Convergence and Data Dependence Results for AK Iterative Algorithm

Abstract

This paper concentrates on studying convergence and data dependence of AK iteration for the class of maps which was introduced by Berinde.  Also, we will support our results with an example.

Keywords

• Class of Maps,Convergence of Iteration,Data Dependency,Fixed Point,Iteration

References

1. 1. Berinde, V.; On the convergence of the Ishikawa iteration in the class of quasi contractive operators. Acta Mathematica Universitatis Comenianae, 2004, 73(1), 119-126.
2. 2. Berinde, V.; Iterative approximation of fixed points; Springer: Berlin, Germany, 2007.
3. 3. Schu, J.; Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bulletin of the Australian Mathematical Society, 1991, 43(1), 153-159.
4. 4. Ullah, K., and Arshad, M.; On di¤erent results for new three step iteration process in Banach spaces. SpringerPlus 2016, 5(1), 1616.
5. 5. Zamfirescu, T.; Fix point theorems in metric spaces. Archiv der Mathematik 1972, 23(1), 292-298.
6. 6. Berinde, V.; General constructive fixed point theorems for Ciric-type almost contractions in metric spaces. Carpathian Jour-nal of Mathematics. 2008, 24(2), 10-19.
7. 7. Berinde, V., and Pacurar M.; Fixed points and continuity of almost contractions. Fixed Point Theory, 2008, 9(1), 23-34.
8. 8. Karakaya, V., Doğan, K., Gürsoy, F., and Ertürk, M.; Fixed point of a new three-step iteration algorithm under contractivelike operators over normed spaces. Abstract and Applied Analysis, 2013, 2013, 1-9.

9. 9. Khan, A. R., Gürsoy, F., and Karakaya, V.; Jungck-Khan itera-tive scheme and higher convergence rate. International Journal of Computer Mathematics 2016, 93(12), 1029-2105.
10. 10. Gürsoy, F., Karakaya, V., and Rhoades, B. E.; Some conver-gence and stability results for the Kirk multistep and Kirk-SP fixed point iterative algorithms. Abstract and Applied Analysis, 2014, 2014, 1-12.
11. 11. Xu, B., and Noor, M. A.; Fixed-point iterations for asymptoti-cally nonexpansive mappings in Banach spaces. Journal of Mathe-matical Analysis and Applications, 2002, 67(2), 444-453.
12. 12. Weng, X.; Fixed point iteration for local strictly pseudo-contractive mapping. Proceedings of the American Mathematical Society, 1991, 113(3), 727-731.
13. 13. Rhoades, B. E.; Comments on two fixed point iteration met-hods. Journal of Mathematical Analysis and Applications 1976, 56(3), 741-750.
14. 14. Tan, K.K., and Xu, H.; Fixed point iteration processes for asymptotically nonexpansive mappings. Proceedings of the Ameri-can Mathematical Society, 1994, 122(3), 733-739.
15. 15. Şoltuz, Ş. M., and Grosan, T.; Data dependence for Ishikawa iteration when dealing with contractive-like operators. Fixed Point Theory and Applications, 2008, 1, 1-7.
16. 16. Şoltuz, Ş. M.; Data dependence for Mann iteration. Octogon Mathematical Magazine 2001, 9(2), 825-828.
17. 17. Gürsoy, F., Karakaya, V., and Rhoades, B E.; Data dependence results of new multistep and S-iterative schemes for contractive-like operators. Fixed Point Theory and Applications, 2013, 76, 1-12.
18. 18. Şahin, A., and Başarır M.; Convergence and data dependence results of an iteration process in a hyperbolic space. Filomat, 2016, 30(3), 569-582.
19. 19. Khan, A. R., Gürsoy, F., and Kumar V.; Stability and data dependence results for the Jungck-Khan iterative scheme. Turkish Journal of Mathematics 2016, 40, 631-640.