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Fan-Hemicontinuity for the gradient of the norm in Hilbert space

  • Marcel Bogdan
Published/Copyright: December 4, 2022
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Abstract

It was claimed in [Sadeqi, I.—Salehi Paydar, M.: A comparative study of Ky Fan hemicontinuity and Brezis pseudomonotonicity of mappings and existence results, J. Optim. Theory Appl. 165(2) (2015), 344–358] that the gradient of a convex Gâteaux differentiable function is Fanhemicontinuous. The aim of the present paper is to correct this implication by exemplifying for in a Hilbert space. This goal was achieved by considering its domain to be scalarly-positive (see Definition 3). For a perturbation of a Fan-hemicontinuous operator with , weak compactness for the set of solutions of the variational inequality problem is obtained.

MSC 2010: 47H05; 49J93; 49J40; 90C47

Acknowledgement

I thank with gratitude Professor Ştefan Cobzaş for his remarks and suggestions made during the preparation of this paper. The author thanks the reviewers for their reports on the manuscript.

  1. (Communicated by Marek Balcerzak)

References

[1] BOGDAN, M. — KOLUMBÁN, J.: On nonlinear variational inequalities, Nonlinear Anal. 67 (2007), 2272–2282.10.1016/j.na.2006.08.035Search in Google Scholar

[2] BOGDAN, M.: Comments on the solutions set of equilibrium problems governed by topological pseudomonotone bifunctions. In: L. Moldovan, A. Gligor (eds.): The 15th International Conference Interdisciplinarity in Engineering. Inter-Eng 2021, Lect. Notes Netw. Syst., vol. 386, Springer, Cham, 2022, pp. 697–703.10.1007/978-3-030-93817-8_62Search in Google Scholar

[3] BRÉZIS, H.: Equations et inéquations nonlinéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble) 18 (1968), 115–175.10.5802/aif.280Search in Google Scholar

[4] CHEN, E.: On the semi-monotone operator theory and applications, J. Math. Anal. Appl. 231 (1999), 177–192.10.1006/jmaa.1998.6245Search in Google Scholar

[5] ERNST, E. — THÉRA, M.: A converse to the Lions-Stampacchia theorem, ESAIM Control Optim. Calc. Var. 15 (2009), 810–817.10.1051/cocv:2008054Search in Google Scholar

[6] FAN, K.: A minmax inequality and applications, Inequalities III. In: Proc. Third Sympos., Academic Press, San Diego, 1972, pp. 103-113.Search in Google Scholar

[7] GALEWSKI, M.: Basic Monotonicity Methods with Some Applications, Birkhäuser, 2021.10.1007/978-3-030-75308-5Search in Google Scholar

[8] GWINNER, J.: Note on pseudomonotone functions, regularization, and relaxed coerciveness, Nonlinear Anal. 30 (1997), 4217–4227.10.1016/S0362-546X(97)00390-8Search in Google Scholar

[9] HARTMAN, P. — STAMPACCHIA, G.: On some non linear elliptic differential functional equations, Acta Math. 115 (1966), 271–310.10.1007/BF02392210Search in Google Scholar

[10] KASSAY, G. — RĂDULESCU, V. D.: Equilibrium Problems and Applications. Math. Sci. Eng., Academic Press, 2019.Search in Google Scholar

[11] KASYANOV, P. O. — MEL’NIK, V. S. — TOSCANO, S.: Solutions of Cauchy and periodic problems for evolution inclusions equations with multivalued wλ0-pseudomonotone maps, J. Differential Equations 249 (2010), 1258–1287.10.1016/j.jde.2010.05.008Search in Google Scholar

[12] KENMOCHI, N.: Pseudomonotone operators and nonlinear elliptic boundary value problems, J. Math. Soc. Japan. 27 (1975), 121–149.10.2969/jmsj/02710121Search in Google Scholar

[13] LERAY, J. — LIONS, J.-L.: Quelques résultats de Vis̆ik sur les problémes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France 93 (1965), 97–107.10.24033/bsmf.1617Search in Google Scholar

[14] MAUGERI, A. — RACITI, F.: On existence theorems for monotone and nonmonotone variational inequalities, J. Convex Anal. 16 (2009), 899–911.Search in Google Scholar

[15] MIGORSKI, A. — OCHAL, A.: Hemivariational inequalities for stationary Navier-Stokes equations, J. Math. Anal. Appl. 306 (2005), 197–217.10.1016/j.jmaa.2004.12.033Search in Google Scholar

[16] NANIEWICZ, Z.: Some economic type problems with applications, Set-Valued Var. Anal. 19(3) (2011), 417–456.10.1007/s11228-010-0167-3Search in Google Scholar

[17] PASCALI, D.: Operatori Neliniari. Editura Academiei R.S.R., Bucureşti, Romania (in Romanian) 1974.Search in Google Scholar

[18] ROUBÍČEK, T.: Nonlinear Differential Equations with Applications, Birkhäuser Verlag, Basel, Switzerland, 2005.Search in Google Scholar

[19] SADEQI, I. — SALEHI PAYDAR, M.: A comparative study of Ky Fan hemicontinuity and Brezis pseudomonotonicity of mappings and existence results, J. Optim. Theory Appl. 165(2) (2015), 344–358.10.1007/s10957-014-0618-3Search in Google Scholar

[20] SHOWALTER, R. E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Math. Surveys Monogr. 49, American Math. Society, USA, 1997.Search in Google Scholar

[21] STAMPACCHIA, G.: Variational Inequalities. In: Proc. Internat. Congr. of Math., Nice, 1970, pp. 877–883.Search in Google Scholar

[22] STECK, D.: Brezis pseudomonotonicity is strictly weaker than Ky Fan hemicontinuity, J. Optim. Theory Appl. 181(1) (2019), 318–323.10.1007/s10957-018-1435-xSearch in Google Scholar

[23] XIAO, Y. — SOFONEA, M.: On the optimal control of variational-hemivariational inequalities, J. Math. Anal. Appl. 475 (2019), 364–384.10.1016/j.jmaa.2019.02.046Search in Google Scholar

[24] ZHANG, Y. — HE, Y.: On stably quasimonotone hemivariational inequalities, Nonlinear Anal. 74(10) (2011), 3324–3332.10.1016/j.na.2011.02.009Search in Google Scholar

[25] ZEIDLER, E.: Nonlinear Functional Analysis and its Applications, Vol. III, Springer Verlag, Berlin, 1990.10.1007/978-1-4612-0981-2Search in Google Scholar

Received: 2021-06-24
Accepted: 2021-12-10
Published Online: 2022-12-04
Published in Print: 2022-12-16

© 2022 Mathematical Institute Slovak Academy of Sciences

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