Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
권호 표지
DOI QR코드

DOI QR Code

GENERAL BICONVEX FUNCTIONS AND BIVARIATIONAL-LIKE INEQUALITIES

  • Received : 2021.02.18
  • Accepted : 2021.12.05
  • Published : 2022.03.15
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider and introduce some new concepts of the biconvex functions involving an arbitrary bifunction and function. Some new relationships among various concepts of biconvex functions have been established. We have shown that the optimality conditions for the general biconvex functions can be characterized by a class of bivariational-like inequalities. Auxiliary principle technique is used to propose proximal point methods for solving general bivariational-like inequalities. We also discussed the conversance criteria for the suggested methods under pseudo-monotonicity. Our method of proof is very simple compared with methods. Several special cases are discussed as applications of our main concepts and results. It is a challenging problem to explore the applications of the general bivariational-like inequalities in pure and applied sciences.

Keywords

Acknowledgement

We wish to express our deepest gratitude to our colleagues, students, collaborators and friends, who have direct or indirect contributions in the process of this paper.

References

  1. A. Ben-Isreal and B. Mond, What is invexity? J. Austral Math. Soc. Ser. B, 28(1) (1986), 1-9. https://doi.org/10.1017/S0334270000005142
  2. L.M. Bregman, The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys., 7 (1967), 200-217. https://doi.org/10.1016/0041-5553(67)90040-7
  3. G. Cristescu and L. Lupsa, Non-Connected Convexities and Applications, Kluwer Academic Publisher, Dordrechet, 2002.
  4. R. Glowinski, J.L. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.
  5. L.O. Jolaoso, M. Aphane and S.H. Khan, Two Bregman projection methods for solving variational inequality problems in Hilbert spaces with applications to signal processing, Symmetry, 12 (2020); doi:10.3390/sym12122007.
  6. S.A. Khan and F. Suhel, Vector variational-like inequalities with pseudo semi-monotone mappings, Nonlinear Funct. Anal. Appl., 16(2) (2011), 191-200.
  7. J.L. Lions and G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493-519, https://doi.org/10.1002/cpa.3160200302
  8. C.P. Niculescu and L.E. Persson, Convex Functions and Their Applications, Springer-Verlag, New York, 2018.
  9. M.A. Noor. On Variational Inequalities, PhD Thesis, Brunel University, London, U. K., 1975.
  10. M.A. Noor, Variational-like inequalities, Optimization, 30 (1994), 323-333. https://doi.org/10.1080/02331939408843995
  11. M.A. Noor, New approximation schemes for generalvariational inequalities, J. Math. Anal. Appl., 151 (2000), 217-229. https://doi.org/10.1006/jmaa.2000.7042
  12. M.A. Noor, Some developments in general variational inequalities, Appl. Math. Comput., 251 (2004), 199-277. https://doi.org/10.1016/S0096-3003(03)00558-7
  13. M.A. Noor, Invex equilibrium problems, J. Math. Anal. Appl., 302 (2005), 463-475. https://doi.org/10.1016/j.jmaa.2004.08.014
  14. M.A. Noor, Fundamentals of equilibrium problems, Math. Inequal. Appl., 9(3) (2006), 529-566.
  15. M.A. Noor, K.I. Noor and E. Al-Said, Auxiliary Principle Technique for Solving Bifunction Variational Inequalities, J. Optim. Theory Appl., 149 (2011), 441-445. https://doi.org/10.1007/s10957-010-9785-z
  16. M.A. Noor, K.I. Noor and Th.M. Rassias, New trends in general variational inequalities, Acta Appl. Math., 170(1) (2020), 981-1046. https://doi.org/10.1007/s10440-020-00366-2
  17. M.A, Noor and K.I. Noor, Higher order strongly general convex functions and variational inequalities, AIMS Math., 5(4) (2020), 3646-3663. https://doi.org/10.3934/math.2020236
  18. M.A, Noor and K.I. Noor, General biconvex functions and bivariational inequalities, Numer. Algebr. Contro, Optim., 12 (2022): doi: 10.3934/naco.2021041.
  19. M.A. Noor and K.I. Noor, Higher order strongly biconvex functions and biequilibrium problems, Adv. Lin. Algeb, Matrix Theory, 11 (2021), 31-53. https://doi.org/10.4236/alamt.2021.112004
  20. M.A. Noor and K.I. Noor, Strongly general bivariational inequalities, Trans. J. Math. Mech., 13(102) (2021), 45-58.
  21. M.A. Noor, K.I. Noor and Th.M. Rassias, Some aspects of variational inequalities, J. Appl. Math. Comput., 47 (1993), 485-512.
  22. J. Pecaric, F. Proschan and Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1992.
  23. G. Stampacchia, Formes bilineaires coercivites sur les ensembles convexes, C. R. Acad. Paris, 258 (1964), 4413-4416.
  24. P. Sunthrayuth and P. Cholamjiak, Modified extragradient method with Bregman distance for variational inequalities, Appl. Anal., 2020, doi.org/10.1080/00036811.2020.1757078.
  25. H. Yu, S. Wu and C.Y. Jung, Solvability of a system of generalized nonlinear mixed variational-like inequalities, Nonlinear Funct. Anal. Appl., 23(1) (2018), 181-203. https://doi.org/10.22771/NFAA.2018.23.01.14
  26. 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 (1996), 714-726. https://doi.org/10.1137/S1052623494250415