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Keywords:
Stability results; Picard and Mann iteration processes
Stability results; Picard and Mann iteration processes
Summary:
In this paper, we obtain some stability results for the Picard iteration process for one and two metrics in complete metric space by using different contractive definitions which are more general than those of Berinde [Berinde, V.: On the stability of some fixed point procedures. Bul. Stiint. Univ. Baia Mare, Ser. B, Matematica–Informatica 18, 1 (2002), 7–14.], Imoru and Olatinwo [Imoru, C. O., Olatinwo, M. O.: On the stability of Picard and Mann iteration processes. Carpathian J. Math. 19, 2 (2003), 155–160.] some others listed in the reference section. The results generalize and unify some of the results of Harder and Hicks [Harder, A. M., Hicks, T. L.: Stability results for fixed point iteration procedures. Math. Japonica 33, 5 (1988), 693–706.], Rhoades [Rhoades, B. E.: Fixed point theorems and stability results for fixed point iteration procedures. Indian J. Pure Appl. Math. 21, 1 (1990), 1–9.], [Rhoades, B. E.: Fixed point theorems and stability results for fixed point iteration procedures II. Indian J. Pure Appl. Math. 24, 11 (1993), 691–703.], Osilike [Osilike, M. O.: Some stability results for fixed point iteration procedures. J. Nigerian Math. Soc. Vol. 14/15 (1995), 17–29.], Berinde [Berinde, V.: On the stability of some fixed point procedures. Bul. Stiint. Univ. Baia Mare, Ser. B, Matematica–Informatica 18, 1 (2002), 7–14.], Imoru and Olatinwo [Imoru, C. O., Olatinwo, M. O.: On the stability of Picard and Mann iteration processes. Carpathian J. Math. 19, 2 (2003), 155–160.] as well as Imoru et al [Imoru, C. O., Olatinwo, M. O., Owojori, O. O.: On the stability of Picard and Mann iteration procedures. J. Appl. Func. Diff. Eqns. 1, 1 (2006), 71–80.].
References:
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