Research article Special Issues

Common fixed point of nonlinear contractive mappings

  • Received: 18 August 2022 Revised: 22 September 2022 Accepted: 25 September 2022 Published: 09 October 2022
  • MSC : 47H10, 54H25

  • The purpose of this paper is to study the existence of a common fixed point for a pair of mappings without assumption of the contractive coefficient being fixed and less than 1. By replacing the fixed contractive coefficient with a nonlinear contractive function, we establish a unique common fixed point theorem for a pair of asymptotically regular self-mappings with either orbital continuity or $ q $-continuity in a metric space. Moreover, by the asymptotical regularity of two approximate mappings, we prove that a pair of nonexpansive and continuous self-mappings, which are defined on a nonempty closed convex subset of a Banach space, have a common fixed point. Some examples are given to illustrate that our results are extensions of a recent result in the existing literature.

    Citation: Hui Huang, Xue Qian. Common fixed point of nonlinear contractive mappings[J]. AIMS Mathematics, 2023, 8(1): 607-621. doi: 10.3934/math.2023028

    Related Papers:

  • The purpose of this paper is to study the existence of a common fixed point for a pair of mappings without assumption of the contractive coefficient being fixed and less than 1. By replacing the fixed contractive coefficient with a nonlinear contractive function, we establish a unique common fixed point theorem for a pair of asymptotically regular self-mappings with either orbital continuity or $ q $-continuity in a metric space. Moreover, by the asymptotical regularity of two approximate mappings, we prove that a pair of nonexpansive and continuous self-mappings, which are defined on a nonempty closed convex subset of a Banach space, have a common fixed point. Some examples are given to illustrate that our results are extensions of a recent result in the existing literature.



    加载中


    [1] P. Agarwal, M. Jleli, B. Samet, Fixed point theory in metric spaces, Springer, 2018. https://doi.org/10.1007/978-981-13-2913-5
    [2] R. K. Bisht, A note on the fixed point theorem of G$\acute{\hbox{o}}$rnicki, J. Fix. Point Theory A., 21 (2019), 54. https://doi.org/10.1007/s11784-019-0695-x doi: 10.1007/s11784-019-0695-x
    [3] R. K. Bisht, N. K. Singh, On asymptotic regularity and common fixed points, J. Anal., 28 (2020), 847–852. https://doi.org/10.1007/s41478-019-00213-0 doi: 10.1007/s41478-019-00213-0
    [4] N. Boonsatit, G. Rajchakit, R. Sriraman, C. P. Lim, P. Agarwal, Finite-/fixed-time synchronization of delayed Clifford-valued recurrent neural networks, Adv. Differ. Equ., 2021 (2021), 276. https://doi.org/10.1186/s13662-021-03438-1 doi: 10.1186/s13662-021-03438-1
    [5] F. E. Browder, W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Am. Math. Soc., 72 (1966), 571–575. https://doi.org/10.1090/S0002-9904-1966-11544-6 doi: 10.1090/S0002-9904-1966-11544-6
    [6] S. Carl, S. Heikkil$\ddot{\hbox{a}}$, Fixed point theory in ordered sets and applications, Springer, 2011. https://doi.org/10.1007/978-1-4419-7585-0
    [7] L. Ćirić, On contraction type mappings, Math. Balk., 1 (1971), 52–57.
    [8] M. A. Geraghty, On contractive mappings, Proc. Am. Math. Soc., 40 (1973), 604–608. https://doi.org/10.1090/S0002-9939-1973-0334176-5 doi: 10.1090/S0002-9939-1973-0334176-5
    [9] J. Górnicki, Remarks on asymptotic regularity and fixed points, J. Fix. Point Theory A., 21 (2019), 29. https://doi.org/10.1007/s11784-019-0668-0 doi: 10.1007/s11784-019-0668-0
    [10] A. Granas, J. Dugundji, Fixed point theory, Springer, 2003. https://doi.org/10.1007/978-0-387-21593-8
    [11] H. A. Hammad, P. Agarwal, J. L. G. Guirao, Applications to boundary value problems and homotopy theory via tripled fixed point techniques in partially metric spaces, Math., 9 (2021), 1–22. https://doi.org/10.3390/math9162012 doi: 10.3390/math9162012
    [12] S. Hassan, M. D. Sen, P. Agarwal, Q. Ali, A. Hussain, A new faster iterative scheme for numerical fixed points estimation of Suzuki's generalized nonexpansive mappings, Math. Probl. Eng., 2020 (2020), 1–9. https://doi.org/10.1155/2020/3863819 doi: 10.1155/2020/3863819
    [13] H. Huang, J. Zou, A note on the paper "a common fixed point theorem with applications", J. Optimiz. Theory Appl., 168 (2016), 1087–1090. https://doi.org/10.1007/s10957-015-0836-3 doi: 10.1007/s10957-015-0836-3
    [14] A. R. Khan, D. M. Oyetunbi, On some mappings with a unique common fixed point, J. Fix. Point Theory A., 22 (2020), 47. https://doi.org/10.1007/s11784-020-00781-w doi: 10.1007/s11784-020-00781-w
    [15] A. Pant, R. P. Pant, Fixed points and continuity of contractive maps, Filomat, 31 (2017), 3501–3506. https://doi.org/10.2298/FIL1711501P doi: 10.2298/FIL1711501P
    [16] T. Rasham, P. Agarwal, L. S. Abbasi, S. Jain, A study of some new multivalued fixed point results in a modular like metric space with graph, J. Anal., 30 (2022), 833–844. https://doi.org/10.1007/s41478-021-00372-z doi: 10.1007/s41478-021-00372-z
    [17] B. Wang, H. Jahanshahi, C. Volos, S. Bekiros, A. Yusuf, P. Agarwal, A. A. Aly, Control of a symmetric chaotic supply chain system using a new fixed-time super-twisting sliding mode technique subject to control input limitations, Symmetry, 13 (2021), 1257. https://doi.org/10.3390/sym13071257 doi: 10.3390/sym13071257
    [18] H. K. Xu, An iterative approach to quadratic optimization, J. Optimiz. Theory Appl., 116 (2003), 659–678. https://doi.org/10.1023/A:1023073621589 doi: 10.1023/A:1023073621589
  • 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

Metrics

Article views(1391) PDF downloads(138) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog