Iterative solution of split equilibrium and fixed point problems in real Hilbert spaces

Volume 14, Issue 5, pp 359--371 http://dx.doi.org/10.22436/jnsa.014.05.06

Publication Date: April 19, 2021 Submission Date: January 22, 2021 Revision Date: February 27, 2021 Accteptance Date: March 18, 2021

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 889 Downloads
• 1772 Views

Authors

J. N. Ezeora - Department of Mathematics and Statistics, University of Port Harcpourt, Nigeria. P. C. Jackreece - Department of Mathematics and Statistics, University of Port Harcpourt, Nigeria.

Abstract

In this article, we introduce a hybrid iteration involving inertial-term for split equilibrium problem and fixed point for a finite family of asymptotically strictly pseudocontractive mappings. We prove that the sequence converges strongly to a solution of split equilibrium problem and a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings. The results proved extend and improve recent results of Chang et al. [S. S. Chang, H. W. J. Lee, C. K. Chan, L. Wang, L. J. Qin, Appl. Math. Comput., \(\bf 219\) (2013), 10416--10424], Dewangan et al. [R. Dewangan, B. S. Thakur, M. Postolache, J. Inequal. Appl., \(\bf 2014\) (2014), 11 pages], and many others.

Share and Cite

Keywords

• Total asymptotically strict pseudocontractive mapping
• split equilibrium problem
• fixed point problem
• inertial-step
• bounded linear operator

MSC

•  47H09
•  47H10
•  49M05
•  54H25

References

• [1] Y. I. Alber, C. E. Chidume, H. Zegeye, Approximating fixed points of total asymptotically nonexpansive mappings, Fixed Point Theory Appl., 2006 (2006), 20 pages
  • View Article


• [2] P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optimization, 62 (2013), 271--283
  • View Article


• [3] K. Aoyama, F. Kohsaka, W. Takahashi, Three generalizations of firmly nonexpansive mappings: their relations and continuity properties, J. Nonlinear Convex Anal., 10 (2009), 131--147
  • Google Scholar


• [4] Y. Arfat, P. Kumam, P. Sa Ngiamsunthorn, M. A. A. Khan, H. Sarwar, H. Fukhar-ud-Din, Approximation results for split equilibrium problems and fixed point problems of nonexpansive semigroup in Hilbert spaces, Adv. Difference Equ., 2020 (2020), 21 pages
  • View Article


• [5] A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183--202
  • View Article


• [6] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student., 63 (1994), 123--145
  • View Article


• [7] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197--228
  • View Article


• [8] C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problems, Inverse Problem, 18 (2002), 441--453
  • View Article


• [9] L.-C. Ceng, J.-C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math., 214 (2008), 186--201
  • View Article


• [10] Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353--2365
  • View Article


• [11] Y. Censor, T. Elfving, A multi-projection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221--239
  • View Article


• [12] Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multi-set split feasibility problem and its applications, Inverse Problem, 21 (2005), 2071--2084
  • View Article


• [13] Y. Censor, A. Motov, A. Segal, Pertured projections and subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), 1244--1256
  • View Article


• [14] S. S. Chang, H. W. J. Lee, C. K. Chan, L. Wang, L. J. Qin, Split feasibility problem for Quasi-nonexpansive multi-valued mappings and total asymptotically strict pseudo contractive mappings, Appl. Math. Comput., 219 (2013), 10416--10424
  • View Article


• [15] C. Chidume, Geometric properties of Banach spaces and nonlinear iterations, Springer Verlag Series: Lecture Notes in Mathematics, Berlin (2009)
  • Google Scholar


• [16] P. L. Combettes, The convex feasibility problem in image recovery, Adv. Imaging Electron Phys., 95 (1996), 155--270
  • View Article


• [17] P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117--136
  • View Article


• [18] R. Dewangan, B. S. Thakur, M. Postolache, A hybrid iteration for asymptotically strictly pseudocontractive mappings, J. Inequal. Appl.,, 2014 (2014), 11 pages
  • View Article


• [19] J. N. Ezeora, Iterative Solution of Fixed Points Problem, System of Generalized Mixed Equilibrium Problems and Variational Inclusion Problems, Thai J. Math., 12 (2014), 223--244
  • View Article


• [20] J. N. Ezeora, R. C. Ogbonna, Split feasibilty problem for countable family of multi-valued nonlinear mappings, Mathematicki Vesnik, 70 (2018), 233--242
  • View Article


• [21] D. V. Hieu, An inertial-like proximal algorithm for equilibrium problems, Math. Methods Oper. Res., 88 (2018), 399--415
  • View Article


• [22] C. Izuchukwu, F. O. Isiogugu, C. C. Okeke, A new viscosity-type iteration for a finite family of split variational inclusion and fixed point problems between Hilbert and Banach spaces, J. Inequal. Appl., 2019 (2019), 33 pages
  • View Article


• [23] C. Izuchukwu, C. C. Okeke, F. O. Isiogugu, A viscosity iterative technique for split variational inclusion and fixed point problems between a Hilbert space and a Banach space, J. Fixed Point Theory Appl., 20 (2018), 25 pages
  • View Article


• [24] Q. Liu, Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear Anal., 26 (1996), 1835--1842
  • View Article


• [25] L. V. Long, D. V. Thong, V. T. Dung, New algorithms for the split variational inclusion problems and application to split feasibility problems, Optimization, 68 (2019), 2335--2363
  • View Article


• [26] Z. Ma, L. Wang, An algorithm with strong convergence for the split common fixed point problem of total asymptotically strict pseudocontraction mappings,, J. Inequal. Appl., 2015 (2015), 13 pages
  • View Article


• [27] E. Masad, S. Reich, A note on the multiple-set split feasibility problem in Hilbert spaces, J. Nonlinear Convex Anal., 8 (2007), 367--371
  • Google Scholar


• [28] S. Plubtieng, K. Sombut, Weak convergence theorems for a system of mixed equilibrium problems and nonspreading mappings in a Hilbert space, J. Inequal. Appl., 2010 (2010), 12 pages
  • View Article


• [29] B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Ussr Comput. Math. Math. Phys., 4 (1964), 1--17
  • View Article


• [30] Y. Shehu, Fixed Point Solutions of Generalized Equilibrium Problems for Nonexpansive Mappings, J. Comput. Appl. Math., 234 (2010), 892--898
  • View Article


• [31] K. Sitthithakerngkiet, J. Deepho, J. Martınez-Moreno, P. Kumam, Convergence analysis of a general iterative algorithm for finding a common solution of split variational inclusion and optimization problems, Numer. Algorithms, 79 (2018), 801--824
  • View Article


• [32] S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal., 69 (2008), 1025--1033
  • View Article


• [33] C. Udriste, Convex Functions and Optimization Methods on Riemannian Manifolds, Kluwer Academic Publishers Group, Dordrecht (1994)
  • View Article


• [34] H.-K. Xu, Iterative methods for split feasibility problem in infinite dimensional Hilbert spaces, Inverse Problems, 26 (2010), 17 pages
  • View Article


• [35] L. Yang, S.-S. Chang, Y. J. Cho, J. K. Kim, Multiple-set split feasibility problems for total asymp-totically strict pseudocontractions mappings, Fixed Point Theory Appl., 2011 (2011), 1--11
  • View Article


• [36] J. Zhao, D. Hou, A self-adaptive iterative algorithm for the split common fixed point problems, Numer. Algorithms, 82 (2019), 1047--1063
  • View Article