Inertial hybrid self-adaptive subgradient extragradient method for fixed point of quasi$-\phi-$nonexpansive multivalued mappings and equilibrium problem

Öz

In this paper, we propose a new inertial self-adaptive subgradient extragradient algorithm for approximating common solution in the set of pseudomonotone equilibrium problems and the set of fixed point of finite family of quasi$-\phi-$nonexpansive multivalued mappings in real uniformly convex Banach spaces and uniformly smooth Banach spaces. The step size n is chosen self adaptively and estimates of Lipschizt-type constants are dispensed with. Strong convergence of the iterative scheme is established. Our results generalizes and improves several recent results anouced in the literature.

Anahtar Kelimeler

• Pseudomonotone Equilibrium problem
• Inertial self adaptive hybrid method
• Multivalued quasi$-\phi-$nonexpansive mapping
• Banach spaces

Kaynakça

1. [1] P.N. Anh, A hybrid extragradient method extented to fixed point problems and equilibrium problems, Opt., (2011) DOI:10.1080/02331934.2011.607497.
2. [2] Y.I. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, In: Theory and applications of nonlinear operators of accretive and monotone type. Lecture notes, Pure Appl. Math., (1996), pp. 15-50.
3. [3] B. Ali and M.H. Harbau, Convergence theorems for Bregman K-mappings and mixed equilibrium problems in reflexive Banach spaces, J. Function spaces, (2016) Doi:10.1155/2016/5161682.
4. [4] B. Ali, J.N. Ezeora and M.S. lawan, Inertial algorithm for solving generalized mixed equilibrium problems in Banah spaces, PanAmerican Journal, 29 (2019), 64-83.
5. [5] E. Blum and W. Oettli, From Optimization and Variational inequalities to Equilibrium Problems, Mathematics Students, 63 (1994), 123-145.
6. [6] A. Brondsted and R.T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc., 16 (1965), 605-611.
7. [7] Y. Censor, A. Gibali and S. Riech, The subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl. 148 (2011), 318-335.
8. [8] C.E. Chidume, S.I. Ikechukwu and A. Adamu, Inertial algorithm for approximating a common fixed point for finite family of relatively nonexpansive maps, Fixed Point Theory Appl. 9(2018), Doi:10.1186/s13663-018-0634-3.

9. [9] C.E. Chidume and M.O. Nnakwe, Convergence theorems of subgradient extragradient algorithm for solving variational inequalities and convex feasibility problems, Fixed Point Theory Appl. 16 (2018), Doi:10.1186/s13663-018-0641-4.
10. [10] I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Kluwer, Dordrecht, (1990) Doi:10.1007/978-94-009-212-4.
11. [11] W. Cholamjiak, P. Cholamjiak and S. Suantai, An inertial forwardâ-backward splitting method for solving inclusion problems in Hilbert spaces. Journal of Fixed Point Theory Appl., 20(1), (2018), 1-17.
12. [12] W. Cholamjiak, S.A. Khan, D. Yambangwai and K.R. Kazmi, Strong convergence analysis of common variational inclusion problems involving an inertial parallel monotone hybrid method for a novel application to image restoration. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matematicas, 114(2), (2020), 1-20.
13. [13] W. Cholamjiak, N. Pholasa and S. Suantai, A modified inertial shrinking projection method for solving inclusion problems and quasi-nonexpansive multivalued mappings. Computational and Applied Mathematics, 37(5), (2018), 5750-5774.
14. [14] V. Dadashi, O.S. Iyiola and Y. Shehu, The subgradient extragradient method for pseudomonotone equilibrium problems, Optim., (2019), Doi:10.1080/02331934.2019.1625899.
15. [15] P. Daniele, F. Giannessi and A.Maugeri, Equilibrium problems and Variational models, Kluwer Academic, Dordrecht, The Netherlands, 2003.
16. [16] Q.L. Dong, H.B. Yuan, Y.J. Cho and T.M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Opt. Letters, (2016), Doi:10.1007/s11590-016-1102-9.
17. [17] G.Z. Eskandani, M. Raeisi and T.M. Rassias, A hybrid extragradient method for solving pseudomonotone equilibrium problems using Bregman distance, J. Fixed Point Theory Appl., (2018), Doi:10.1007/s11784-018-0611-9
18. [18] C. Garodia, A new fixed point algorithm for finding the solution of a delay differential equation, AIMS Mathematics, 5 (4), (2020), 3182-3200.
19. [19] C. Garodia and I. Uddin, A new iterative method for solving split feasibility problem, J. Applied Anal. Compt., 10 (3), (2020), 986-1004.
20. [20] K. Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Ad-vanced Mathematics, 28 Cambridge University Press, Cambridge, 1990.
21. [21] D.V. Hieu, Halpern subgradient extragradient method extended to equilibrium problems, RACSAM, (2016) Doi:10,1007/s13398-016-0328-9.
22. [22] D.V. Hieu, Convergence analysis of a new algorithm for strongly pseudomonotone equilibrium problems, Numer. Algorithm, 77(4), (2018), 983-1001.
23. [23] D.V. Hieu and J. J. Strodoit, Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces, J. Fixed Point Theory Appl. (2018) Doi:10.1007/s11784-018-0608-4
24. [24] J.B. Hiriart-Urruty, Subdifferential calculus in Convex analysis and Optimization, Pitman, London, (1982), 43-92.
25. [25] Z. Jouymandi and F. Moradlou, Extragradient and linesearch algorithms for solving equilibrium problems and fixed point problems in Banach spaces, arXiv:1606.01615[Math].
26. [26] S. Kamimura and W. Takahashi, Strong convergence of proximal-type algorithm in a Banach spaces, SIAM J. Optim., 13 (2002), 938-945.
27. [27] S.A. Khan, S. Suantai and W. Cholamjiak, Shrinking projection methods involving inertial forward-backward split- ting methods for inclusion problems. Revista de la Real Academia de Ciencias Exactas, Fi­sicas y Naturales. Serie A. Matematicas, 113(2), (2019), 645-656.
28. [28] I. V. Konnov, Equilibrium models and variationa inequalities, Elsevier, Amsterdam, (2007).
29. [29] A. Phon-on, N. Makaje, A. Sama-Ae, and K. Khongraphan, An inertial S-iteration process, Fixed Point Theory Appl., (2019), Doi;10.1186/s13663-019-0654-7.
30. [30] B.T. Polyak, Some method of speeding up the convergence of the iteration methods, USSR Comput. Math. Phys., 4 (1964) 1-17.
31. [31] H. Rehman, P. Kumam, W. Kumam, M. Shutaywi and W. Jirakitpuwan, The inertial subgradient extragradient method for a class of pseudo-monotone equilibrium problems, Symmetry, (2020), Doi;10.3390/sym12030463.
32. [32] P. Santos and S. Scheimberg, An inexact subgradient algorithms for equilibrium problems, Comput. Math. Appl., 30 (2011), 91-107.
33. [33] Y. Shehu, Iterative approximations for zeros of sum of accretive operators in Banach spaces, J. Function spaces, (2016), Article ID 5973468.
34. [34] J.J. Strodiot, V.H. Nguyen and P.T. Vuong, Stong convergence of two hybrid extragradient methods for solving equilibrium problems and fixed point problems, Vietnam J. Math., 40(2), (2013), 371-389.
35. [35] S. Takahashi and W. Takahashi, Viscosity Approximation methods for Equilibrium problems and ?xed point problems in Hilbert Spaces, J. Math. Anal. and Appl., 331(2007), 506-515.
36. [36] W. Takahashi and K. Zembayash, Strong convergence theorem by new hybrid method for equilibrium problems and reletively nonexpansive mappings, Fixed point Theory and Appl., (2008),Doi. 1155/2008/528476.
37. [37] B. Tan, S. Xu and S. Li, Modi?ed inertial hybrid and shrinking projection algorithm for solving fixed point problems, Mathematics, (2020), 8, 236, Doi:10.3390/math8020236.
38. [38] J.V. Tiel, Convex analysis: An introduction, Wiley, New York, (1984).
39. [39] Q.D. Tran, L. D. Muu and H. V. Nguyen, Extragradient algorithms extended to equilibrium problems, Journal of Optim. 57 (2008) 749-776.
40. [40] N.T. Vinh, and L. D. Muu, Inertial extragradient algorithm for solving equilibrium problem, Acta Math. Vietnamica, (2019), Doi:10.1007/s40306-019-00338-1.
41. [41] D.J. Wen, Weak and Strong convergence of hybrid subgradient method for pseudomonotone equilibrium problem and multivalued nonexpansive mappings, Fixed point theory and appli, 2014:232.
42. [42] J. Yang and H. Liu, The subgradient extragradient method extended to pseudomonotone equilibrium problems and ?xed point problems in Hilbert space, Optim. Letters, (2019), Doi:10.1007/s11590-019-01474-1.
43. [43] S. Zhang, L. Wang and Y.Zhao, Multi-valued quasi-φ-asymptotically nonexpansive semi-groups and strong convergence theorems in Banach spaces, Acta Math. Sci., 33(B), (2013), 589-599.
44. [44] H. Zegeye, E.U. Ofoedu and N. Shahzad, Convergence theorems for equilibrium problems, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings, Appl. Math. Comput., 216 (2010), 3439-3449.