Some stability results in complete metric space

Memudu Olaposi Olatinwo

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2009)

  • Volume: 48, Issue: 1, page 83-92
  • ISSN: 0231-9721

Abstract

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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.].

How to cite

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Olatinwo, Memudu Olaposi. "Some stability results in complete metric space." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 48.1 (2009): 83-92. <http://eudml.org/doc/35189>.

@article{Olatinwo2009,
abstract = {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.].},
author = {Olatinwo, Memudu Olaposi},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Stability results; Picard and Mann iteration processes; metric space; contractive mapping; fixed point; Picard iteration; stability},
language = {eng},
number = {1},
pages = {83-92},
publisher = {Palacký University Olomouc},
title = {Some stability results in complete metric space},
url = {http://eudml.org/doc/35189},
volume = {48},
year = {2009},
}

TY - JOUR
AU - Olatinwo, Memudu Olaposi
TI - Some stability results in complete metric space
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2009
PB - Palacký University Olomouc
VL - 48
IS - 1
SP - 83
EP - 92
AB - 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.].
LA - eng
KW - Stability results; Picard and Mann iteration processes; metric space; contractive mapping; fixed point; Picard iteration; stability
UR - http://eudml.org/doc/35189
ER -

References

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  1. Berinde, V., On the stability of some fixed point procedures, Bul. Stiint. Univ. Baia Mare, Ser. B, Matematica–Informatica 18, 1 (2002), 7–14. (2002) Zbl1031.47030MR2014277
  2. Berinde, V., Iterative Approximation of Fixed Points, Editura Efemeride, Baia Mare, Romania, 2002. (2002) Zbl1036.47037MR1995230
  3. Berinde, V., A priori and a posteriori error estimates for a class of ϕ -contractions, Bulletins for Applied Mathematics 90-B (1999), 183–192. (1999) 
  4. Harder, A. M., Hicks, T. L., Stability results for fixed point iteration procedures, Math. Japonica 33, 5 (1988), 693–706. (1988) Zbl0655.47045MR0972379
  5. Imoru, C. O., Olatinwo, M. O., On the stability of Picard and Mann iteration processes, Carpathian J. Math. 19, 2 (2003), 155–160. (2003) Zbl1086.47512MR2069844
  6. 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. (2006) MR2293939
  7. Jachymski, J. R., An extension of A. Ostrowski’s theorem on the round-off stability of iterations, Aequationes Math. 53 (1997), 242–253. (1997) Zbl0885.47023MR1444177
  8. Osilike, M. O., Some stability results for fixed point iteration procedures, J. Nigerian Math. Soc. Vol. 14/15 (1995), 17–29. (1995) MR1775011
  9. Osilike, M. O., Udomene, A., Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings, Indian J. Pure Appl. Math. 30, 12 (1999), 1229–1234. (1999) Zbl0955.47038MR1729212
  10. Rhoades, B. E., Fixed point theorems and stability results for fixed point iteration procedures., Indian J. Pure Appl. Math. 21, 1 (1990), 1–9. (1990) Zbl0692.54027MR1048010
  11. Rhoades, B. E., Some fixed point iteration procedures, Internat. J. Math. and Math. Sci. 14, 1 (1991), 1–16. (1991) Zbl0716.47030
  12. 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. (1993) Zbl0794.54048MR1251180
  13. Singh, S. L., Bhatnagar, C., Mishra, S. N., Stability of Jungck-type iterative procedures, Internat. J. Math. & Math. Sc. 19 (2005), 3035–3043. (2005) Zbl1117.26005MR2206082
  14. Zeidler, E., Nonlinear Functional Analysis and its Applications, Fixed-Point Theorems I, Springer-Verlag, New York, 1986. (1986) MR0816732

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