Some Iterative Schemes for Solving Mixed Equilibrium Variational-like Inequalities

  • Muhammad Aslam Noor Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
  • Hayat Ali Department of Sciences and Humanities, FAST National University FSD Campus, Faisalabad, Pakistan
  • Khalida Inayat Noor Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
Keywords: mixed variational-like inequality, bifunction, auxiliary principle technique, iterative methods, convergence


Some new types of equilibrium variational-like inequalities are considered, which is called the bifunction mixed equilibrium variational-like inequalities. The auxiliary principle technique is used to construct some iterative schemes to solve these new equilibrium variational-like inequalities. Convergence of the suggested schemes is discussed under relaxed conditions. Several special cases are discussed as applications of the main results. The ideas and techniques may be starting point for future research.


C. Baiocchi and A. Capelo, Variational and Quasi-Variational inequalities, J. Wiley & Sons, New York, London, 1984.

A. Ben-Israel and B. Mond, What is invexity?, J. Austral. Math. Soc. Ser. E 28 (1986), 1-9.

M. I. Bloach, M. A. Noor and K. I. Noor, Well-posedness of triequilibrium-like problems, Int. J. Anal. Appl. 20 (2022), 3.

R. Glowinski, J. L. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.

M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80 (1981), 545-550.

M. A. Hanson and B. Mond, Convex transformable programming problems and invexity, J. Inf. Opt. Sci. 8 (1987), 201-207.

S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl. 189 (1995), 901-908.

M. A. Noor, Some recent advances in variational inequalities, Part I: Basic concepts, New Zealand J. Math. 26(1997), 53-80.

M. A. Noor, Preinvex functions and variational inequalities, J. Natur. Geom. 9 (1998), 63-76.

M. A. Noor, Variational-like inequalities, Optimization 3 (1994), 323-330.

M. A. Noor, Some developments in general vatiational inequalities, Appl. Math. Comput. 152 (2004), 199-277.

M. A. Noor, Merit functions for variational-like inequalities, Math. Inequal. Appl. 1 (2000), 117-128.

M. A. Noor, Fundamental of equlibrium problems, Math. Inequal. Appl. 9 (2006), 529-566.

M. A. Noor, K. I. Noor and M. I. Baloch, Auxiliary principle technqiue for strongly mixed variational-like inequalities, Politehn. Univ. Bucharest Sci. Bull. Ser. A 80 (2018), 93-100.

M. A. Noor, K. I. Noor and M. Th. Rassias, New trends in general variational inequalities, Acta Appl. Math. 170 (2020), 981-1064.

M. A. Noor, K. I. Noor, M. U. Awan and S. Khan, Hermite-Hadamard inequalities for s-Godunova-Levin preinvex functions, J. Adv. Math. Stud. 7(2) (2014), 12-19.

M. A. Noor, K. I. Noor, M. U. Awan and J. Li, On Hermite-Hadamard inequalities for h-preinvex functions, Filomat 28(7) (2014), 1463-1474.

M. Patriksson, Nonlinear Programming and Variational Inequality Problems: A Unified Approach, Kluwer Academic Publisher, Dordrecht, Holand, 1999.

G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413-4416.

T. Weir and B. Mond, Preinvex functions in multiobjective optimization, J. Math. Anal. Appl. 136 (1988), 29-38.

D. L. Zhu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM J. Optim. 6(3) (1996), 714-726.

How to Cite
Noor, M. A., Ali, H., & Noor, K. I. (2022). Some Iterative Schemes for Solving Mixed Equilibrium Variational-like Inequalities. Earthline Journal of Mathematical Sciences, 10(1), 67-84.