Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Convergence Theorem for Finding Common Fixed Points of N-generalized Bregman Nonspreading Mapping and Solutions of Equilibrium Problems in Banach Spaces

  • Received : 2019.05.04
  • Accepted : 2020.08.04
  • Published : 2021.09.30
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Abstract

In this paper, we study some fixed point properties of n-generalized Bregman nonspreading mappings in reflexive Banach space. We introduce a hybrid iterative scheme for finding a common solution for a countable family of equilibrium problems and fixed point problems in reflexive Banach space. Further, we give some applications and numerical example to show the importance and demonstrate the performance of our algorithm. The results in this paper extend and generalize many related results in the literature.

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Acknowledgement

The authors sincerely thank the reviewer for his careful reading, constructive comments and fruitful suggestions that substantially improved the manuscript. The first author acknowledges with thanks the bursary and financial support from Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS) Doctoral Bursary. The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF.

References

  1. H. A. Abass, K. O. Aremu, L. O. Jolaoso and O. T. Mewomo, An inertial forward-backward splitting method for approximating solutions of certain optimization problems, J. Nonlinear Funct. Anal., 2020(2020), Art. ID 6, 20 pp.
  2. T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, Modified inertia subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization(2020), DOI:10.1080/02331934.2020.1723586.
  3. B. Ali, M. H. Harbau and L. H. Yusuf, Existence theorems for attractive points of semigroups of Bregman generalized nonspreading mappings in Banach spaces, Adv. Oper. Theory, 2(3)(2017), 257-268.
  4. K. O. Aremu, C. Izuchukwu and G. C. Ugwunnadi, O. T. Mewomo, On the proximal point algorithm and demimetric mappings in CAT(0) spaces, Demonstr. Math., 51(2018), 277-294. https://doi.org/10.1515/dema-2018-0022
  5. H. H. Bauschke and J. M. Borwein, Legendre functions and the method of random Bregman projection, J. Convex Anal., 4(1997), 27-67.
  6. H. H. Bauschke, J. M. Boorwein and P. L. Combettes, Essential smoothness, essential strict convexity and Legendre functions in Banach space, Comm. Contemp. Math, 3(2001), 615-647. https://doi.org/10.1142/S0219199701000524
  7. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math Stud., 63(1994), 123-145.
  8. J. M. Borwein, S. Reich and S. Sabach, A characterization of Bregman firmly nonexpansive operators using a new monotonicity concept, J. Nonlinear Convex Anal, 12(2011), 161-184.
  9. D. Butnariu and A. N. Iusem, Totally convex functions for fixed points computational and infinite dimensional optimization, Kluwer Academic Publishers, Dordrecht, The Netherland, (2000).
  10. D. Butnariu, E. Resmerita, Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal., (2006), 1-39, Article ID: 84919.
  11. P. Daniele, F. Giannessi, and A. Mougeri,(eds.), Equilibrium problems and variational models. Nonconvex optimization and its application, vol. 68. Kluwer Academic Publications, Norwell (2003).
  12. N. Hussain, E. Naraghirad and A. Alotaibi, Existence of common fixed points using Bregman nonexpansive retracts and Bregman functions in Banach spaces, Fixed Point Theory Appl., 2013, 2013:113. https://doi.org/10.1186/1687-1812-2013-113
  13. C. Izuchukwu, G. C. Ugwunnadi, O. T. Mewomo, A. R. Khan, and M. Abbas, Proximal-type algorithms for split minimization problem in p-uniformly convex metric space, Numer. Algorithms, 82(3)(2019), 909-935. https://doi.org/10.1007/s11075-018-0633-9
  14. l. O. Jolaoso, O. T. Alakoya, A. Taiwo and O. T. Mewomo, A parallel combination extragradient method with Armijo line searching for finding common solution of finite families of equilibrium and fixed point problems, Rend. Circ. Mat. Palermo II, (2019), DOI:10.1007/s12215-019-00431-2
  15. L. O. Jolaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo, Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space, Optimization, (2020), DOI:10.1080/02331934.2020.1716752.
  16. L. O. Jolaoso and O. T. Mewomo, On generalized Bregman nonspreading mappings and zero points of maximal monotone operator in a reflexive Banach space, Port. Math., 76(2019), 229-258. doi: 10.4171/PM/2034.
  17. L. O. Jolaoso, K. O. Oyewole, C. C. Okeke and O. T. Mewomo, A unified algorithm for solving split generalized mixed equilibrium problem and fixed point of nonspreading mapping in Hilbert space, Demonstr. Math., 51(2018), 211-232. https://doi.org/10.1515/dema-2018-0015
  18. L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39(1)(2020), Art. 38, 1-28. https://doi.org/10.1007/s40314-019-0964-8
  19. L. O. Jolaoso, A. Taiwo, T. O. Alakoya, and O. T. Mewomo, Strong convergence theorem for solving pseudo-monotone variational inequality problem using projection method in a reflexive Banach space, J. Optim. Theory Appl., (2020), DOI: 10.1007/s10957-020-01672-3.
  20. G. Kassay, S. Reich and S. Sabach, Iterative methods for solving system of variational inequalities in reflexive Banach spaces, SIAM J. Optim., 21(4)(2011), 1319-1344. https://doi.org/10.1137/110820002
  21. K. R. Kazmi, and R. Ali, Common solution to an equilibrium problem and a fixed point problem for an asymptotically quasi-φ-nonexpansive mapping in intermediate sense, RACSAM, 111(2017), 877889.
  22. K. R. Kazmi, R. Ali and S. Yousuf, Generalized equilibrium and fixed point problems for Bregman relatively nonexpansive mappings in Banach spaces, J. Fixed Point Theory Appl., 20:151(2018), doi: 10.1007/s11784-018-0627-1.
  23. F. Kohsaka, Existence of fixed points of nonspreading mappings with Bregman distance, In: Nonlinear Mathematics for Uncertainty and its Applications, Advances in Intelligent and Soft Computing, Vol. 100, 49(2011), pp. 403-410. https://doi.org/10.1007/978-3-642-22833-9_49
  24. F. Kohsaka and W. Takahashi, Proximal point algorithms with Bregman functions in Banach space, J. Nonlinear Convex Anal., 6(2005), 505523.
  25. L. -J. Lin, W. Takahashi and Z. -T. Yu, Attractive point theorems for generalized nonspreading mappings in Banach spaces, J. nonlinear and convex analysis, 14(1)(2013), 1-20.
  26. A. Moudafi, Second order differential proximal methods for equilibrium problems, J. Inequal. Pure Appl. Math., 4(1)(2003), 17.
  27. E. Naraghirad and J. -C. Yao, Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2013(2013), Art. No. 141.
  28. F. U. Ogbuisi and O. T. Mewomo, On split generalized mixed equilibrium problems and fixed point problems with no prior knowledge of operator norm, J. Fixed Point Theory Appl., 19(3)(2016), 2109-2128. https://doi.org/10.1007/s11784-016-0397-6
  29. F. U. Ogbuisi and O. T. Mewomo, Convergence analysis of common solution of certain nonlinear problems, Fixed Point Theory, 19(1)(2018), 335-358. https://doi.org/10.24193/fpt-ro.2018.1.26
  30. G. N. Ogwo, C. Izuchukwu, K. O. Aremu and O. T. Mewomo, A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space, Bull. Belg. Math. Soc. Simon Stevin, 27(2020), 127-152. https://doi.org/10.36045/bbms/1590199308
  31. O. K. Oyewole, H. A. Abass and O. T. Mewomo, Strong convergence algorithm for a fixed point constraint split null point problem, Rend. Circ. Mat. Palermo II, (2020), DOI:10.1007/s12215-020-00505-6.
  32. K. O. Oyewole, L. O. Jolaoso, C. Izuchuwu and O. T. Mewomo, On approximation of common solution of finite family of mixed equilibrium problems with µ-η relaxed monotone operator in a Banach space, Politehin. Uni. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 80(1)(2018), 175-190.
  33. S. Reich and S. Sabach, A strong convergence theorem for proximal type- algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal., 10(2009), 471-485.
  34. S. Reich and S. Sabach, Two strong convergence theorem for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim., 31(13)(2010), 22-44. https://doi.org/10.1080/01630560903499852
  35. S. Reich and S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces, Nonlinear Anal., 73(1)(2010), 122-135. https://doi.org/10.1016/j.na.2010.03.005
  36. S. Reich and S. Sabach, Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces, Contemp. Math., 568(2012), 225240.
  37. S. Sabach, Products of finite many resolvents of maximal monotone mappings in reflexive Banach space, SIAM J. Optim, 21(2011), 1289-1308. https://doi.org/10.1137/100799873
  38. A. Taiwo, T. O. Alakoya and O. T. Mewomo, Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces, Numer. Algorithms, (2020), DOI: 10.1007/s11075-020-00937-2.
  39. A. Taiwo, L. O. Jolaoso and O. T. Mewomo, A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces, Comput. Appl. Math., 38(2)(2019), Article 77.
  40. A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Parallel hybrid algorithm for solving pseudomonotone equilibrium and Split Common Fixed point problems, Bull. Malays. Math. Sci. Soc., 43(2020), 1893-1918. https://doi.org/10.1007/s40840-019-00781-1
  41. A. Taiwo, L. O. Jolaoso and O. T. Mewomo, Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert Spaces, J. Ind. Manag. Optim., (2020), DOI:10.3934/jimo.2020092.
  42. A. Taiwo, L. O. Jolaoso and O. T. Mewomo, General alternative regularization method for solving Split Equality Common Fixed Point Problem for quasipseudocontractive mappings in Hilbert spaces, Ricerche Mat., (2019), DOI:10.1007/s11587-019-00460-0.
  43. W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and its Applications, Yokohama Publishers, Yokohama, 2000.
  44. W. Takahashi, N. -C. Wong and J. -C. Yao, Fixed point theorems for three new nonlinear mappings in Banach spaces, J. Nonlinear Convex Anal., 13(2012), 368-381.
  45. W. Takahashi, N. -C. Wong and J. -C. Yao, Fixed point theorems and convergence theorems for generalized nonspreading mappings in Banach spaces, J. Fixed Point Theory and Appl., 11(2012), 159-183. https://doi.org/10.1007/s11784-012-0074-3
  46. W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mmapings in Banach spaces, Nonlinear Anal., 70(2009) 45-57. https://doi.org/10.1016/j.na.2007.11.031