Search Advanced

AIMS Mathematics

2023, Volume 8, Issue 12: 28582-28600. doi: 10.3934/math.20231463
Previous Article Next Article
Research article Special Issues

Approximating fixed points of demicontractive mappings in metric spaces by geodesic averaged perturbation techniques

  • 1.
    Center of Excellence in Theoretical and Computational Science (TaCS-CoE) & KMUTT Fixed Point Research Laboratory, Room SCL 802, Fixed Point Laboratory, Science Laboratory Building, Departments of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
  • 2.
    Sule Lamido University Kafin Hausa, Department of Mathematics, Faculty of Natural and Applied Sciences, P.M.B 048, Jigawa, Nigeria
  • 3.
    Department of Mathematics and Computer Science, North University Center at Baia Mare, Technical University of Cluj-Napoca, Victoriei 76, 430122 Baia Mare, Romania
  • 4.
    Academy of Romanian Scientists, Ilfov Str. No. 3, 050045 Bucharest, Romania
  • Received: 23 August 2023 Revised: 21 September 2023 Accepted: 09 October 2023 Published: 20 October 2023

  • MSC : 47H05, 47H10, 47H25

  • In this article, we introduce the fundamentals of the theory of demicontractive mappings in metric spaces and expose the key concepts and tools for building a constructive approach to approximating the fixed points of demicontractive mappings in this setting. By using an appropriate geodesic averaged perturbation technique, we obtained strong convergence and $ \Delta $-convergence theorems for a Krasnoselskij-Mann type iterative algorithm to approximate the fixed points of quasi-nonexpansive mappings within the framework of CAT(0) spaces. The main results obtained in this nonlinear setting are natural extensions of the classical results from linear settings (Hilbert and Banach spaces) for both quasi-nonexpansive mappings and demicontractive mappings. We applied our results to solving an equilibrium problem in CAT(0) spaces and showed how we can approximate the equilibrium points by using our fixed point results. In this context we also provided a numerical example in the case of a demicontractive mapping that is not a quasi-nonexpansive mapping and highlighted the convergence pattern of the algorithm in Table 1. It is important to note that the numerical example is set in non-Hilbert CAT(0) spaces.

    Citation: Sani Salisu, Vasile Berinde, Songpon Sriwongsa, Poom Kumam. Approximating fixed points of demicontractive mappings in metric spaces by geodesic averaged perturbation techniques[J]. AIMS Mathematics, 2023, 8(12): 28582-28600. doi: 10.3934/math.20231463

    Related Papers:

  • In this article, we introduce the fundamentals of the theory of demicontractive mappings in metric spaces and expose the key concepts and tools for building a constructive approach to approximating the fixed points of demicontractive mappings in this setting. By using an appropriate geodesic averaged perturbation technique, we obtained strong convergence and $ \Delta $-convergence theorems for a Krasnoselskij-Mann type iterative algorithm to approximate the fixed points of quasi-nonexpansive mappings within the framework of CAT(0) spaces. The main results obtained in this nonlinear setting are natural extensions of the classical results from linear settings (Hilbert and Banach spaces) for both quasi-nonexpansive mappings and demicontractive mappings. We applied our results to solving an equilibrium problem in CAT(0) spaces and showed how we can approximate the equilibrium points by using our fixed point results. In this context we also provided a numerical example in the case of a demicontractive mapping that is not a quasi-nonexpansive mapping and highlighted the convergence pattern of the algorithm in Table 1. It is important to note that the numerical example is set in non-Hilbert CAT(0) spaces.


    加载中


    [1] A. Adamu, A. Adam, Approximation of solutions of split equality fixed point problems with applications, Carpathian J. Math., 37 (2021), 381–392. https://doi.org/10.37193/CJM.2021.03.02 doi: 10.37193/CJM.2021.03.02
    [2] R. Agarwal, D. O'Regan, D. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8 (2007), 61–79.
    [3] K. Aremu, L. Jolaoso, C. Izuchukwu, O. Mewomo, Approximation of common solution of finite family of monotone inclusion and fixed point problems for demicontractive multivalued mappings in CAT(0) spaces, Ricerche Mat., 69 (2020), 13–34. https://doi.org/10.1007/s11587-019-00446-y doi: 10.1007/s11587-019-00446-y
    [4] T. Bantaojai, C. Garodia, I. Uddin, N. Pakkaranang, P. Yimmuang, A novel iterative approach for solving common fixed point problems in geodesic spaces with convergence analysis, Carpathian J. Math., 37 (2021), 145–160. https://doi.org/10.37193/CJM.2021.02.01 doi: 10.37193/CJM.2021.02.01
    [5] I. Berg, I. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195–218. https://doi.org/10.1007/s10711-008-9243-3 doi: 10.1007/s10711-008-9243-3
    [6] V. Berinde, Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), 293–304. https://doi.org/10.37193/cjm.2019.03.04 doi: 10.37193/cjm.2019.03.04
    [7] V. Berinde, M. Păcurar, Approximating fixed points of enriched contractions in Banach spaces, J. Fixed Point Theory Appl., 22 (2020), 38. https://doi.org/10.1007/s11784-020-0769-9 doi: 10.1007/s11784-020-0769-9
    [8] V. Berinde, Approximating fixed points of demicontractive mappings via the quasi-nonexpansive case, Carpathian J. Math., 39 (2023), 73–84. https://doi.org/10.37193/cjm.2023.01.04 doi: 10.37193/cjm.2023.01.04
    [9] V. Berinde, M. Pǎcurar, Fixed points theorems for unsaturated and saturated classes of contractive mappings in Banach spaces, Symmetry, 13 (2021), 713. https://doi.org/10.3390/sym13040713 doi: 10.3390/sym13040713
    [10] M. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Berlin: Springer, 1999. https://doi.org/10.1007/978-3-662-12494-9
    [11] F. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc., 74 (1968), 660–665. https://doi.org/10.1090/s0002-9904-1968-11983-4 doi: 10.1090/s0002-9904-1968-11983-4
    [12] F. Browder, W. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc., 72 (1966), 571–575. https://doi.org/10.1090/s0002-9904-1966-11544-6 doi: 10.1090/s0002-9904-1966-11544-6
    [13] F. Browder, W. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228. https://doi.org/10.1016/0022-247x(67)90085-6 doi: 10.1016/0022-247x(67)90085-6
    [14] F. Bruhat, J. Tits, Groupes réductifs sur un corps local, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 41 (1972), 5–251. https://doi.org/10.1007/bf02715544 doi: 10.1007/bf02715544
    [15] K. Calderón, M. Khamsi, J. Martínez-Moreno, Perturbed approximations of fixed points of nonexpansive mappings in $\rm CAT_p(0)$ spaces, Carpathian J. Math., 37 (2021), 65–79. https://doi.org/10.37193/CJM.2021.01.07 doi: 10.37193/CJM.2021.01.07
    [16] M. Choban, About convex structures on metric spaces, Carpathian J. Math., 38 (2022), 391–404. https://doi.org/10.37193/cjm.2022.02.10 doi: 10.37193/cjm.2022.02.10
    [17] S. Dhompongsa, W. Kirk, B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 65 (2006), 762–772. https://doi.org/10.1016/j.na.2005.09.044 doi: 10.1016/j.na.2005.09.044
    [18] S. Dhompongsa, B. Panyanak, On $\Delta$-convergence theorems in CAT $(0)$ spaces, Comput. Math. Appl., 56 (2008), 2572–2579. https://doi.org/10.1016/j.camwa.2008.05.036 doi: 10.1016/j.camwa.2008.05.036
    [19] W. Dotson, Fixed points of quasi-nonexpansive mappings, J. Aust. Math. Soc., 13 (1972), 167–170. https://doi.org/10.1017/s144678870001123x doi: 10.1017/s144678870001123x
    [20] M. Eslamian, General algorithms for split common fixed point problem of demicontractive mappings, Optimization, 65 (2016), 443–465. https://doi.org/10.1080/02331934.2015.1053883 doi: 10.1080/02331934.2015.1053883
    [21] Q. Fan, J. Peng, H. He, Weak and strong convergence theorems for the split common fixed point problem with demicontractive operators, Optimization, 70 (2021), 1409–1423. https://doi.org/10.1080/02331934.2021.1890074 doi: 10.1080/02331934.2021.1890074
    [22] A. Hanjing, S. Suantai, The split fixed point problem for demicontractive mappings and applications, Fixed Point Theory, 21 (2020), 507–524. https://doi.org/10.24193/fpt-ro.2020.2.37 doi: 10.24193/fpt-ro.2020.2.37
    [23] A. Hanjing, S. Suantai, Y. Cho, Hybrid inertial accelerated extragradient algorithms for split pseudomonotone equilibrium problems and fixed point problems of demicontractive mappings, Filomat, 37 (2023), 1607–1623. https://doi.org/10.2298/FIL2305607H doi: 10.2298/FIL2305607H
    [24] T. Hicks, J. Kubicek, On the Mann iteration process in a Hilbert space, J. Math. Anal. Appl., 59 (1977), 498–504. https://doi.org/10.1016/0022-247x(77)90076-2 doi: 10.1016/0022-247x(77)90076-2
    [25] A. Inuwa, P. Chaipunya, P. Kumam, S. Salisu, Equilibrium problems and proximal algorithm using tangent space products, Carpathian J. Math., in press.
    [26] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150. https://doi.org/10.1090/s0002-9939-1974-0336469-5 doi: 10.1090/s0002-9939-1974-0336469-5
    [27] S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59 (1976), 65–71. https://doi.org/10.1090/s0002-9939-1976-0412909-x doi: 10.1090/s0002-9939-1976-0412909-x
    [28] P. Jailoka, S. Suantai, Split null point problems and fixed point problems for demicontractive multivalued mappings, Mediterr. J. Math., 15 (2018), 204. https://doi.org/10.1007/s00009-018-1251-4 doi: 10.1007/s00009-018-1251-4
    [29] P. Jailoka, S. Suantai, Split common fixed point and null point problems for demicontractive operators in Hilbert spaces, Optim. Method. Softw., 34 (2019), 248–263. https://doi.org/10.1080/10556788.2017.1359265 doi: 10.1080/10556788.2017.1359265
    [30] P. Jailoka, S. Suantai, The split common fixed point problem for multivalued demicontractive mappings and its applications, RACSAM, 113 (2019), 689–706. https://doi.org/10.1007/s13398-018-0496-x doi: 10.1007/s13398-018-0496-x
    [31] K. Juagwon, W. Phuengrattana, Iterative approaches for solving equilibrium problems, zero point problems and fixed point problems in Hadamard spaces, Comp. Appl. Math., 42 (2023), 75. https://doi.org/10.1007/s40314-023-02209-w doi: 10.1007/s40314-023-02209-w
    [32] Y. Kimura, Resolvents of equilibrium problems on a complete geodesic space with curvature bounded above, Carpathian J. Math., 37 (2021), 463–476. https://doi.org/10.37193/CJM.2021.03.09 doi: 10.37193/CJM.2021.03.09
    [33] W. Kirk, N. Shahzad, Fixed point theory in distance spaces, Cham: Springer, 2014. https://doi.org/10.1007/978-3-319-10927-5
    [34] M. Krasnosel'skii, Two remarks on the method of successive approximations, Uspekhi Matematicheskikh Nauk, 10 (1955), 123–127.
    [35] W. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510. https://doi.org/10.1090/s0002-9939-1953-0054846-3 doi: 10.1090/s0002-9939-1953-0054846-3
    [36] M. Minjibir, S. Salisu, Strong and $\Delta$-convergence theorems for a countable family of multivalued demicontractive maps in Hadamard spaces, Nonlinear Functional Analysis and Applications, 27 (2022), 45–58. https://doi.org/10.22771/nfaa.2022.27.01.03 doi: 10.22771/nfaa.2022.27.01.03
    [37] C. Moore, Iterative approximation fixed points of demicontractive maps, Proceedings of The Abdus Salam International Centre for Theoretical Physics, 1998, 1–12.
    [38] Ş. Mǎruşter, Sur le calcul des zéros d'un opérateur discontinu par itération, Can. Math. Bull., 16 (1973), 541–544. https://doi.org/10.4153/cmb-1973-088-7 doi: 10.4153/cmb-1973-088-7
    [39] Ş. Mǎruşter, The solution by iteration of nonlinear equations in Hilbert spaces, Proc. Amer. Math. Soc., 63 (1977), 69–73. https://doi.org/10.1090/s0002-9939-1977-0636944-2 doi: 10.1090/s0002-9939-1977-0636944-2
    [40] M. Nnakwe, J. Ezeora, Strong convergence theorems for variational inequalities and fixed point problems in Banach spaces, Carpathian J. Math., 37 (2021), 477–487. https://doi.org/10.37193/cjm.2021.03.10 doi: 10.37193/cjm.2021.03.10
    [41] M. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217–229. https://doi.org/10.1006/jmaa.2000.7042 doi: 10.1006/jmaa.2000.7042
    [42] F. Ogbuisi, F. Isiogugu, A new iterative algorithm for pseudomonotone equilibrium problem and a finite family of demicontractive mappings, Abstr. Appl. Anal., 2020 (2020), 3183529. https://doi.org/10.1155/2020/3183529 doi: 10.1155/2020/3183529
    [43] A. Padcharoen, P. Kumam, Y. Cho, Split common fixed point problems for demicontractive operators, Numer. Algor., 82 (2019), 297–320. https://doi.org/10.1007/s11075-018-0605-0 doi: 10.1007/s11075-018-0605-0
    [44] S. Panja, M. Saha, R. Bisht, Existence of common fixed points of non-linear semigroups of enriched Kannan contractive mappings, Carpathian J. Math., 38 (2022), 169–178. https://doi.org/10.37193/cjm.2022.01.14 doi: 10.37193/cjm.2022.01.14
    [45] S. Salisu, M. Minjibir, P. Kumam, S. Sriwongsa, Convergence theorems for fixed points in CATp(0) spaces, J. Appl. Math. Comput., 69 (2023), 631–650. https://doi.org/10.1007/s12190-022-01763-6 doi: 10.1007/s12190-022-01763-6
    [46] S. Salisu, P. Kumam, S. Sriwongsa, On fixed points of enriched contractions and enriched nonexpansive mappings, Carpathian J. Math., 39 (2022), 237–254. https://doi.org/10.37193/cjm.2023.01.16 doi: 10.37193/cjm.2023.01.16
    [47] S. Salisu, P. Kumam, S. Sriwongsa, Strong convergence theorems for fixed point of multi-valued mappings in Hadamard spaces, J. Inequal. Appl., 2022 (2022), 143. https://doi.org/10.1186/s13660-022-02870-5 doi: 10.1186/s13660-022-02870-5
    [48] S. Salisu, P. Kumam, S. Sriwongsa, Convergence theorems for monotone vector field inclusions and minimization problems in Hadamard spaces, Anal. Geom. Metr. Space., 11 (2023), 20220150. https://doi.org/10.1515/agms-2022-0150 doi: 10.1515/agms-2022-0150
    [49] S. Suantai, P. Jailoka, A self-adaptive algorithm for split null point problems and fixed point problems for demicontractive multivalued mappings, Acta Appl. Math., 170 (2020), 883–901. https://doi.org/10.1007/s10440-020-00362-6 doi: 10.1007/s10440-020-00362-6
    [50] R. Suparatulatorn, P. Cholamjiak, S. Suantai, Self-adaptive algorithms with inertial effects for solving the split problem of the demicontractive operators, RACSAM, 114 (2020), 40. https://doi.org/10.1007/s13398-019-00737-x doi: 10.1007/s13398-019-00737-x
    [51] A. Wang, J. Zhao, Self-adaptive iterative algorithms for the split common fixed point problem with demicontractive operators, J. Nonlinear Var. Anal., 5 (2021), 573–587.
    [52] Y. Wang, X. Fang, J. Guan, T. Kim, On split null point and common fixed point problems for multivalued demicontractive mappings, Optimization, 70 (2021), 1121–1140. https://doi.org/10.1080/02331934.2020.1764952 doi: 10.1080/02331934.2020.1764952
    [53] F. Wang, The split feasibility problem with multiple output sets for demicontractive mappings, J. Optim. Theory Appl., 195 (2022), 837–853. https://doi.org/10.1007/s10957-022-02096-x doi: 10.1007/s10957-022-02096-x
    [54] J. Xiao, Y. Wang, A viscosity method with inertial effects for split common fixed point problems of demicontractive mappings, J. Nonlinear Funct. Anal., 2022 (2022), 17. https://doi.org/10.23952/jnfa.2022.17 doi: 10.23952/jnfa.2022.17
    [55] Y. Yao, Y. Liou, M. Postolache, Self-adaptive algorithms for the split problem of the demicontractive operators, Optimization, 67 (2018), 1309–1319. https://doi.org/10.1080/02331934.2017.1390747 doi: 10.1080/02331934.2017.1390747
    [56] Y. Yao, J. Yao, Y. Liou, M. Postolache, Iterative algorithms for split common fixed points of demicontractive operators without priori knowledge of operator norms, Carpathian J. Math., 34 (2018), 459–466. https://doi.org/10.37193/cjm.2018.03.23 doi: 10.37193/cjm.2018.03.23
    [57] C. Zhang, Y. Li, Y. Wang, On solving split generalized equilibrium problems with trifunctions and fixed point problems of demicontractive multi-valued mappings, J. Nonlinear Convex Anal., 21 (2020), 2027–2042.
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
2.2 3

Metrics

Article views(628) PDF downloads(89) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(1)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog