Convergence Results for Jungck-type Iterative Processes in Convex Metric Spaces

Memudu Olaposi Olatinwo

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2012)

  • Volume: 51, Issue: 1, page 79-87
  • ISSN: 0231-9721

Abstract

top
In this paper, the convergence results of [V. Berinde; A convergence theorem for Mann iteration in the class of Zamfirescu operators, Analele Universitatii de Vest, Timisoara, Seria Matematica-Informatica 45 (1) (2007), 33–41], [V. Berinde; On the convergence of Mann iteration for a class of quasi-contractive operators, Preprint, North University of Baia Mare (2003)] and [V. Berinde; On the Convergence of the Ishikawa Iteration in the Class of Quasi-contractive Operators, Acta Math. Univ. Comenianae 73 (1) (2004), 119–126] are extended from arbitrary Banach space setting to the convex metric space by weakening further the conditions on the parameter sequence { α n } ⊂ [ 0 , 1 ] . We establish the convergence of Jungck–Mann and Jungck–Ishikawa iterative processes for two nonselfmappings in a convex metric space setting by employing a general contractive condition. Similar results are also deduced for the Mann and Ishikawa iterations. Our results generalize, extend and improve a multitude of results in the literature including those of Berinde mentioned above.

How to cite

top

Olatinwo, Memudu Olaposi. "Convergence Results for Jungck-type Iterative Processes in Convex Metric Spaces." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 51.1 (2012): 79-87. <http://eudml.org/doc/246525>.

@article{Olatinwo2012,
abstract = {In this paper, the convergence results of [V. Berinde; A convergence theorem for Mann iteration in the class of Zamfirescu operators, Analele Universitatii de Vest, Timisoara, Seria Matematica-Informatica 45 (1) (2007), 33–41], [V. Berinde; On the convergence of Mann iteration for a class of quasi-contractive operators, Preprint, North University of Baia Mare (2003)] and [V. Berinde; On the Convergence of the Ishikawa Iteration in the Class of Quasi-contractive Operators, Acta Math. Univ. Comenianae 73 (1) (2004), 119–126] are extended from arbitrary Banach space setting to the convex metric space by weakening further the conditions on the parameter sequence $\lbrace \alpha _n\rbrace \subset [0,1]$. We establish the convergence of Jungck–Mann and Jungck–Ishikawa iterative processes for two nonselfmappings in a convex metric space setting by employing a general contractive condition. Similar results are also deduced for the Mann and Ishikawa iterations. Our results generalize, extend and improve a multitude of results in the literature including those of Berinde mentioned above.},
author = {Olatinwo, Memudu Olaposi},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {arbitrary Banach space setting; Jungck–Mann and Jungck–Ishikawa iterative processes; convex metric space; arbitrary Banach space setting; Jungck-Mann and Jungck-Ishikawa iterative processes; convex metric space},
language = {eng},
number = {1},
pages = {79-87},
publisher = {Palacký University Olomouc},
title = {Convergence Results for Jungck-type Iterative Processes in Convex Metric Spaces},
url = {http://eudml.org/doc/246525},
volume = {51},
year = {2012},
}

TY - JOUR
AU - Olatinwo, Memudu Olaposi
TI - Convergence Results for Jungck-type Iterative Processes in Convex Metric Spaces
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2012
PB - Palacký University Olomouc
VL - 51
IS - 1
SP - 79
EP - 87
AB - In this paper, the convergence results of [V. Berinde; A convergence theorem for Mann iteration in the class of Zamfirescu operators, Analele Universitatii de Vest, Timisoara, Seria Matematica-Informatica 45 (1) (2007), 33–41], [V. Berinde; On the convergence of Mann iteration for a class of quasi-contractive operators, Preprint, North University of Baia Mare (2003)] and [V. Berinde; On the Convergence of the Ishikawa Iteration in the Class of Quasi-contractive Operators, Acta Math. Univ. Comenianae 73 (1) (2004), 119–126] are extended from arbitrary Banach space setting to the convex metric space by weakening further the conditions on the parameter sequence $\lbrace \alpha _n\rbrace \subset [0,1]$. We establish the convergence of Jungck–Mann and Jungck–Ishikawa iterative processes for two nonselfmappings in a convex metric space setting by employing a general contractive condition. Similar results are also deduced for the Mann and Ishikawa iterations. Our results generalize, extend and improve a multitude of results in the literature including those of Berinde mentioned above.
LA - eng
KW - arbitrary Banach space setting; Jungck–Mann and Jungck–Ishikawa iterative processes; convex metric space; arbitrary Banach space setting; Jungck-Mann and Jungck-Ishikawa iterative processes; convex metric space
UR - http://eudml.org/doc/246525
ER -

References

top
  1. Agarwal, R. P., O’Regan, D., Sahu, D. R., Fixed Point Theory for Lipschitzian-type Mappings with Applications, Topological Fixed Point Theory and Its Applications, Vol. 6 6 Springer Science & Bussiness Media, 2009. (2009) Zbl1176.47037MR2508013
  2. Banach, S., Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, Fund. Math. 3 (1922), 133–181. (1922) 
  3. Beg, I., Structure of the set of fixed points of nonexpansive mappings on convex metric spaces, Ann. Univ. Marie Curie-Sklodowska Sec. 52, Sec. A (1998), 7–14. (1998) Zbl1004.54031MR1728052
  4. Beg, I., Inequalities in metric spaces with applications, Topological Methods in Nonlinear Analysis 17 (2001), 183–190. (2001) Zbl0998.47040MR1846986
  5. Berinde, V., On the convergence of Mann iteration for a class of quasi-contractive operators, North University of Baia Mare, Preprint, 2003. (2003) 
  6. Berinde, V., On the convergence of the Ishikawa iteration in the class of quasi-contractive operators, Acta Math. Univ. Comenianae 73, 1 (2004), 119–126. (2004) Zbl1100.47054MR2076050
  7. Berinde, V., Iterative Approximation of Fixed Points, Springer-Verlag, Berlin–Heidelberg, 2007. (2007) Zbl1165.47047MR2323613
  8. Berinde, V., A convergence theorem for Mann iteration in the class of Zamfirescu operators, Analele Universitatii de Vest, Timisoara, Seria Matematica-Informatica 45, 1 (2007), 33–41. (2007) Zbl1164.47065MR2371084
  9. Berinde, V., Some remarks on a fixed point theorem for Ciric-type almost contractions, Carpathian J. Math. 25, 2 (2009), 157–162. (2009) Zbl1249.54078MR2731191
  10. Chatterjea, S. K., Fixed-point theorems, C. R. Acad. Bulgare Sci. 10 (1972), 727–730. (1972) Zbl0274.54033MR0324493
  11. Ciric, Lj. B., Generalized contractions and fixed point theorems, Publ. Inst. Math. (Beograd) 12, 26 (1971), 19–26. (1971) Zbl0234.54029MR0309092
  12. Ciric, Lj. B., 10.2307/2040075, Proc. Amer. Math. Soc. 45 (1974), 267–273. (1974) Zbl0291.54056MR0356011DOI10.2307/2040075
  13. Ciric, L., On some discontinuous fixed point theorems on convex metric spaces, Czechoslovak Math. J. 43 (1993), 319–326. (1993) MR1211753
  14. Guay, M. D., Singh, K. L., Whitfield, J. H. M., Fxed point theorems for nonexpansive mappings in convex metric spaces, In: Singh, S. P., Barry, J. H. (eds.) Proceedings of Conference on Nonlinear Analysis 80, Marcel Dekker Inc., New York, 1982, 179–189. (1982) MR0689554
  15. Ishikawa, S., 10.1090/S0002-9939-1974-0336469-5, Proc. Amer. Math. Soc. 44, 1 (1974), 147–150. (1974) MR0336469DOI10.1090/S0002-9939-1974-0336469-5
  16. Jungck, G., 10.2307/2318216, Amer. Math. Monthly 83, 4 (1976), 261–263. (1976) Zbl0321.54025MR0400196DOI10.2307/2318216
  17. Kannan, R., Some results on fixed points, Bull. Calcutta Math. Soc. 10 (1968), 71–76. (1968) Zbl0209.27104MR0257837
  18. Kannan, R., Some results on fixed points III, Fund. Math. 70, 2 (1971), 169–177. (1971) Zbl0246.47065MR0283649
  19. Kannan, R., 10.1016/0022-247X(73)90218-7, J. Math. Anal. Appl. 41 (1973), 430–438. (1973) Zbl0261.47037MR0320837DOI10.1016/0022-247X(73)90218-7
  20. Mann, W. R., 10.1090/S0002-9939-1953-0054846-3, Proc. Amer. Math. Soc. 44 (1953), 506–510. (1953) Zbl0050.11603MR0054846DOI10.1090/S0002-9939-1953-0054846-3
  21. Olatinwo, M. O., Some stability and strong convergence results for the Jungck-Ishikawa iteration process, Creative Math. Inf. 17 (2008), 33–42. (2008) Zbl1199.47282MR2409230
  22. Popescu, O., Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Mathematical Communications 12 (2007), 195–202. (2007) Zbl1153.47055MR2382455
  23. Rhoades, B. E., 10.1090/S0002-9947-1974-0348565-1, Trans. Amer. Math. Soc. 196 (1974), 161–176. (1974) Zbl0267.47032MR0348565DOI10.1090/S0002-9947-1974-0348565-1
  24. Rhoades, B. E., 10.1016/0022-247X(76)90038-X, J. Math. Anal. Appl. 56, 2 (1976), 741–750. (1976) Zbl0353.47029MR0430880DOI10.1016/0022-247X(76)90038-X
  25. Rus, I. A., Generalized Contractions and Applications, Cluj Univ. Press, Cluj Napoca, 2001. (2001) Zbl0968.54029MR1947742
  26. Rus, I. A., Petrusel, A., Petrusel, G., Fixed Point Theory, 1950–2000, Romanian Contributions, House of the Book of Science, Cluj Napoca, 2002. (2002) Zbl1005.54037MR1947195
  27. Shimizu, T., Takahashi, W., Fixed point theorems in certain convex metric spaces, Math. Japon. 37 (1992), 855–859. (1992) Zbl0764.47030MR1186552
  28. Singh, S. L., Bhatnagar, C., Mishra, S. N., 10.1155/IJMMS.2005.3035, Internat. J. Math. & Math. Sc. 19 (2005), 3035–3043. (2005) Zbl1117.26005MR2206082DOI10.1155/IJMMS.2005.3035
  29. Takahashi, W., 10.2996/kmj/1138846111, Kodai Math. Sem. Rep. 22 (1970), 142–149. (1970) MR0267565DOI10.2996/kmj/1138846111
  30. Zamfirescu, T., 10.1007/BF01304884, Arch. Math. 23 (1972), 292–298. (1972) Zbl0239.54030MR0310859DOI10.1007/BF01304884

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.