Some new results on convergence, stability and data dependence in n-normed spaces

Year 2020, Volume: 69 Issue: 1, 112 - 122, 30.06.2020

Kadri Doğan , Faik Gürsoy , Vatan Karakaya , Safeer Hussain Khan

https://doi.org/10.31801/cfsuasmas.560471

Cited By: 1

Abstract

We introduce a new contractive condition and a new iterative

method in n0normed space setting. We employ both of these to

study convergence, stability, and data dependence. The results presented

here extend and improve some recent results announced in the existing

literature.

Keywords

Iterative methods, convergence, stability, data dependence, quasi--contractive operators

References

• Misiak, A., n-inner product spaces, Math. Nachr., vol. 140, (1989), 299--319.
• Gähler, S., Linear 2-normietre räume, Math. Nachr., vol. 28, (1965), 1--43.
• Dutta, H., Some iterated convergence and fixed point theorems in real linear n-normed spaces, Miskolc Mathematical Notes, vol. 15, no. 2, (2014), 423-437.
• Gunawan, H. and Mashadi, M., On n-normed spaces, Int. J. Math. Math. Sci., vol. 2, (2001), 631--639.
• Chugh, R. and Sushma, L., On Generalized n-inner product spaces, Novi Sad J. Math., vol. 41, no. 2, (2011), 73-80.
• Owojori, O. O. and Imoru, C. O., New iteration methods for pseudocontractive and accretive operators in arbitrary banach spaces, Kragujevac J. Math., vol. 25, (2003), 97--110.
• Thianwan, S., Common fixed points of new iterations for two asymptotically nonexpansive nonself mappings in a banach space, J. Comput. Appl. Math., vol. 224, (2009), 688--695.
• Raphael, P. and Pulickakunnel, S., Fixed point theorems in normed linear spaces using a generalized Z-type condition, Kragujevac J. Math., vol. 36, no. 2, (2012), 207--214.
• Khan, S.H., Approximating Fixed Points by a Two-Step Iterative Algorithm,World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical and Quantum Engineering, vol. 8, no. 6, (2014), 939-941.
• Bosede, A.O. and Rhoades, B.E., Stability of Picard and Mann iteration for a general class of functions, J. Adv. Math. Studies, vol. 3, no. 2, (2010), 23-25.
• Weng, X., Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc., vol. 113, (1991), 727--731.
• Soltuz, S. M. and Grosan, T., Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl., vol. 2008, (2008), 7 pages.
• Urabe, M., Convergence of numerical iteration in solution of equations, Journal of Science of the Hiroshima University A, vol.19, (1956), 479--489.
• Ostrowski, A. M., The round-off stability of iterations, Z. Angew. Math. Mech., vol. 47, (1967), 77-81.
• Harder, A. M. and Hicks, T. L., Stability results for xed point iteration procedures, Math. Jap., vol. 33, no. 5, (1988), 693-706.
• Bosede, A. O.,Stability of Noor Iteration for a General Class of Functions in Banach Spaces, Acta Univ. Palacki. Olomuc., Fac. Rer. Nat., Mathematica, vol. 51, no. 2, (2012), 19--25.
• Khan, A. R., Kumar, V. and Hussain, N. Analytical and numerical treatment of Jungck-type iterative schemes, Appl. Math. Comput., vol. 231, (2014), 521-535.
• Imoru, C. O. and Olatinwo, M. O., On the stability of Picard and Mann iteration processes, Carp. J. Math., vol. 19, no. 2, (2003), 155-160.
• Gürsoy, F., Karakaya, V. and Rhoades, B. E., Some convergence and stability results for the Kirk multistep and Kirk-SP fixed point iterative algorithms, Abstr. Appl. Anal., vol. 2014, (2014), 12 pages.
• Osilike, M. O., Stability of the Mann and Ishikawa iteration procedures for-strong pseudocontractions and nonlinearequations of the-strongly accretive type, J. Math. Anal. Appl., vol. 227, no. 2, (1998), 319-334.
• Verinde, V. B., Summable almost stability of fixed point iteration procedures, Carpathian J. Math., vol. 19, (2003), 81-88.
• Ishikawa, S., Fixed points by a new iteration method, Proc. Amer. Math. Soc., vol. 44, (1974), 147--150.
• Mann, W. R., Mean value in iteration, Proc. Amer. Math. Soc., vol. 4, (1953), 506--510.
• Agarwal, R. P., O' Regan, D. and Sahu, D. R., Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., vol. 8, 2007, 61-79.
• Gürsoy, F., Karakaya, V. and Rhoades, B.E., Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl., vol. 76, (2013), 12 pages.