Korean J. Math. Vol. 30 No. 4 (2022) pp.665-678
DOI: https://doi.org/10.11568/kjm.2022.30.4.665

A new iteration method for fixed point of nonexpansive mapping in uniformly convex Banach space

Main Article Content

Omprakash Sahu Sahu
Amitabh Banerjee
Niyati Gurudwan

Abstract

The aim of this paper is to introduce a new iterative process and show that our iteration scheme is faster than other existing iteration schemes with the help of numerical examples. Next, we have established convergence and stability results for the approximation of fixed points of the contractive-like mapping in the framework of uniformly convex Banach space. In addition, we have established some convergence results for the approximation of the fixed points of a nonexpansive mapping.



Article Details

References

[1] A.G.Sanatee, L.Rathour, V.N.Mishra and V.Dewangan, Some fixed point theorems in regular modular metric spaces and application to Caratheodory’s type anti-periodic boundary value problem, The Journal of Analysis (2022), 1–14, DOI: https://doi.org/10.1007/ s41478-022-00469-z. Google Scholar

[2] A.M.Harder, Fixed point theory and stability results for fixed point iteration procedures. PhD thesis, University of Missouri-Rolla, Missouri, MO, USA, 1987. Google Scholar

[3] B.S.Thakur, D.Thakur and M.Postolache, A new iterative scheme for numerical reckoning fixed points of suzuki’s generalized nonexpansive mappings, App. Math. Comp. 275 (2016), 147–155. Google Scholar

[4] C.O.Imoru and M.O.Olantiwo , On the stability of Picard and Mann iteration process, Carpath. J. Math. 19 (2) (2003), 155–160. Google Scholar

[5] D.P.Shukla, S.K.Tiwari and A.Gautam, Common fixed point theorems for two steps iterative scheme using nonexpansive mappings, Ultra scientist 22 (3) (2010), 867–870. Google Scholar

[6] D.P.Shukla and S.K.Tiwari, Common fixed point theorems for three steps iterative scheme using nonexpansive mappings, Napier Indian Advanced Research Journal of Science 8 (2012), 5–8. Google Scholar

[7] H.F.Senter and W.G.Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (2) (1974), 375–380. Google Scholar

[8] J.Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), 153–159. Google Scholar

[9] K.Goebel and W.A.Kirk, Topic in metric fixed point theory, Cambridge University Press, 1990. Google Scholar

[10] K.Ullah and M.Arshad, Numerical reckoning fixed points for Suzuki generalized nonexpansive mappings via new iteration process, Filomat 32 (1) (2018), 187–196. Google Scholar

[11] L.N.Mishra, S.K.Tiwari and V.N.Mishra, Fixed point theorems for generalized weakly S- contractive mappings in partial metric spaces, J. Appl. Anal. Comput. 5 (5) (2015), 600–612. Google Scholar

[12] L.N.Mishra, S.K.Tiwari, V.N.Mishra and I.A.Khan, Unique fixed point theorems for generalized contractive mappings in partial metric spaces, J. Funct. Spaces 2015, Art. ID. 960827, 8pp. Google Scholar

[13] L.N.Mishra, V.Dewangan, V.N.Mishra and H. Amrulloh, Coupled best proximity point theorems for mixed g-monotone mappings in partially ordered metric spaces, J. Math. Comput. Sci. 11 (5) (2021), 6168–6192. Google Scholar

[14] M.Abbas, T.Nazir, A new faster iteration process applied to constrained minimation and feasi- bility problems, Matematicki Vesnik, vol. 66 (2) (2014), 223–234. Google Scholar

[15] M.A.Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217–229. Google Scholar

[16] N.Sharma, L.N.Mishra, V.N.Mishra and H.Almusawa, Endpoint approximation of standard three-step multi-valued iteration algorithm for nonexpansive mappings, Appl. Math. Inf. Sci.15 (1) (2021), 73–81. Google Scholar

[17] N.Sharma, L.N.Mishra, V.N.Mishra and S.Pandey, Solution of Delay Differential equation via N1v iteration algorithm, European J. Pure Appl. Math. 13 (5) (2020), 1110–1130. Google Scholar

[18] N.Sharma, L.N.Mishra, S.N.Mishra and V.N.Mishra, Empirical study of new iterative algorithm for generalized nonexpansive operators, J Math Comput Sci. 25 (3) (2022), 284–295. Google Scholar

[19] P.Shahi, L.Rathour and V.N.Mishra, Expansive Fixed Point Theorems for tri-simulation functions, J. Eng. Exact Sci. –jCEC, 8 (3) (2022), 14303–01e. Google Scholar

[20] R.P.Agarwal, D.O’Regan and D.R.Sahu, Iterative construction of fixed points of nearly asymptoticall nonexpansive mappings, J. Nonlinear Convex Anal. 8 (1) (2007), 61–79. Google Scholar

[21] S.Hassan, M.De la Sen, P.Agrawal, Q.Ali and A. Hussain, A new faster iterative scheme for numerical fixed points estimation of suzuki’s generalized non-expansive mappings, Math. Probl. Eng. 2020, Art. ID 3863819, 9pp. Google Scholar

[22] S.Ishikawa, Fixed points by new iteration method, Proc. Amer. Math. Soc. 149 (1974), 147–150. Google Scholar

[23] S.B.Nadlor, Multivalued contraction mappings, Pac. J. Math. 30 (1969), 475–488. Google Scholar