Some convergence results using a new iterative algorithm in CAT(0) space
Year 2022,
Volume: 5 Issue: 3, 263 - 272, 30.09.2022
Anju Panwar
,
Pinki Lamba
,
Santosh Kumar
Abstract
This paper presents a new iterative algorithm for approximating the invariant points of Suzuki’s generalized nonexpansive maps. Some strong convergence theorems are developed in the context of CAT(0) space. We also included examples to demonstrate the proposed algorithm’s convergence nature. Lastly, the stability of the said iterative algorithm is discussed to validate the results
Supporting Institution
None
References
- [1] T. Abdeljawad, K. Ullah, J. Ahmad, M. Sen, and M. N. Khan, Some convergence results for a class of generalized
nonexpansive mappings in Banach spaces, Advances in Mathematical Physics, (2021), 2021, 6 pages. Article ID 8837317.
- [2] J. Ahmad, K. Ullah, M. Arshad and Z. Ma, A new iterative method for suzuki mappings in Banach spaces, Journal of
Mathematics,(2021), 2021, 7 pages. Article ID 6622931.
- [3] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, In:Graduate studies in Math., Amer. Math. Soc.,
Providence, Rhode Island, (2001).
- [4] F. Bruhat and J. Tits, Groups rekductifss sur un corps local. I. DonneKes radicielles valueKes, Publ. Math. Inst. Hautes
EKtudes Sci., 41,(1972), 5-251.
- [5] M. R. Bridson, and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften,
319 (1999), Springer, Berlin.
- [6] M. Gromov, Hyperbolic Groups. Essays in group theory, Math. Sci. Res. Inst. Publ., 8 (1987), Springer, New York.
- [7] A. Ghiura, Convergence of modi?ed Picard-Mann hybrid iteration process for nearly nonexpansive mappings, International
Journal of Mathematics Trends and Technology, 66 (12) (2020), 37-43.
- [8] A. M. Harder, Fixed point theory and stability results for fixed points iteration procedures. Ph. D. Thesis, (1987), University
of MissouriRolla.
- [9] A. M. Harder and T. L. Hicks, Stability results for fixed point iteration procedures, Math. Japonica, 33(5) (1988), 693-706.
- [10] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44(1974), 147-150.
- [11] W. A. Kirk, Geodesic geometry and fixed point theory. In Seminar of Mathematical Analysis (Malaga/Seville,2002/2003),
Colecc. Abierta. University Seville Secretary of Publications, Seville, Spain, 64 (2003), 195-225.
- [12] W. A. Kirk, Geodesic geometry and ?xed point theory II. In International Conference on Fixed point Theory and Apllications, Yokohama Publishers, Yokohama, Japan, 2004: pp.113-142.
- [13] W. R. Mann, Mean value methods in Iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.
- [14] A. Pansuwan and W. Sintunavarat, The new hybrid iterative algorithm for numerical reckoning fixed points of Suzuki's
generalized nonexpansive mappings with numerical experiments, Thai Journal of Mathematics, 19(1) (2021), 157-168.
- [15] T. Suzuki, Fixed points theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal.
Appl., 340 (2008), 1088-1095.
- [16] B.S. Thakur, B. Thakur and M. Postolache, A new iterative scheme for numerical reckoning ?xed points of Suzuki's
generalized nonexpansive mappings, Appl. Math. Comput., 275 (2016), 147-155.
- [17] T. Thianwan, Mixed type algorithms for asymptotically nonexpansive mappings in hyperbolic spaces, European Journal
of Pure and Applied Mathematics, 14(3) (2021), 650-665.
- [18] K. Ullah and M. Arshad, New iteration process and numerical reckoning ?xed points in Banach spaces, U.P.B. Sci. Bull.,
Series A, 79(4) (2017), 113-122.
- [19] K. Ullah and M. Arshad, Numerical reckoning ?xed points for Suzuki's generalized nonexpansive mappings via new iteration
process, Filomat, 32(1) (2018), 187-196.
- [20] X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc., 113(1991), 727-731.
- [21] B. Xu and M. A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces. Journal of
Mathematical Analysis and Applications, 267(2) (2002), 444-453.
- [22] Z. Xue, The convergence of ?xed point for a kind of weak contraction. Nonlinear Func. Anal. Appl., 21(3) (2016), 497-500.
- [23] D. Yambangwai and T. Thianwan, ∆-Convergence and strong convergence for asymptotically nonexpansive mappings on
a CAT(0) space, Thai Journal of Mathematics, 19(3) (2021), 813-826.
Year 2022,
Volume: 5 Issue: 3, 263 - 272, 30.09.2022
Anju Panwar
,
Pinki Lamba
,
Santosh Kumar
References
- [1] T. Abdeljawad, K. Ullah, J. Ahmad, M. Sen, and M. N. Khan, Some convergence results for a class of generalized
nonexpansive mappings in Banach spaces, Advances in Mathematical Physics, (2021), 2021, 6 pages. Article ID 8837317.
- [2] J. Ahmad, K. Ullah, M. Arshad and Z. Ma, A new iterative method for suzuki mappings in Banach spaces, Journal of
Mathematics,(2021), 2021, 7 pages. Article ID 6622931.
- [3] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, In:Graduate studies in Math., Amer. Math. Soc.,
Providence, Rhode Island, (2001).
- [4] F. Bruhat and J. Tits, Groups rekductifss sur un corps local. I. DonneKes radicielles valueKes, Publ. Math. Inst. Hautes
EKtudes Sci., 41,(1972), 5-251.
- [5] M. R. Bridson, and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften,
319 (1999), Springer, Berlin.
- [6] M. Gromov, Hyperbolic Groups. Essays in group theory, Math. Sci. Res. Inst. Publ., 8 (1987), Springer, New York.
- [7] A. Ghiura, Convergence of modi?ed Picard-Mann hybrid iteration process for nearly nonexpansive mappings, International
Journal of Mathematics Trends and Technology, 66 (12) (2020), 37-43.
- [8] A. M. Harder, Fixed point theory and stability results for fixed points iteration procedures. Ph. D. Thesis, (1987), University
of MissouriRolla.
- [9] A. M. Harder and T. L. Hicks, Stability results for fixed point iteration procedures, Math. Japonica, 33(5) (1988), 693-706.
- [10] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44(1974), 147-150.
- [11] W. A. Kirk, Geodesic geometry and fixed point theory. In Seminar of Mathematical Analysis (Malaga/Seville,2002/2003),
Colecc. Abierta. University Seville Secretary of Publications, Seville, Spain, 64 (2003), 195-225.
- [12] W. A. Kirk, Geodesic geometry and ?xed point theory II. In International Conference on Fixed point Theory and Apllications, Yokohama Publishers, Yokohama, Japan, 2004: pp.113-142.
- [13] W. R. Mann, Mean value methods in Iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.
- [14] A. Pansuwan and W. Sintunavarat, The new hybrid iterative algorithm for numerical reckoning fixed points of Suzuki's
generalized nonexpansive mappings with numerical experiments, Thai Journal of Mathematics, 19(1) (2021), 157-168.
- [15] T. Suzuki, Fixed points theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal.
Appl., 340 (2008), 1088-1095.
- [16] B.S. Thakur, B. Thakur and M. Postolache, A new iterative scheme for numerical reckoning ?xed points of Suzuki's
generalized nonexpansive mappings, Appl. Math. Comput., 275 (2016), 147-155.
- [17] T. Thianwan, Mixed type algorithms for asymptotically nonexpansive mappings in hyperbolic spaces, European Journal
of Pure and Applied Mathematics, 14(3) (2021), 650-665.
- [18] K. Ullah and M. Arshad, New iteration process and numerical reckoning ?xed points in Banach spaces, U.P.B. Sci. Bull.,
Series A, 79(4) (2017), 113-122.
- [19] K. Ullah and M. Arshad, Numerical reckoning ?xed points for Suzuki's generalized nonexpansive mappings via new iteration
process, Filomat, 32(1) (2018), 187-196.
- [20] X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc., 113(1991), 727-731.
- [21] B. Xu and M. A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces. Journal of
Mathematical Analysis and Applications, 267(2) (2002), 444-453.
- [22] Z. Xue, The convergence of ?xed point for a kind of weak contraction. Nonlinear Func. Anal. Appl., 21(3) (2016), 497-500.
- [23] D. Yambangwai and T. Thianwan, ∆-Convergence and strong convergence for asymptotically nonexpansive mappings on
a CAT(0) space, Thai Journal of Mathematics, 19(3) (2021), 813-826.