Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CONSTRUCTION OF A SOLUTION OF SPLIT EQUALITY VARIATIONAL INEQUALITY PROBLEM FOR PSEUDOMONOTONE MAPPINGS IN BANACH SPACES

  • Received : 2021.07.15
  • Accepted : 2022.02.10
  • Published : 2022.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

The purpose of this paper is to introduce an iterative algorithm for approximating a solution of split equality variational inequality problem for pseudomonotone mappings in the setting of Banach spaces. Under certain conditions, we prove a strong convergence theorem for the iterative scheme produced by the method in real reflexive Banach spaces. The assumption that the mappings are uniformly continuous and sequentially weakly continuous on bounded subsets of Banach spaces are dispensed with. In addition, we present an application of our main results to find solutions of split equality minimum point problems for convex functions in real reflexive Banach spaces. Finally, we provide a numerical example which supports our main result. Our results improve and generalize many of the results in the literature.

Keywords

References

  1. H. Attouch, J. Bolte, P. Redont, and A. Soubeyran, Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and PDE's, J. Convex Anal. 15 (2008), no. 3, 485-506.
  2. H. Attouch, A. Cabot, P. Frankel, and J. Peypouquet, Alternating proximal algorithms for linearly constrained variational inequalities: application to domain decomposition for PDE's, Nonlinear Anal. 74 (2011), no. 18, 7455-7473. https://doi.org/10.1016/j.na.2011.07.066
  3. J.-B. Baillon and G. Haddad, Quelques proprietes des operateurs angle-bornes et n-cycliquement monotones, Israel J. Math. 26 (1977), no. 2, 137-150. https://doi.org/10.1007/BF03007664
  4. H. H. Bauschke and J. M. Borwein, Legendre functions and the method of random Bregman projections, J. Convex Anal. 4 (1997), no. 1, 27-67.
  5. H. H. Bauschke, J. M. Borwein, and P. L. Combettes, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math. 3 (2001), no. 4, 615-647. https://doi.org/10.1142/S0219199701000524
  6. J. Y. Bello Cruz and A. N. Iusem, A strongly convergent direct method for monotone variational inequalities in Hilbert spaces, Numer. Funct. Anal. Optim. 30 (2009), no. 1-2, 23-36. https://doi.org/10.1080/01630560902735223
  7. O. A. Boikanyo and H. Zegeye, Split equality variational inequality problems for pseudomonotone mappings in Banach spaces, Stud. Univ. Babe,s-Bolyai Math. 66 (2021), no. 1, 139-158. https://doi.org/10.24193/subbmath.2021.1.13
  8. J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000. https://doi.org/10.1007/978-1-4612-1394-9
  9. D. Butnariu and A. N. Iusem, Totally convex functions for fixed points computation and infinite dimensional optimization, Applied Optimization, 40, Kluwer Academic Publishers, Dordrecht, 2000. https://doi.org/10.1007/978-94-011-4066-9
  10. D. Butnariu and E. Resmerita, Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. 2006 (2006), Art. ID 84919, 39 pp. https://doi.org/10.1155/AAA/2006/84919
  11. G. Cai and S. Bu, An iterative algorithm for a general system of variational inequalities and fixed point problems in q-uniformly smooth Banach spaces, Optim. Lett. 7 (2013), no. 2, 267-287. https://doi.org/10.1007/s11590-011-0415-y
  12. Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms 8 (1994), no. 2-4, 221-239. https://doi.org/10.1007/BF02142692
  13. Y. Censor, A. Gibali, and S. Reich, The split variational inequality problem, The Technion-Israel Institute of Technology, Haifa arXiv:1009.3780 (2010).
  14. Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34 (1981), no. 3, 321-353. https://doi.org/10.1007/BF00934676
  15. P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math. 115 (1966), 271-310. https://doi.org/10.1007/BF02392210
  16. H. Iiduka, W. Takahashi, and M. Toyoda, Approximation of solutions of variational inequalities for monotone mappings, PanAmer. Math. J. 14 (2004), no. 2, 49-61.
  17. A. N. Iusem and M. Nasri, Korpelevich's method for variational inequality problems in Banach spaces, J. Global Optim. 50 (2011), no. 1, 59-76. https://doi.org/10.1007/s10898-010-9613-x
  18. K. M. T. Kwelegano, O. A. Boikanyo, and H. Zegeye, An iterative method for split equality variational inequality problems for non-Lipschitz pseudomonotone mappings, Rendiconti del Circolo Matematico di Palermo Series 2 (2021), 1-24. https://doi.org/10.1007/s12215-021-00608-8
  19. P.-E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), no. 7-8, 899-912. https://doi.org/10.1007/s11228-008-0102-z
  20. J. Mashreghi and M. Nasri, Forcing strong convergence of Korpelevich's method in Banach spaces with its applications in game theory, Nonlinear Anal. 72 (2010), no. 3-4, 2086-2099. https://doi.org/10.1016/j.na.2009.10.009
  21. A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal. 79 (2013), 117-121. https://doi.org/10.1016/j.na.2012.11.013
  22. A. Moudafi, Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal. 15 (2014), no. 4, 809-818.
  23. N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006), no. 1, 191-201. https://doi.org/10.1007/s10957-005-7564-z
  24. E. Naraghirad and J.-C. Yao, Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2013 (2013), 141, 43 pp. https://doi.org/10.1186/1687-1812-2013-141
  25. Y. Nesterov, Introductory lectures on convex optimization, Applied Optimization, 87, Kluwer Academic Publishers, Boston, MA, 2004. https://doi.org/10.1007/978-1-4419-8853-9
  26. M. A. Noor, A class of new iterative methods for general mixed variational inequalities, Math. Comput. Modelling 31 (2000), no. 13, 11-19. https://doi.org/10.1016/S0895-7177(00)00108-4
  27. R. R. Phelps, Convex functions, monotone operators and differentiability, second edition, Lecture Notes in Mathematics, 1364, Springer-Verlag, Berlin, 1993.
  28. S. Reich and S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces, Nonlinear Anal. 73 (2010), no. 1, 122-135. https://doi.org/10.1016/j.na.2010.03.005
  29. S. Reich and S. Sabach, A projection method for solving nonlinear problems in reflexive Bnanch spaces, J. Fixed Point Theory Appl. (2011), 101-116.
  30. R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209-216. http://projecteuclid.org/euclid.pjm/1102977253 https://doi.org/10.2140/pjm.1970.33.209
  31. P. Senakka and P. Cholamjiak, Approximation method for solving fixed point problem of Bregman strongly nonexpansive mappings in reflexive Banach spaces, Ric. Mat. 65 (2016), no. 1, 209-220. https://doi.org/10.1007/s11587-016-0262-3
  32. P. T. Vuong and Y. Shehu, Convergence of an extragradient-type method for variational inequality with applications to optimal control problems, Numer. Algorithms 81 (2019), no. 1, 269-291. https://doi.org/10.1007/s11075-018-0547-6
  33. G. B. Wega and H. Zegeye, Split equality methods for a solution of monotone inclusion problems in Hilbert spaces, Linear Nonlinear Anal. 5 (2019), no. 3, 495-516.
  34. G. B. Wega and H. Zegeye, Convergence results of forward-backward method for a zero of the sum of maximally monotone mappings in Banach spaces, Comput. Appl. Math. 39 (2020), no. 3, Paper No. 223, 16 pp. https://doi.org/10.1007/s40314-020-01246-z
  35. G. B. Wega and H. Zegeye, Convergence theorems of common solutions of variational inequality and f-fixed point problems in Banach spaces, Appl. Set-Valued Anal. Optim. 3 (2021), no. 1, 55-73.
  36. H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), no. 1, 279-291. https://doi.org/10.1016/j.jmaa.2004.04.059
  37. C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. https://doi.org/10.1142/9789812777096
  38. H. Zegeye, E. U. Ofoedu, and N. Shahzad, Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings, Appl. Math. Comput. 216 (2010), no. 12, 3439-3449. https://doi.org/10.1016/j.amc.2010.02.054
  39. H. Zegeye and N. Shahzad, A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems, Nonlinear Anal. 74 (2011), no. 1, 263-272. https://doi.org/10.1016/j.na.2010.08.040
  40. H. Zegeye and N. Shahzad, Construction of a common solution of a finite family of variational inequality problems for monotone mappings, J. Nonlinear Sci. Appl. 9 (2016), no. 4, 1645-1657. https://doi.org/10.22436/jnsa.009.04.21
  41. H. Zegeye and G. B. Wega, Approximation of a common f-fixed point of f-pseudocontractive mappings in Banach spaces, Rend. Circ. Mat. Palermo (2) 70 (2021), no. 3, 1139-1162. https://doi.org/10.1007/s12215-020-00549-8