on Fan-hemicontinuity can be extended to an arbitrary reflexive Banach space. In this matter it is natural to consider the duality map and its topological properties. Based on the weak-weak sequential continuity of the generalized duality map, the discussed property holds on $\ell^p$, $1 < p < +\infty$ and does not hold on $L^p$, $p\neq 2$, thus restricting the space setting where weak compactness results are applicable.References
Y.I. Alber, The regularization method for variational inequalities with nonsmooth unbounded operators in Banach space, Appl. Math. Lett. 6 (1993), no. 4, 63–68, DOI: 10.1016/0893-9659(93)90125-7.
Y.I. Alber, D. Butnariu and G. Kassay, Convergence and stability of a regularization method for maximal monotone inclusions and its applications to convex optimization, Variational Analysis and Applications (F. Giannessi and A. Maugeri, eds.), Kluwer Acad. Publ., Dordrecht, 2004.
M. Bianchi, G. Kassay and R. Pini, Regularization of Brézis pseudomonotone variational inequalities, Set-Valued Var. Anal. 29 (2021), 175–190, DOI: 10.1007/s11228-02000543-3.
M. Bianchi, G. Kassay and R. Pini, Brézis pseudomonotone bifunctions and quasi equilibrium problems via penalization, J. Global Optim. 82 (2022), 483–498, DOI: 10.1007/s10898-021-01088-x
M. Bogdan, Fan-hemicontinuity for the gradient of the norm in Hilbert space, Math. Slovaca 6 (2022), 1567–1572, DOI: 10.1515/ms-2022-0107.
M. Bogdan, Comments on the solutions set of equilibrium problems governed by topological pseudomonotone bifunctions, Lecture notes in Networks and Systems, Springer Nature Switzerland AG, 2022, pp. 696–703.
M. Bogdan and J. Kolumbán, Some regularities for parametric equilibrium problems, J. Global Optim. 44 (2009), 481–492, DOI: 10.1007/s10898-008-9345-3.
H. Brézis, Equations et inéquations nonlinéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble) 18 (1968), 115–175, DOI: 10.5802/aif.280.
F.E. Browder, Multi-valued monotone nonlinear mappings and duality mappings in Banach spaces, Trans. Amer. Math. Soc. 118 (1965), 338–351, DOI 10.1090/S0002-99471965-0180884-9.
F.E. Browder, Fixed point theorems for nonlinear semicontractive mappings in Banach spaces, Arch. Rational Mech. Anal. 21 (1966), 259–269, DOI: 10.1007/BF00282247.
F.E. Browder, Nonlinear variational inequalities and maximal monotone mappings in Banach spaces, Math. Ann. 183 (1969), 213–231, DOI: 10.1007/BF01351381.
S. Carl, V.K. Le and D. Motreanu, Existence and comparison principles for general quasilinear variationalhemivariational inequalities, J. Math. Anal. Appl. 302 (2005), no. 1, 65–83, DOI: 10.1016/j.jmaa.2004.08.011.
O. Chadli, Q.H. Ansari, S. Al-Homidan and M. Alshahrani, Optimal control of problems governed by mixed quasi-equilibrium problems under monotonicity-type conditions with applications, Appl. Math. Opt. 83 (2021), 2185–2209, DOI: 10.1007/s00245-01909623-9.
Y-Q. Chen, On the semi-monotone operator theory and applications, J. Math. Anal. Appl. 231 (1999), 177–192, DOI: 10.1006/jmaa.1998.6245.
L. Chen, Y-H. Dai and Z. Wei, Sufficient conditions for existence of global minimizers of functions on Hilbert spaces, J. Global Optim. 84 (2022), 137–147, DOI: 10.1007/s10898022-01133-3.
I. Ciorănescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, 1990.
J. Diblik, G. Galewski, V.D. Rădulescu and Z. S̆marda, Multiplicity of solutions for nonlinear coercive problems, J. Math. Anal. Appl. 528 (2023), 127473 , 13 pp., DOI: 10.1016/j.jmaa.2023.127473.
G. Dincă, Duality mappings on infinite dimensional reflexive and smooth Banach spaces are not compact, Bull. de l’Académie Royale de Belgique 15 (2004), 33–40, DOI: 10.3406/barb.2004.28406.
G. Dincă, P. Jebelean and J. Mahwin, Variational and topological methods for Dirichlet problems with p-laplacian, Port. Math. 58 (2001), 339–378, https://www.emis.de/journals/PM/58f3/4.html.
M. Galewski, Basic Monotonicity Methods with Some Applications. Birkhäuser, 2021.
C.M. Gariboldi, A. Ochal, M. Sofonea and D. Tarzia, A convergence criterion for elliptic variational inequalities, Appl. Anal. (2023), to appear, DOI: 10.1080/00036811.2023.2268636.
L. Gasiński and P. Winkert, Existence and uniqueness results for double phase problems with convection term, J. Differential Equations 268 (2020), 4183–4193, DOI: 10.1016/j.jde.2019.10.022.
J. Gwinner and N. Ovcharova, From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics, Optimization 64 (2015), no. 8, 1–20, DOI: 10.1080/02331934.2014.1001758.
Ph. Hartman and G. Stampacchia, On some non linear elliptic differential functional equations, Acta Math. 115 (1966), 271–310, DOI: 10.1007/BF02392210.
M. Hintermüller and C.N. Rautenberg, On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities, Port. Math. 74 (2017), 1–35, DOI: 10.4171/PM/1991.
S. Hu and N. Papageorgiu, Handbook of Multivalued Analysis, Vol. 1, Theory, Kluwer Academic Publishers, Boston, 1997.
G. Kassay and M. Miholca, Existence results for variational inequalities with surjectivity consequences related to generalized monotone operators, J. Optim. Theory Appl. 159 (2013), 721–740, DOI: 10.1007/s10957-013-0383-8.
G. Kassay and V.D. Rădulescu, Equilibrium Problems and Applications. Mathematics in Science and Engineering, Academic Press, 2019.
B.T. Kien, M.M. Wong, N.C. Wong and J.C. Yao, Solution existence of variational inequalities with pseudomonotone operators in the sense of Brézis, J. Optim. Theory Appl. 140 (2009), 249–263, DOI: 10.1007/s10957-008-9446-7.
R. Landes and V. Mustonen, On pseudo-monotone operators and nonlinear noncoercive variational problems on unbounded domains, Math. Ann. 248 (1980), 241–246, DOI: 10.1007/BF01420527.
J. Leray and J.L. Lions, Quelques résultats de Vis̆ik sur les problémes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France 93 (1965), 97–107, http://www.numdam.org/item/SJL 1964 1 1 0/.
Y. Liu, Z. Liu, C-F. Wen and J.-C. Yao, Existence of solutions for non-coercive variational-hemivariational inequalities involving the nonlocal fractional p-Laplacian, Optimization 71 (2022), no. 3, 485–503, DOI: 10.1080/02331934.2020.1808643.
Z. Naniewicz, On some nonconvex variational problems related to hemivariational inequalities, Nonlinear Anal. 13 (1989), 87–100, DOI: 10.1016/0362-546X(89)90037-0.
Z. Naniewicz and P.D. Panagiatopoulos, Mathematical Theory of Hemivariational Inequalities, Marcel Dekker, Inc., 1985.
D. Pascali, Nonlinear Operators, R.S.R. Academy Publishing House, Bucharest, Romania, 1974. (Romanian)
V. Popescu, Optimal Control for Equations of Evolution, R.S.R. Academy Publishing House, Bucharest, Romania, 1988. (Romanian)
T. Roubic̆ek Nonlinear Differential Equations with Applications, Birkhäuser Verlag, Basel, Switzerland, 2005.
I. Sadeqi and M. Salehi Paydar, A comparative study of Ky Fan hemicontinuity and Brézis pseudomonotonicity of mappings and existence results, J. Optim. Theory Appl. 165 (2015), no. 2, 344–358, DOI: 10.1007/s10957-014-0618-x.
R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, vol. 49, Amer. Math. Soc., USA, 1997.
G. Stampacchia, Variational Inequalities, Actes, Congrés intern, Math. 2 (1970), 877–883, https://hal.science/hal-01378842/document.
D. Steck, Brezis pseudomonotonicity is strictly weaker than Ky Fan hemicontinuity, J. Optim. Theory Appl. 181 (2019), no. 1, 318–323, DOI: 10.1007/s10957-018-1435-x.
Y. Xiao and M. Sofonea, On the optimal control of variational-hemivariational inequalities, J. Math. Anal. Appl. 475 (2019), no. 1, 364–384, DOI: 10.1016/j.jmaa.2019.02.046.
H.-K. Xu, T.H. Kim and X. Yin, Weak continuity of the normalized duality map J. Nonlinear Convex Anal. 15 (2014), no. 3, 595–604, http://www.ybook.co.jp/online2/opjnca/vol15/p595.html.
C. Zălinescu, Convex Analysis and its Applications, World Scientific, Singapore, 2002.
E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. III, Variational Methods and Optimization, Springer Verlag, Berlin, Germany, 1985.
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