AIMS Mathematics, 2020, 5(4): 3646-3663. doi: 10.3934/math.2020236.

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Higher order strongly general convex functions and variational inequalities

Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan

In this paper, we define and consider some new concepts of the higher order strongly general convex functions with respect to an arbitrary function. Some properties of the higher order strongly general convex functions are investigated under suitable conditions. It is shown that the optimality conditions of the higher order strongly general convex functions are characterized by a class of variational inequalities, which is called the higher order strongly variational inequality. Auxiliary principle technique is used to suggest an implicit method for solving strongly general variational inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. It is shown that the parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine convex functions. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.
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Keywords higher order convex functions; variational inequalities; parallelogram laws

Citation: Muhammad Aslam Noor, Khalida Inayat Noor. Higher order strongly general convex functions and variational inequalities. AIMS Mathematics, 2020, 5(4): 3646-3663. doi: 10.3934/math.2020236


  • 1. B. B. Mohsen, M. A. Noor, K. I. Noor, et al. Strongly convex functions of higher order involving bifunction, Mathematics, 7 (2019), 1028.
  • 2. G. H. Lin, M. Fukushima, Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints, J. Optimiz. Theory App., 118 (2003), 67-80.    
  • 3. B. T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl., 7 (1966), 287-290.
  • 4. S. Karamardian, The nonlinear complementarity problems with applications, Part 2, J. Optimiz. Theory App., 4 (1969), 167-181.    
  • 5. M. U. Awan, M. A. Noor, M. V. Mihai, et al. On approximately harmonic h-convex functions depending on a given function, Filomat, 33 (2019), 3783-3793.    
  • 6. M. U. Awan, M. A. Noor, T. S. Du, et al. New refinemnts of fractional Hermite-Hadamard inequality, RACSAM, 113 (2019), 21-29.    
  • 7. A. Azcar, J. Gimnez, K. Nikodem, et al. On strongly midconvex functions, Opuscula Math., 31 (2010), 15-26.
  • 8. M. Adamek, On a problem connected with strongly convex functions, Math. Inequal. Appl., 19 (2015), 1287-1293.
  • 9. H. Angulo, J. Gimenez, A. M. Moeos, et al. On strongly h-convex functions, Ann. Funct. Anal., 2 (2010), 85-91.
  • 10. C. P. Niculescu, L. E. Persson, Convex Functions and Their Applications, Springer-Verlag, New York, 2018.
  • 11. K. Nikodem, Z. S. Pales, Characterizations of inner product spaces by strongly convex functions, Banach J. Math. Anal., 1 (2011), 83-87.
  • 12. D. L. Zu, P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, Siam J. Optimiz., 6 (1996), 714-726.    
  • 13. M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput., 251 (2004), 199-277.
  • 14. M. A. Noor, Fundamentals of equilibrium problems, Math. Inequal. Appl., 9 (2006), 529-566.
  • 15. M. A. Noor, Differentiable non-convex functions and general variational inequalities, Appl. Math. Comput., 199 (2008), 623-630.
  • 16. G. Cristescu, M. Găianu, Shape properties of Noor's convex sets, Proc. Twelfth Symp. Math. Appl.
  • Timisoara, (2009), 1-13, 17. G. Cristescu, L. Lupsa, Non Connected convexities and applications, Kluwer Academic Publisher, Dordrechet, 2002.
  • 18. E. A. Youness, E-convex sets, E-convex functions and E-convex programming, J. Optimiz. Theory App., 102 (1999), 439-450.    
  • 19. G. Stampacchia, Formes bilieaires coercives sur les ensembles convexes, Comput. Rend. l'Acad. Sci. Paris, 258 (1964), 4413-4416.
  • 20. J. L. Lions, G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493-519.    
  • 21. R. Glowinski, J. L. Lions, R. Tremlier, Numerical analysis of variational inequalities, NortHolland, Amsterdam, Holland, 1980.
  • 22. M. A. Noor, Extended general variational inequalities, Appl. Math. Lett., 22 (2009), 182-186.    
  • 23. W. L. Bynum, Weak parallelogram laws for Banach spaces, Can. Math. Bull., 19 (1976), 269-275.    
  • 24. R. Cheng, C. B. Harris, Duality of the weak parallelogram laws on Banach spaces, J. Math. Anal. Appl., 404 (2013), 64-70.    
  • 25. R. Cheng, W. T. Ross, Weak parallelogram laws on Banach spaces and applications to prediction, Period. Math. Hung., 71 (2015), 45-58.
  • 26. G. H. Toader, Some generalizations of the convexity, Proc. Colloq. Approx. Optim. Cluj-Naploca (Romania), (1984), 329-338.
  • 27. M. A. Noor, K. I. Noor, On strongly exponentially preinvex functions, U. P. B. Sci. Bull. Series A, 81 (2019), 75-84.
  • 28. M. A. Noor, K. I. Noor, Strongly exopnetially convex functions and their properties, J. Advan. Math. Stud., 12 (2019), 177-185.
  • 29. M. A. Noor, K. I. Noor, On generalized strongly convex functions involving bifunction, Appl. Math. Inform. Sci., 13 (2019), 411-416.    
  • 30. W. Oettli, M. Thera, On maximal monotonicity of perturbed mapping, B. Unione Mat. Ital., 7 (1995), 47-55.
  • 31. G. Qu, N. Li, On the exponentially stability of primal-dual gradeint dynamics, IEEE Control Syst. Lett., 3 (2019), 43-48.    
  • 32. J. Pecric, F. Proschan, Y. I. Tong, Convex functions, partial ordering and statistical applications, Academic Press, New York, USA, 1992.
  • 33. H. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal-Theor., 16 (1991), 1127-1138.    


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