**Title:** Some classes of higher order general convex functions and variational inequalities

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-21-00064; **Volume 5 / Issue 3 / Year 2023**, Pages 1-15

**Document Type:** Research Paper

**Author(s):** Muhammad Aslam Noor ^{a} , Khalida Inayat Noor ^{b}

^{a}Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan

^{b}Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan

Received: 14 November 2021, Accepted: 8 January 2022, Published: 12 March 2022.

**Corresponding Author:** Muhammad Aslam Noor (Email address: noormaslam@gmail.com)

**Full Text:** PDF

**Abstract**

Some new classes the higher order convex functions with respect to an arbitrary function are introduced and studied. Properties of the general functions are investigated. Higher order general variational inequalities are considered. Several important problems such as are deduced as special cases. Several iterative schemes are proposed. Convergence of the proposed methods are analyzed. Parallelogram laws are derived as applications. Results obtained can be viewed as important refinement of the known results.

**Keywords:** General convex functions, variational inequalities, parallelogram laws

**References:**

- M. Adamek,
*On a problem connected with strongly convex functions,*Math. Inequal. Appl.**19 (4)**, 1287–1293, 2016. - O. Alabdali, A. Guessab and G. Schmeisser,
*Characterization of uniform convexity for differntiable functions*, Appl. Anal. Discrete Math.**13**, 721–732, 2019. - H. Angulo, J. Gimenez, A. M. Moeos and K. Nikodem,
*On strongly*, Ann. Funct. Anal.*h*-convex functions**2 (2)**, 85–91, 2011. - M. U. Awan, M. A. Noor, T.-S. Du and K. I. Noor,
*New refinemnts of fractional Hermite-Hadamard inequality*, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM**113**, 21–29, 2019. - W. L. Bynum,
*Weak parallelogram laws for Banach spaces*, Canad. Math. Bull.**19**, 269–275. 1976. - R. Cheng and C. B. Harris,
*Duality of the weak parallelogram laws on Banach spaces*, J. Math. Anal. Appl.**404**, 64–70, 2013. - R. Cheng and W. T. Ross,
*Weak parallelogram laws on Banach spaces and applications to prediction*, Period. Math. Hung.**71**, 45–58, 2015. - G. Cristescu and M. Găianu,
*Shape properties of Noors convex sets*, Proceed. of the Twelfth Symposium of Mathematics and its Applications, Timisoara, 1-13, 2009. - G. Cristescu and L. Lupsa,
*Non connected convexities and applications*, Kluwer Academic Publisher, Dordrechet, 2002. - R. Glowinski, J. L. Lions and R. Tremileres,
*Numerical analysis of variational inequalities*, NortHolland, Amsterdam, Holland, 1981. - S. Karamardian,
*The nonlinear complementarity problems with applications*, Part 2, J. Optim. Theory Appl.**4 (3)**, 167–181, 1969. - J. L. Lions and G. Stampacchia,
*Variational inequalities,*Comm. Pure Appl. Math.**20**, 491–512, 1967. - G. H. Lin and M. Fukushima,
*Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints*, J. Optim. Theory Appl.**118 (1)**, 67–80, 2003. - B. B. Mohsen, M. A. Noor, K. I. Noor and M. Postolache,
*Strongly convex functions of higher order involving bifunction*, Mathematics**7 (11)**, Article ID: 1028, 2019. - C. P. Niculescu and L. E. Persson,
*Convex functions and their applications*, Springer-Verlag, New York, 2018. - K. Nikodem and Z. S. Pales,
*Characterizations of inner product spaces by strongly convex functions,*Banach J. Math. Anal.**1**, 83–87, 2011. - M. A. Noor,
*On variational inequalities,*PhD Thesis, Brunel University, London, U. K. 1975. - M. A. Noor,
*General variational inequalities,*Appl. Math. Lett.**1 (2)**, 119–122, 1988. - M. A. Noor,
*Quasi variational inequalities,*Appl. Math. Lett.**1 (4)**, 367–370, 1988. - M. A. Noor,
*New approximation schemes for general variational inequalities,*J. Math. Anal. Appl.**251 (1)**, 217–229, 2000. - M. A. Noor,
*Some developments in general variational inequalities,*Appl. Math. Comput.**152 (1)**, 199–277, 2004. - M. A. Noor,
*Fundamentals of equilibrium problems,*Math. Inequal. Appl.**9 (3)**, 529–566, 2006. - M. A. Noor,
*Differentiable non-convex functions and general variational inequalities,*Appl. Math. Comput.**199 (2)**, 623–630, 2008. - M. A. Noor,
*Extended general variational inequalities,*Appl. Math. Lett.**22 (2)**, 182–186, 2009. - M. A. Noor and K. I. Noor,
*On strongly exponentially preinvex functions,*Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys.**81 (4)**, 75–84, 2019. - M. A. Noor and K. I. Noor,
*Strongly exopnetially convex functions and their properties*, J. Adv. Math. Stud.**12 (2)**, 177–185, 2019. - M. A. Noor and K. I. Noor,
*On generalized strongly convex functions involving bifunction*, Appl. Math. Inf. Sci.**13 (3)**, 411–416, 2019. - M. A. Noor and K. I. Noor,
*Higher order strongly general convex functions and variational inequalities,*AIMS Math.**5**, 3646–3663, 2020. - M. A. Noor and K. I. Noor,
*Properties of higher order preinvex functions,*Numer. Algebra Control Optim.**11 (3)**, 431–441, 2021. - M. A. Noor and K. I. Noor,
*New classes of preinvex functions and variational-like inequalities,*Filomat**35 (6)**, 2081–2097, 2021. - M. A. Noor and K. I. Noor,
*Exponentially general convex functions and variational inequalities,*In: Inequalities with Generalized Convex Functions and Applications (Editors: M. U. Awan and G, Cristescu), Springer, 2022. - M. A. Noor and K. I. Noor,
*Nonconvex bifunction general variational inequalities*, Montes Taurus J. Pure Appl. Math.**4 (3)**, 1–8, 2022. - M. A. Noor and K. I. Noor,
*Iterative algorithms for solving nonlinear quasi-variational inequalities*, Montes Taurus J. Pure Appl. Math.**4 (1)**, 44–58, 2022. - M. A. Noor and K. I. Noor,
*Some new trends in mixed variational inequalities*, J. Adv. Math. Stud., (Accepted). - M. A. Noor and K. I. Noor,
*New classes of higher order variational-like inequalities*, To appear in: Rad Hazu. Matemat. Znanosti, 2022. - M. A. Noor and K. I. Noor,
*General biconvex functions and bivariational inequalities*, Numer. Algebra Control Optim.; Doi: 10.3934/naco.2021041. - M. A. Noor, K. I. Noor and H. M. Y. Al-Bayatti,
*Higher order variational inequalities,*Inform. Sci. Lett.**11**, 1–5, 2022. - M. A. Noor, K. I. Noor and Th. M. Rassias,
*New trends in general variational inequalities,*Acta Appl. Math.**170 (1)**, 981–1064, 2020. - M. A. Noor, K. I. Noor and Th. M. Rassias,
*Some aspects of variational inequalities,*J. Comput. Appl. Math.**47**, 493–512, 1993. - M. A. Noor, K. I. Noor, A. Hamdi and E. H. El-Shemas,
*On difference of two monotone operators,*Optim. Lett.**3**, 329–335, 2009. - G. Qu and N. Li.
*On the exponentially stability of primal-dual gradeint dynamics,*IEEE Control Syst. Lett.**3 (1)**, 43–48, 2019. - J. Pecric, F. Proschan and Y. I. Tong,
*Convex functions, partial ordering and statistical applications*, Academic Press, New York, USA, 1992. - B. T. Polyak,
*Existence theorems and convergence of minimizing sequences in extremum problems with restrictions*, Soviet mathematics – doklady**7**, 2–75, 1966. - G. Stampacchia,
*Formes bilieaires coercives sur les ensembles convexes,*C. R. Math. Acad. Sci. Paris,**258**, 4413–4416, 1964. - G. H. Toader,
*Some generalizations of the convexity,*Proceedings of the Colloquium on Approximation and Optimization: Cluj-Napoca (Romania), 329–338, 1984 - H-K, Xu,
*Inequalities in Banach spaces with applications,*Nonlinear Anal Theory Methods Appl.**16 (12)**, 1127–1138, 1991 - E. A. Youness,
J. Optim. Theory Appl.*E*-convex sets,*E*-convex functions and*E*-convex programming,**102**, 439–450, 1999. - D. L. Zu and P. Marcotte,
*Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities,*SIAM J. Optim.**6 (3)**, 714–726, 1996.