International Journal of Nonlinear Analysis and Applications

On fixed point approximation method for finite family of $k$-strictly pseudo-contractive mappings and pseudomonotone equilibrium problem in Hadamard space.

Document Type : Research Paper

Authors

1 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

2 DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa

3 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, P.O. Box 60, 0204, South Africa

4 Department of Mathematics, Usmanu Danfodiyo University Sokoto, PMB 2346, Sokoto State, Nigeria

Abstract

In this paper, we first introduce the Halpern iteration process for approximating the solution of the fixed point problem of a finite family of $k$-strictly pseudo-contractive mappings in Hadamard spaces. We also propose an extra gradient Halpern iterative algorithm for approximating a common solution of a finite family of $k_j$-strictly pseudocontractive mappings and a pseudomonotone equilibrium problem in Hadamard space. We prove a strong convergence result without imposing any strict (compactness) conditions for approximating the solutions to the aforementioned problems. We state some consequences of our results and display some numerical examples to show the performance of our results. Our results improve and generalize many recent results in the literature.

Keywords

[1] H. A. Abass, C. Izuchukwu and K. O. Aremu, A common solution of family of minimization and fixed point problem for multi-valued type-one demicontractive-type mappings, Adv. Nonlinear Var. Inequal. 21 (2018), no.2, 94–108.
[2] 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 problem, J. Nonlinear Funct. Anal. 2020 (2020), Article ID 6.
[3] H.A. Abass, A.A. Mebawondu, K.O. Aremu and O.K. Oyewole, Generalized viscosity approximation method for minimization and fixed point problems of quasi-pseudocontractive mappings in Hadamard spaces, Asian European J. Math. 2250188, https://doi.org/10.1142/S1793557122501881.
[4] M. Abbas, V. Parvaneh and A. Razani, Periodic points of T-Ciric generalized contraction mappings in ordered metric spaces, Georgian Math. J. 19 (2012), 597–610.
[5] P.N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optim. 62 (2013), no. 2, 271–283.
[6] K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), 2350–2360.
[7] K. O. Aremu, H. A. Abass, C. Izuchukwu and O. T. Mewomo, A viscosity type algorithm for an infinitely many countable family of (f, g)-generalized k-strictly pseudononspreading mappings inn CAT(0) spaces, Anal. 40 (2020), no. 1, 19–37.
[8] K.O. Aremu, C. Izuchukwu, G.C. Ugwunnadi and O.T. Mewomo, On the proximal point algorithm and demimetric mappings in CAT(0) spaces, Demonstr. Math. 51 (2018), 277–294.
[9] I.D. Berg and I.G. Nikolaev, Quasilinearization and curvature of Alexandrov spaces, Geom. Dedicata 133 (2008), 195 218.
[10] F.E. Browder an W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967), 197–228.
[11] C.E. Chidume, A.U. Bello and P. Ndambomve, Strong and △− convergence theorems for common fixed points of a finite family of multi-valued demicontractive mappings in CAT(0) spaces, Abstr. Appl. Anal. 2014 (2014), 6 pages.
[12] S.Y. Cho and S.M. Kang, Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process, Appl. Math. Lett. 24 (2011), 224–228.
[13] P. Cholamjiak, A.A.N. Abdou, Y.J. Cho, Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces, Fixed Point Theory Appl. 2015 (2015), 227.
[14] H. Dehghan and J. Rooin, Metric projection and convergence theorems for nonexpansive mapping in Hadamard spaces, arXiv:1410.1137VI [math.FA], 5 Oct. 2014.
[15] H. Dehghan and J. Rooin, A characterization of metric projection in CAT(0) spaces, Int. Conf. Funct. Equ. Geo. Funct. Appl. Payame University, Tabriz, 2012, pp. 41—43.
[16] B.C. Dhage, G.P. Kample and R.G. Metkar, On generalized Merlin-Hardy integral transformations, Eng-Math. Lett. 2 (2013), 67-80.
[17] S. Dhompongsa, W.A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 64 (2006), 762–772.
[18] S. Dhompongsa and B. Panyanak, On ∆-convergence theorems in CAT(0) spaces, Comp. Math. Appl. 56 (2008), 2572–257.
[19] A. Gharajelo and H. Dehghan, Convergence theorems for strict pseudo-contractions in CAT(0) space, Filomat 31 (2017), no. 7, 1967–1971.
[20] D.V. Hieu, Common solution to pseudomonotone equilibrium problems, Bull. Iran. Math. Soc. 42 (2016), no. 5, 1207 1219.
[21] N. Hussain, J.R. Roshan, V. Parvaneh and A. Latif, A unification of G-metric, partial metric, and b-metric spaces, Abstr. Appl. Anal. 2014 (2014), Article ID 180698, 1–14.
[22] A.N. Iusem and V. Mohebbi, Convergence analysis of the extragradient method for equilibrium problems in Hadamard spaces, Comput. Appl. Math., 39 (2020), no. 2, 1–21.
[23] C. Izuchukwu, H. A. Abass and O.T. Mewomo, Viscosity approximation method for solving minimization problem and fixed point problem for nonexpansive multivalued mapping in CAT (0) spaces, Ann. Acad. Rom. Sci. Ser. Math. Appl. 11 (2019), no. 1.
[24] C. Izuchukwu, A. A. Mebawondu, K. O. Aremu, H. A. Abass and O. T. Mewomo, Viscosity iterative techniques for approximating a common zero of monotone operators in a Hadamard space, Rend. Circ. Mat. Palermo Series 2 69 (2020), no. 2, 475–495.
[25] J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comment. Math. Helv. 70 (1995), 659–673.
[26] B. A. Kakavandi and M. Amini, Duality and subdifferential for convex functions on complete CAT(0) metric spaces, Nonlinear Anal. 73 (2010), 3450–3455.
[27] A. Kangtunyakarn, S. Suantai, Strong convergence of a new iterative scheme for a finite family of strictpseudocontraction, Comput. Math. Appl. 60 (2010), 680–694.
[28] H. Khatibzadeh and V. Mohebbi, Monotone and pseudo-monotone equilibrium problems in Hadamard spaces, J. Aust. Math. Soc. 110 (2021), no. 2, 220–242.
[29] H. Khatibzadeh and V. Mohebbi, Approximating solutions of equilibrium problems in Hadamard spaces, Miskolc Math Notes 20 (2019), 281–297
[30] Y. Kimura and S. Saejung, Strong convergence for a common fixed points of two different generalizations of cutter operators, Linear Nonlinear Anal. 1 (2015), 53–65.
[31] W.A. Kirk, Geodesic geometry and fixed point theory, Seminar of Math. Anal. (Malaga/Seville, 2002/2003), Univ. Sevilla Secr. Publ., Seville, 2003, pp. 195–225.
[32] W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), 3689–3696.
[33] Y.C. Liou, Computing the fixed points of strictly pseudocontractive mappings by the implicit abd explicit iterations, Abstr. Appl. Anal. 2012 (2012), Article ID 315835, 1–13.
[34] P.E. Maing’e, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), no. 7, 899–912.
[35] B. Martinet, Regularisation variationelles par approximations successives, Rev. Fr. Inform. Rech. Oper.,- 4 (1970), 154–158.
[36] T.J. Markin, Fixed points, selections and best approximations for multi-valued mappings in R-trees, Nonlinear Anal. 67 (2007), 2712–2716.
[37] R. Moharami and· G. Zamani Eskandani, An extragradient algorithm for solving equilibrium problem and zero point problem in Hadamard spaces, RACSAM 114 (2020), 154.
[38] S. Park, A review of the KKM theory on ϕA-space or GFC spaces, Adv. Fixed Point Theory 3 (2013), 355–382.
[39] R. Suparatulatorn, P. Cholamjiak and S. Suantai, On solving the minimization problem and the fixed point problem for nonexpansive mappings in CAT(0) space, Optim. Meth. Software 32 (2017), no. 1, 182–192.
[40] X. Qin, M. Shang and Y. Qing, Common fixed points of a family of strictly pseudocontractive mappings, Fixed Point Theory Appl. 2013 (2013), 298.
[41] J. Tits, A theorem of Liekolchin for trees. Contributions to algebra: A collection of papers dedicated dedicated to Ellis Kolchin, Academic Press, New York, 1977.
[42] L.Q. Thuy, C.F. Wen, J.C. Yao and T.N. Hai, An extragradient-like parallel method for pseudomonotone equilibrium problems and semigroup nonexpansive mappings, Miskolc Math. Notes 19 (2018), no. 2, 1185.
[43] Z.M. Wang, Convergence theorems on total asymptotically pseudocontractive mapping, J. Math. Comput. Sc. 3 (2013), 788–798.
[44] H.K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc. 66 (2002), 240–256.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 1
January 2023
Pages 11-24
  • Receive Date: 27 October 2021
  • Revise Date: 25 April 2022
  • Accept Date: 29 April 2022