Abstract
We study a weaker and more natural notion of stability called weak \(w^{2}\)-stability to get an insight in the corresponding results obtained by Măruşter and Măruşter (J Comput Appl Math 276:110–116, 2015) and Wang (J Comput Appl Math 285:226–229, 2015). A data dependence result for fixed points of strongly demicontractive operators is also established. Some illustrative examples are given to validate results obtained herein.
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Abbas, M., Ali, B., Butt, A.R.: Existence and data dependence of the fixed points of generalized contraction mappings with applications. Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Math. RACSAM 109, 603–621 (2015)
Alfuraidan, M.R., Khamsi, M.A.: Fibonacci–Mann iteration for monotone asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. (2017). doi:10.1017/S0004972717000120
Berinde, V.: Iterative Approximation of Fixed Points. Springer, Berlin (2007)
Berinde, V.: On the stability of some fixed point procedures. Bul. Ştiinţ. Univ. Baia Mare Ser. B, Matematică-Informatică 18, 7–14 (2002)
Berinde, V.: Summable almost stability of fixed point iteration procedures. Carpathian J. Math. 19, 81–88 (2003)
Cardinali, T., Rubbioni, P.: A generalization of the Caristi fixed point theorem in metric spaces. Fixed Point Theory 11, 3–10 (2010)
Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)
Dehaish, B.I., Khamsi, M.A., Khan, A.R.: Mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces. J. Math. Anal. Appl. 397, 861–868 (2013)
Espínola, R., Petruşel, A.: Existence and data dependence of fixed points for multivalued operators on gauge spaces. J. Math. Anal. Appl. 309, 420–432 (2005)
Gürsoy, F., Karakaya, V., Rhoades, B.E.: Data dependence results of new multi-step and S-iterative schemes for contractive-like operators. Fixed Point Theory Appl. 2013(1), 1–12 (2013)
Harder, A.M., Hicks, T.L.: Stability results for fixed point iteration procedures. Math. Jpn. 33, 693–706 (1988)
Ivan, D., Leuştean, L.: A rate of asymptotic regularity for the Mann iteration of \(\kappa -\)strict pseudo-contractions. Numer. Funct. Anal. Optim. 36, 792–798 (2015)
Karakaya, V., Gürsoy, F., Ertürk, M.: Some convergence and data dependence results for various fixed point iterative methods. Kuwait J. Sci. 43, 112–128 (2016)
Khan, A.R., Gürsoy, F., Karakaya, V.: Jungck–Khan iterative scheme and higher convergence rate. Int. J. Comput. Math. 93, 2092–2105 (2016)
Khan, A.R., Gürsoy, F., Kumar, V.: Stability and data dependence results for Jungck–Khan iterative scheme. Turkish J. Math. 40, 631–640 (2016)
Khan, A.R., Kumar, V., Hussain, N.: Analytical and numerical treatment of Jungck-type iterative schemes. Appl. Math. Comput. 231, 521–535 (2014)
Liu, Z., Kang, S.M., Shim, S.H.: Almost stability of the Mann iteration method with errors for strictly hemi-contractive operators in smooth Banach spaces. J. Korean Math. Soc. 40, 29–40 (2003)
Mann, W.R.: Mean value methods in iterations. Proc. Am. Math. Soc. 4, 506–510 (1953)
Măruşter, L., Măruşter, Ş.: On the error estimation and T-stability of the Mann iteration. J. Comput. Appl. Math. 276, 110–116 (2015)
Măruşter, L., Măruşter, Ş.: Strong convergence of the Mann iteration for \(\alpha -\)demicontractive mappings. Math. Comput. Model. 54, 2486–2492 (2011)
Măruşter, Ş., Rus, I.A.: Kannan contractions and strongly demicontractive mappings. Creat. Math. Inform. 24, 173–182 (2015)
Ostrowski, A.M.: The round-off stability of iterations. ZAMM Z. Angew. Math. Mech. 47, 77–81 (1967)
Rus, I.A., Petruşel, A., Sîntamarian, A.: Data dependence of the fixed point set of some multivalued weakly Picard operators. Nonlinear Anal. 52, 1947–1959 (2003)
Soltuz, S.M., Grosan, T.: Data dependence for Ishikawa iteration when dealing with contractive like operators. Fixed Point Theory Appl. 2008, 1–7 (2008)
Timis, I.: On the weak stability of Picard iteration for some contractive type mappings and coincidence theorems. Int. J. Comput. Appl. 37, 9–13 (2012)
Urabe, M.: Convergence of numerical iteration in solution of equations. J. Sci. Hiroshima Univ. A 19, 479–489 (1956)
Wang, C.: A note on the error estimation of the Mann iteration. J. Comput. Appl. Math. 285, 226–229 (2015)
Wang, C., Zhang, C.Z.: Approximating common fixed points for a pair of generalized nonlinear mappings in convex metric space. J. Nonlinear Sci. Appl. 9, 1–7 (2016)
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The authors would like to thank the anonymous reviewers for their constructive comments to improve quality and presentation of the paper.
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Gürsoy, F., Khan, A.R., Ertürk, M. et al. Weak \(w^{2}\)-stability and data dependence of Mann iteration method in Hilbert spaces. RACSAM 113, 11–20 (2019). https://doi.org/10.1007/s13398-017-0447-y
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DOI: https://doi.org/10.1007/s13398-017-0447-y