In this paper, we introduce and study some new classes of biconvex functions with respect to an arbitrary function and a bifunction, which are called the higher order strongly biconvex functions. These functions are nonconvex functions and include the biconvex function, convex functions, and k-convex as special cases. We study some properties of the higher order strongly biconvex functions. Several parallelogram laws for inner product spaces are obtained as novel applications of the higher order strongly biconvex affine functions. It is shown that the minimum of generalized biconvex functions on the k-biconvex sets can be characterized by a class of equilibrium problems, which is called the higher order strongly biequilibrium problems. Using the auxiliary technique involving the Bregman functions, several new inertial type methods for solving the higher order strongly biequilibrium problem are suggested and investigated. Convergence analysis of the proposed methods is considered under suitable conditions. Several important special cases are obtained as novel applications of the derived results. Some open problems are also suggested for future research.
References
[1]
Stampacchia, G. (1964) Formes bilineaires coercivites sur les ensembles convexes. Comptes rendus de l’Académie des Sciences Paris, 258, 4413-4416.
[2]
Jolaoso, L.O., Aphane, M. and Khan, S.H. (2020) Two Bregman Projection Methods for Solving Variational Inequality Problems in Hilbert Spaces with Applications to Signal Processing. Symmetry, 12, 2007. https://doi.org/10.3390/sym12122007
[3]
Sunthrayuth, P. and Cholamjiak, P. (2020) Modified Extragradient Method with Bregman Distance for Variational Inequalities. Applicable Analysis.
[4]
Glowinski, R., Lions, J.L. and Tremolieres, R. (1981) Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam.
[5]
Kinderlehrer, D. and Stampaccia, G. (1980) Introduction to Variational Inequalties and Applications. Academic Press, New York.
[6]
Lin, G.H. and Fukushima, M. (2003) Some Exact Penalty Results for Nonlinear Programs and Mathematical Programs with Equilibrium Constraints. Journal of Optimization Theory and Applications, 118, 67-80. https://doi.org/10.1023/A:1024787424532
[7]
Lions, J.L. and Stampacchia, G. (1967) Variational Inequalities. Communications on Pure and Applied Mathematics, 20, 493-519. https://doi.org/10.1002/cpa.3160200302
[8]
Noor, M.A. (1988) General Variational Inequalities. Applied Mathematics Letters, 1, 119-121. https://doi.org/10.1016/0893-9659(88)90054-7
[9]
Noor, M.A. (2000) New Approximation Schemes for General Variational Inequalities. Journal of Mathematical Analysis and Applications, 251, 217-229. https://doi.org/10.1006/jmaa.2000.7042
[10]
Noor, M.A. (2006) Fundamental of Equilibrium Problems. Mathematical Inequalities & Applications, 9, 520-566. https://doi.org/10.7153/mia-09-51
[11]
Noor, M.A. (2004) Some Developments in General Variational Inequalities. Applied Mathematics and Computation, 152, 199-277. https://doi.org/10.1016/S0096-3003(03)00558-7
[12]
Noor, M.A. (1994) Variational-Like Inequalities. Optimization, 30, 323-330. https://doi.org/10.1080/02331939408843995
[13]
Noor, M.A. (2005) Invex Equilibrium Problems. Journal of Mathematical Analysis and Applications, 302, 463-475. https://doi.org/10.1016/j.jmaa.2004.08.014
[14]
Noor, M.A., Noor, K.I. and Al-Said, E. (2011) Auxiliary Principle Technique for Solving Bifunction Variational Inequalities. Journal of Optimization Theory and Applications, 149, 441-445. https://doi.org/10.1007/s10957-010-9785-z
[15]
Noor, M.A., Noor, K.I. and Rassias, M.Th. (2020) New Trends in General Variational Inequalities. Acta Applicandae Mathematicae, 170, 981-1064. https://doi.org/10.1007/s10440-020-00366-2
[16]
Noor, M.A., Noor, K.I. and Rassias, Th.M. (1993) Some Aspects of Variational Inequalities. Journal of Applied Mathematics and Computing, 47, 485-512. https://doi.org/10.1016/0377-0427(93)90058-J
[17]
Zhu, D.L. and Marcotte, P. (1966) Co-Coercvity and Its Role in the Convergence of Iterative Schemes for Solving Variational Inequalities. SIAM Journal on Optimization, 6, 714-726. https://doi.org/10.1137/S1052623494250415
[18]
Blum, E. and Oettli, W. (1994) From Optimization and Variational Inequalities to Equilibrium Problems. Mathematics Student, 63, 123-145.
[19]
Noor, M.A. and Oettli, W. (1994) On General Nonlinear Complementarity Problems and Quasi-Equilibria. Le Matematiche (Catania), 49, 313-331.
[20]
Noor, M.A. and Noor, K.I. (2021) Exponentially Biconvex Functions and Bivariational Inequalties. In: Hazarika, B., Acharjee, S. and Srivastava, H.M., Eds., Advances in Mathematical Analysis and Multidisciplinary Applications, CRC Press, Boca Raton, 23 p.
[21]
Noor, M.A., Noor, K.I. and Rassias, M.T. (2021) Strongly Biconvex Functions and Bi-Variational Inequalities. In: Pardalos, P.M. and Rassias, Th.M., Eds., Mathematical Analysis, Optimization, Approximation and Applications, World Scientific Publishing Company, Singapore, 21 p.
[22]
Noor, M.A. and Noor, K.I. (2021) Properties of Higher Order Preinvex Functions. Numerical Algebra, Control and Optimization, 11, 431-441. https://doi.org/10.3934/naco.2020035
[23]
Micherda, B. and Rajba, T. (2012) On Some Hermite-Hadamard-Fejer Inequalities for (k,h)-Convex Functions. Mathematical Inequalities & Applications, 12, 931-940. https://doi.org/10.7153/mia-15-79
[24]
Hazy, A. (2012) Bernstein-Doetsch Type Results for (k,h)-Convex Functions. Miskolc Mathematical Notes, 13, 325-336. https://doi.org/10.18514/MMN.2012.538
[25]
Crestescu, G., Gaianu, M. and Awan, M.U. (2015) Regularity Properties and Integral Inequalities Related to (k,h1,h2)-Convexity of Functions. Annals of West University of Timisoara—Mathematics and Computer Science, 53, 19-35. https://doi.org/10.1515/awutm-2015-0002
[26]
Noor, M.A. (2006) Some New Classes of Nonconvex Functions. Nonlinear Functional Analysis and Applications, 11, 165-171.
[27]
Bregman, L.M. (1967) The Relaxation Method for Finding Common Points of Convex Sets and Its Application to the Solution of Problems in Convex Programming. USSR Computational Mathematics and Mathematical Physics, 7, 200-217. https://doi.org/10.1016/0041-5553(67)90040-7
[28]
Mohsen, B.B., Noor, M.A., Noor, K.I. and Postolache, M. (2019) Strongly Convex Functions of Higher Order Involving Bifunction. Mathematics, 7, 1028. https://doi.org/10.3390/math7111028
[29]
Alabdali, O., Guessab, A. and Schmeisser, G. (2019) Characterizations of Uniform Convexity for Differentiable Functions. Applicable Analysis and Discrete Mathematics, 13, 721-732. https://doi.org/10.2298/AADM190322029A
[30]
Olbrys, A. (2018) A Support Theorem for Generalized Convexity and Its Applications. Journal of Mathematical Analysis and Applications, 458, 1044-1058. https://doi.org/10.1016/j.jmaa.2017.09.038
[31]
Mako, J. and Pales, Z. (2012) On ø-Convexity. Publicationes Mathematicae Debrecen, 80, 107-126.
[32]
Cristescu, G. and Lupsa, L. (2002) Non-Connected Convexities and Applications. Kluwer Academic Publisher, Dordrecht.
[33]
Niculescu, C.P. and Persson, L.E. (2018) Convex Functions and Their Applications. Springer-Verlag, New York.
[34]
Pecaric, J.E., Proschan, F. and Tong, Y.L. (1992) Convex Functions and Statistical Applications. Academic Press, New York.
[35]
Tiel, J.V. (1984) Convex Analysis. John Wiley and Sons, New York.
[36]
Xu, H.K. (1991) Inequalities in Banach Spaces with Applications. Nonlinear Analysis, Theory, Methods and Applications, 16, 1127-1138. https://doi.org/10.1016/0362-546X(91)90200-K
[37]
Bynum, W.L. (1976) Weak Parallelogram laws for Banach Spaces. Canadian Mathematical Bulletin, 19, 269-275. https://doi.org/10.4153/CMB-1976-042-4
[38]
Cheng, R. and Harris, C.B. (2013) Duality of the Weak Parallelogram Laws on Banach Spaces. Journal of Mathematical Analysis and Applications, 404, 64-70. https://doi.org/10.1016/j.jmaa.2013.02.064
[39]
Cheng, R. and Ross, W.T. (2015) Weak Parallelogram Laws on Banach Spaces and Applications to Prediction. Periodica Mathematica Hungarica, 71, 45-58. https://doi.org/10.1007/s10998-014-0078-4
