Variational inequality for a vector field on Hadamard spaces

Ranjbar, Sajad

doi:10.2478/auom-2023-0013

Abstract

Our purpose is to study the variational inequality problem for a vector field on Hadamard spaces. The existence and uniqueness of the solutions to the variational inequality problem associated with a vector field in Hadamard spaces are studied.

References

[1] B. Ahmadi Kakavandi, M. Amini, Duality and subdifferential for convex functions on complete CAT(0) metric spaces, Nonlinear Anal. 73 (2010), 3450-3455.10.1016/j.na.2010.07.033
Search in Google Scholar Back to article
[2] M. Bačák, Convex Analysis and Optimization in Hadamard Spaces. De Gruyter Series in Nonlinear Analysis and Applications, Germany, 22 (2014).
Search in Google Scholar Back to article
[3] I.D. Berg, I.G. Nikolaev, Quasilinearization and curvature of Alexandrov spaces, Geom. Dedicata 133 (2008), 195-218.10.1007/s10711-008-9243-3
Search in Google Scholar Back to article
[4] M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature. Fundamental Principles of Mathematical Sciences. Springer, Berlin 319 (1999).10.1007/978-3-662-12494-9
Search in Google Scholar Back to article
[5] D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Graduate Studies in Math., 33, Amer. Math. Soc., Providence, RI (2001).10.1090/gsm/033
Search in Google Scholar Back to article
[6] M. Gromov, S.M. Bates, Metric Structures for Riemannian and Non-Riemannian Spaces, with appendices by M. Katz, P. Pansu and S. Semmes, ed. by J. Lafontaine and P. Pansu, Progr. Math. 152, BirkhNauser, Boston (1999).
Search in Google Scholar Back to article
[7] P.T. Harker, J.S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Program. 48, (1990), 161220.
Search in Google Scholar Back to article
[8] J. Jöst, Nonpositive Curvature: Geometric and Analytic Aspects, Lectures Math. ETH ZNurich, BirkhNauser, Basel (1997).
Search in Google Scholar Back to article
[9] D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, 88. Academic Press, Inc. New York-London, 1980.
Search in Google Scholar Back to article
[10] H. Khatibzadeh, S. Ranjbar, A variational inequality in complete CAT(0) spaces, J. fixed point theory appl. 17 (2015) 557-574.10.1007/s11784-015-0245-0
Search in Google Scholar Back to article
[11] N. Hadjisavvas, S. Schaible, Quasimonotone variational inequalities in Banach spaces. J. Optim. Theory Appl. 90 (1996) 95111.
Search in Google Scholar Back to article
[12] J. Li, On the existence of solutions of variational inequalities in Banach spaces, J. Math. Anal. Appl. 295 (2004) 115-126.10.1016/j.jmaa.2004.03.010
Search in Google Scholar Back to article
[13] S.L. Li, C. Li, Y.C. Liou, J.C. Yao, Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 71, (2009), 5695-5706.10.1016/j.na.2009.04.048
Search in Google Scholar Back to article
[14] C. Li, G. López, V. Martín-Márquez, Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, (2009), 663-683.10.1112/jlms/jdn087
Search in Google Scholar Back to article
[15] S.Z. Németh, Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52, (2003), 1491-1498.10.1016/S0362-546X(02)00266-3
Search in Google Scholar Back to article
[16] M. Patriksson, Nonlinear Programming and Variational Inequality Problems: A Unified Approach, Kluwer Academic Publishers, Dordrecht, 1999.10.1007/978-1-4757-2991-7
Search in Google Scholar Back to article
[17] K.T. Sturm, Probability Measures on Metric Spaces of Non-positive Curvature, Heat Kernels and Analysis on Manifolds, Graphs and Metric Spaces (Paris, 2002), 357-390, Contemp. Math. 338, Amer. Math. Soc., Providence, RI, 2003.10.1090/conm/338/06080
Search in Google Scholar Back to article
[18] X. Wang, G. Lpez, C. Li, J.C. Yao, Equilibrium problems on Riemannian manifolds with applications, J. Math. Anal. Appl. 473 (2019) 866-891.10.1016/j.jmaa.2018.12.073
Search in Google Scholar Back to article
[19] J.C. Yao,, Variational Inequalities with Generalized Monotone Operators, Math. Oper. Research, 19 (1994) 691-705.
Search in Google Scholar Back to article

Variational inequality for a vector field on Hadamard spaces

Abstract

Paradigm

My account