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

Year 2021, Volume: 5 Issue: 4, 507 - 522, 30.12.2021

Murtala Harbau

https://doi.org/10.31197/atnaa.822150

Abstract

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.

Keywords

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

References

• [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] 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] 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] 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] E. Blum and W. Oettli, From Optimization and Variational inequalities to Equilibrium Problems, Mathematics Students, 63 (1994), 123-145.
• [6] A. Brondsted and R.T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc., 16 (1965), 605-611.
• [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] 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] 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] I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Kluwer, Dordrecht, (1990) Doi:10.1007/978-94-009-212-4.
• [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] 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] 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] 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] P. Daniele, F. Giannessi and A.Maugeri, Equilibrium problems and Variational models, Kluwer Academic, Dordrecht, The Netherlands, 2003.
• [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] 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] C. Garodia, A new fixed point algorithm for finding the solution of a delay differential equation, AIMS Mathematics, 5 (4), (2020), 3182-3200.
• [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] 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] D.V. Hieu, Halpern subgradient extragradient method extended to equilibrium problems, RACSAM, (2016) Doi:10,1007/s13398-016-0328-9.
• [22] D.V. Hieu, Convergence analysis of a new algorithm for strongly pseudomonotone equilibrium problems, Numer. Algorithm, 77(4), (2018), 983-1001.
• [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] J.B. Hiriart-Urruty, Subdifferential calculus in Convex analysis and Optimization, Pitman, London, (1982), 43-92.
• [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] S. Kamimura and W. Takahashi, Strong convergence of proximal-type algorithm in a Banach spaces, SIAM J. Optim., 13 (2002), 938-945.
• [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] I. V. Konnov, Equilibrium models and variationa inequalities, Elsevier, Amsterdam, (2007).
• [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] B.T. Polyak, Some method of speeding up the convergence of the iteration methods, USSR Comput. Math. Phys., 4 (1964) 1-17.
• [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] P. Santos and S. Scheimberg, An inexact subgradient algorithms for equilibrium problems, Comput. Math. Appl., 30 (2011), 91-107.
• [33] Y. Shehu, Iterative approximations for zeros of sum of accretive operators in Banach spaces, J. Function spaces, (2016), Article ID 5973468.
• [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] 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] 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] 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] J.V. Tiel, Convex analysis: An introduction, Wiley, New York, (1984).
• [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] 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] 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] 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] 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] 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.