Geodesic
Inertial hybrid S-iteration algorithm for fixed point of asymptotically nonexpansive mappings and equilibrium problems in a real Hilbert space
Harbau, M. H. 1 ; Ahmad, A.2
1 Department of Science and Technology Education, Bayero University, Kano, Nigeria
2 Department of Mathematics, Federal College of Education, Katsina, Nigeria
Journal of nonlinear sciences and its applications, Tome 15 (2022) no. 2, p. 123-135.

Voir la notice de l'article provenant de la source International Scientific Research Publications

In this paper, we introduce an inertial hybrid S-iteration algorithm for two asymptotically nonexpansive mappings and equilibrium problems in a real Hilbert space. Strong convergence of the iterative scheme is established. Our results improve and extend many recent results in the literature.
DOI : 10.22436/jnsa.015.02.04
Classification : 47H09, 47J25
Keywords: Asymptotically nonexpansive, inertial S-iteration method, equilibrium problems, fixed point

Harbau, M. H.  1 ; Ahmad, A. 2

1 Department of Science and Technology Education, Bayero University, Kano, Nigeria
2 Department of Mathematics, Federal College of Education, Katsina, Nigeria 
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Harbau, M. H. ; Ahmad, A. Inertial hybrid S-iteration algorithm for fixed point of asymptotically nonexpansive mappings and equilibrium problems in a real Hilbert space. Journal of nonlinear sciences and its applications, Tome 15 (2022) no. 2, p. 123-135. doi : 10.22436/jnsa.015.02.04. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.015.02.04/

[1] Agarwal, R. P.; O'Regan, D.; Sahu, D. R. Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., Volume 8 (2007), pp. 61-79

[2] Ahn, P. K; Hieu, D. V. Parallel Hybrid Iterative Methods for Variational Inequalities, Equilibrium Problems, and Common Fixed Point Problems, Vietnam J. Math., Volume 44 (2016), pp. 351-374

[3] Ali, B.; Harbau, M. H. Convergence theorems for pseudomonotone equilibrium problem, split feasibility problem, and multivalued strictly pseudocontractive mappings, Numer. Funct. Anal. Optim., Volume 40 (2019), pp. 1194-1214 | DOI

[4] Blum, E. From optimization and variational inequalities to equilibrium problems, Math. Student, Volume 63 (1994), pp. 123-145

[5] Bot, R. I.; Csetnek, E. R.; Hendrich, C. Inertial Douglas-Rachford splitting for monotone inclusion problems, Appl. Math. Comput., Volume 256 (2015), pp. 472-487

[6] Ceng, L. C.; Petrusel, A.; Qin, X.; Yao, J. C. Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints, Optimization, Volume 70 (2021), pp. 1337-1358

[7] Ceng, L. C.; Petrusel, A.; Wen, C. F.; Yao, J. C. Inertial-like subgradient extragradient methods for variational inequalities and fixed points of asymptotically nonexpansive and strictly pseudocontractive mappings, Mathematics, Volume 7 (2019), pp. 1-11

[8] Combettes, P. L.; Hirstoaga, S. A. Equilibrium programming in hilbert spaces, J. Nonlinear Convex Anal., Volume 6 (2005), pp. 117-136

[9] Dong, Q. L.; Yuan, H. B.; Cho, Y. J.; Rassias, T. M. Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optim. Lett., Volume 12 (2018), pp. 87-102

[10] Goebel, K.; Kirk, W. A. A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., Volume 35 (1972), pp. 171-174 | DOI

[11] Goebel, K.; Reich, S. Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984

[12] Harbau, M. H. Inertial hybrid self-adaptive subgradient extragradient method for fixed point of quasi-$\phi$-nonexpansive multivalued mappings and equilibrium problem, Adv. Theory Nonlinear Anal. Appl., Volume 5 (2021), pp. 507-522

[13] Harbau, M. H.; Ali, B. Hybrid linesearch algorithm for pseudomonotone equilibrium problem and fixed Points of Bregman quasi asymptotically nonexpansive mappings, J. Linear Topol. Algebra, Volume 10 (2021), pp. 153-177

[14] Hieu, D. V.; Cho, Y. J.; Xiao, Y.-B.; Kumam, P. Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces, Vietnam J. Math., Volume 49 (2020), pp. 1165-1183

[15] Hieu, D. V.; Muu, L. D.; Ahn, P. K. Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algorithms, Volume 73 (2016), pp. 197-217

[16] Hieu, D. V.; Muu, L. D.; Quy, P. K.; Duong, H. N. Regularization extragradient methods for equilibrium programming in Hilbert spaces, Optimization, Volume 70 (2021), pp. 1-31 | DOI

[17] Hieu, D. V.; Strodiot, J. J.; Muu, L. D. Strongly convergent algorithms by using new adaptive regularization parameter for equilibrium problems, J. Comput. Appl. Math., Volume 376 (2020), pp. 1-21

[18] Khan, S. A.; Kazmi, K. R.; Yambangwai, D.; Cholamjiak, W. A hybrid projective method for solving system of equilibrium problems with demicontractive mappings applicable in image restoration problems, Math. Methods Appl. Sci., Volume 43 (2020), pp. 3413-3431

[19] Korpelevich, G. M. The extragradient method for finding saddle points and other problems, Matecon, Volume 12 (1976), pp. 747-756

[20] Lorenz, D. A.; Pock, T. An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vision, Volume 51 (2015), pp. 311-325

[21] Mainge, P.-E. Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., Volume 219 (2008), pp. 223-236

[22] Mann, W. A. Mean value methods in iteration, Proc. Amer. Math. Soc., Volume 4 (1953), pp. 506-510

[23] Martinez-Yanes, C.; Xu, H.-K. Strong convergence of the CQ method for fixed point processes, Nonlinear Anal., Volume 64 (2006), pp. 2400-2411 | DOI

[24] Moudafi, A. Second-order differential proximal methods for equilibrium problems, JIPAM. J. Inequal. Pure Appl. Math., Volume 4 (2003), pp. 1-7

[25] Muu, L. D.; Oettli, W. Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal., Volume 18 (1992), pp. 1159-1166

[26] Oyewole, K. O.; Jolaoso, L. O.; Izuchuwu, C.; Mewomo, O. T. On approximation of common solution of finite family of mixed equilibrium problems with $\mu-\eta$ relaxed monotone operator in a banach space, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., Volume 81 (2019), pp. 19-34

[27] Phon-on, A.; Makaje, N.; Sama-Ae, A.; Khongraphan, K. An inertial $S$-iteration process, Fixed Point Theory Appl., Volume 2019 (2019), pp. 1-14

[28] Picard, E. Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pures Appl., Volume 6 (1890), pp. 145-210 | EuDML

[29] Polyak, B.T. Some methods of speeding up the convergence of iteration methods, U.S.S.R. Comput. Math. Math. Phys., Volume 4 (1964), pp. 1-17 | DOI | Zbl

[30] Sahu, D. R. Applications of the S-iteration process to constrained minimization problems and split feasibility problems, Fixed Point Theory, Volume 12 (2011), pp. 187-204

[31] Shang, M. J.; Su, Y. F.; Qin, X. L. A general iterative method for equilibrium problems and fixed point in Hilbert space, Fixed point Theory Appl., Volume 2007 (2007), pp. 1-9

[32] Suparatulatorn, R.; Cholamjiak, W.; Suantai, S. A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs, Numer. Algorithms, Volume 77 (2018), pp. 479-490 | DOI

[33] Tada, A.; Takahashi, W. Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, In: Nonlinear analysis and convex analysis, Volume 2007 (2007), pp. 609-617

[34] Tada, A.; Takahashi, W. Weak and strong convergence theorems for a nonexpansive mapping and equilibrium problem, J. Optim. Theory Appl., Volume 133 (2007), pp. 359-370

[35] Takahashi, S.; Takahashi, W. Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., Volume 331 (2007), pp. 506-515 | DOI

[36] Takahashi, W.; Takeuchi, Y.; Kubota, R. Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., Volume 341 (2008), pp. 276-286 | DOI

[37] Takahashi, W.; Zembayashi, K. Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., Volume 70 (2009), pp. 45-57

[38] Tan, K.-K.; Xu, H. K. Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., Volume 122 (1994), pp. 733-739

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