International Journal of Nonlinear Analysis and Applications

Perturbed absolute value variational inequalities

Document Type : Research Paper

Authors

Department of Mathematics, University of Jammu, Jammu-180006, India

Abstract

In this paper, we examine the perturbed absolute value variational inequalities (PAVVI), a new class of variational inequalities. For the (PAVVI), some new merit functions are established. We develop the error bounds for (PAVVI) using these merit functions. The results presented here recapture a number of previously established findings in the relevant fields because (PAVVI) include variational inequalities, the absolute value complementarity problem, and systems of absolute value equations as special cases.

Keywords

[1] D. Aussel, R. Correa, and M. Marechal, Gap functions for quasi variational inequalities and generalized Nash equilibrium problems, J. Optim. Theory Appl. 151 (2011), 474–488.
[2] D. Aussel, R. Gupta, and A. Mehra, Gap functions and error bounds for inverse quasi-variational inequality problems, J. Math. Anal. Appl. 407 (2013), 270–280.
[3] S. Batool, M.A. Noor, and K.I. Noor, Absolute value variational inequalities and dynamical systems, Int. J. Anal. Appl. 18 (2020), 337–355.
[4] S. Batool, M.A. Noor, and K.I. Noor, Merit functions for absolute value variational inequalities, AIMS Math. 6 (2021), 12133–12147.
[5] M.I. Bloach and M.A. Noor, Perturbed mixed variational-like inequalities, AIMS Mathematics 5 (2020) 2153-2162.
[6] M. Bohner, A. Kashuri, P. Mohammed, and J.E.N. Valdes, Hermite-Hadamard-type inequalities for conformable integrals, Hacet. J. Math. Stat. 51 (2022), no. 3, 775–786.
[7] S.S. Chang, S. Salahuddin, M. Liu, X.R. Wang, and J.F. Tang, Error bounds for generalized vector inverse quasivariational inequality problems with point to set mappings, AIMS Math. 6 (2020), 1800–1815.
[8] J. Dutta, Gap functions and error bounds for variational and generalized variational inequalities, Vietnam J. Math. 40 (2012), 231–253.
[9] R. Gupta and A. Mehra, Gap functions and error bounds for quasi variational inequalities, J. Glob. Optim. 53 (2012), 737–748.
[10] S.L. Hu and Z.H. Huang, A note on absolute value equations, Optim. Lett. 4 (2010), 417–424.
[11] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, SIAM Publishing Co., Philadelphia, USA, 2000.
[12] T. Kouichi, On gap functions for quasi-variational inequalities, Abstr. Appl. Anal. 2008 (2008), 531361.
[13] G.Y. Li and K.F. Ng, Error bounds of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems, SIAM J. Optim. 20 (2009) 667-690.
[14] J.L. Lions and G. Stampacchia, Variational inequalities, Commun. Pur. Appl. Math. 20 (1967), 493–519.
[15] O.L. Mangasarian, The linear complementarity problem as a separable bilinear program, J. Glob. Optim. 6 (1995), 153–161.
[16] O.L. Mangasarian and R.R. Meyer, Absolute value equations, Linear Algebra Appl. 419 (2006), 359-,367.
[17] M.A. Noor, On variational inequalities, Ph.D Thesis, Brunel University, London, UK, 1975.
[18] M.A. Noor, Merit functions for general variational inequalities, J. Math. Anal. Appl. 316 (2006), 736–752.
[19] M.A. Noor, On merit functions for quasi variational inequalities, J. Math. Inequal. 1 (2007), 259–268.
[20] M.A. Noor and K.I. Noor, From representation theorems to variational inequalities, Computational Mathematics and Variational Analysis, Springer 2020, pp. 261–277.
[21] M.A. Noor, K.I. Noor, and S. Batool, On generalized absolute value equations, UPB Sci. Bull. Series A 80 (2018), 63–70.
[22] B. Qu, C.Y. Wang, and Z.J. Hang, Convergence and error bound of a method for solving variational inequality problems via the generalized D-gap function, J. Optim. Theory App. 119 (2003), 535–552.
[23] O. Prokopyev, On equivalent reformulation for absolute value equations, Comput. Optim. Appl. 44 (2009), 363.
[24] T. Ram and R. Kour, Mixed variational like inequalities involving perturbed operator, J. Math. Comput. Sci. 12 (2022), 1–10.
[25] J. Rohn, A theorem of the alternatives for the equation Ax+B—x—=b, Linear Multilinear. A. 52 (2004), 421–426.
[26] S.K. Sahoo, Y.S. Hamed, P.O. Mohammed, B. Kodamasingh, and K. Nonlaopon, New midpoint type Hermite-Hadamard-Mercer inequalities pertaining to Caputo-Fabrizio fractional operators, Alexandria Engin. J. 65 (2023), 689–698.
[27] M.V. Solodov, Merit functions and error bounds for generalized variational inequalities, J. Math. Anal. Appl. 287 (2003), 405–414.
[28] M.V. Solodov and P. Tseng, Some methods based on the D-gap function for solving monotone variational inequalities, Comput. Optim. Appl. 17 (2000), 255–277.
[29] H.M. Srivastava, M. Tariq, P.O. Mohammed, H. Alrweili, E. Al-Sarairah, and M. De La Sen, On modified integral inequalities for a generalized class of convexity and applications, Axioms 12 (2023), no. 2, 162.
[30] G. Stampacchia, Formes bilineaires coercivities sur less ensembles convexes, C.R. Acad. Sci. Paris 258 (1964), 4413–4416.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 12
December 2023
Pages 13-24
  • Receive Date: 02 March 2023
  • Revise Date: 26 April 2023
  • Accept Date: 02 June 2023