Kannan-type cyclic contraction results in 2 -Menger space

Binayak S. Choudhury; Samir Kumar BHANDARI

Mathematica Bohemica (2016)

  • Volume: 141, Issue: 1, page 37-58
  • ISSN: 0862-7959

Abstract

top
In this paper we establish Kannan-type cyclic contraction results in probabilistic 2-metric spaces. We use two different types of t -norm in our theorems. In our first theorem we use a Hadzic-type t -norm. We use the minimum t -norm in our second theorem. We prove our second theorem by different arguments than the first theorem. A control function is used in our second theorem. These results generalize some existing results in probabilistic 2-metric spaces. Our results are illustrated with an example.

How to cite

top

Choudhury, Binayak S., and BHANDARI, Samir Kumar. "Kannan-type cyclic contraction results in $2$-Menger space." Mathematica Bohemica 141.1 (2016): 37-58. <http://eudml.org/doc/276821>.

@article{Choudhury2016,
abstract = {In this paper we establish Kannan-type cyclic contraction results in probabilistic 2-metric spaces. We use two different types of $t$-norm in our theorems. In our first theorem we use a Hadzic-type $t$-norm. We use the minimum $t$-norm in our second theorem. We prove our second theorem by different arguments than the first theorem. A control function is used in our second theorem. These results generalize some existing results in probabilistic 2-metric spaces. Our results are illustrated with an example.},
author = {Choudhury, Binayak S., BHANDARI, Samir Kumar},
journal = {Mathematica Bohemica},
keywords = {$2$-Menger space; Cauchy sequence; fixed point; $\phi $-function; $\psi $-function; cyclic contraction},
language = {eng},
number = {1},
pages = {37-58},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Kannan-type cyclic contraction results in $2$-Menger space},
url = {http://eudml.org/doc/276821},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Choudhury, Binayak S.
AU - BHANDARI, Samir Kumar
TI - Kannan-type cyclic contraction results in $2$-Menger space
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 1
SP - 37
EP - 58
AB - In this paper we establish Kannan-type cyclic contraction results in probabilistic 2-metric spaces. We use two different types of $t$-norm in our theorems. In our first theorem we use a Hadzic-type $t$-norm. We use the minimum $t$-norm in our second theorem. We prove our second theorem by different arguments than the first theorem. A control function is used in our second theorem. These results generalize some existing results in probabilistic 2-metric spaces. Our results are illustrated with an example.
LA - eng
KW - $2$-Menger space; Cauchy sequence; fixed point; $\phi $-function; $\psi $-function; cyclic contraction
UR - http://eudml.org/doc/276821
ER -

References

top
  1. Banach, S., 10.4064/fm-3-1-133-181, Fundam. Math. 3 French (1922), 133-181. (1922) DOI10.4064/fm-3-1-133-181
  2. Chang, S.-S., Huang, N. J., On the generalized 2-metric spaces and probabilistic 2-metric spaces with applications to fixed point theory, Math. Jap. 34 (1989), 885-900. (1989) Zbl0692.54030MR1025044
  3. Choudhury, B. S., Das, K., 10.1016/j.chaos.2009.04.020, Chaos Solitons Fractals 42 (2009), 3058-3063. (2009) Zbl1198.54072MR2560014DOI10.1016/j.chaos.2009.04.020
  4. Choudhury, B. S., Das, K., 10.4134/CKMS.2009.24.4.529, Commun. Korean Math. Soc. 24 (2009), 529-537. (2009) Zbl1231.54020MR2568990DOI10.4134/CKMS.2009.24.4.529
  5. Choudhury, B. S., Das, K., 10.1007/s10114-007-6509-x, Acta Math. Sin., Engl. Ser. 24 (2008), 1379-1386. (2008) Zbl1155.54026MR2438308DOI10.1007/s10114-007-6509-x
  6. Choudhury, B. S., Das, K., Bhandari, S. K., A fixed point theorem in 2 -Menger space using a control function, Bull. Calcutta Math. Soc. 104 (2012), 21-30. (2012) MR3088824
  7. Choudhury, B. S., Das, K., Bhandari, S. K., Cyclic contraction result in 2 -Menger space, Bull. Int. Math. Virtual Inst. 2 (2012), 223-234. (2012) MR3159041
  8. Choudhury, B. S., Das, K., Bhandari, S. K., A fixed point theorem for Kannan type mappings in 2 -Menger space using a control function, Bull. Math. Anal. Appl. 3 (2011), 141-148. (2011) Zbl1314.47076MR2955353
  9. Choudhury, B. S., Das, K., Bhandari, S. K., A generalized cyclic C -contraction principle in Menger spaces using a control function, Int. J. Appl. Math. 24 (2011), 663-673. (2011) MR2931524
  10. Choudhury, B. S., Das, K., Bhandari, S. K., Fixed point theorem for mappings with cyclic contraction in Menger spaces, Int. J. Pure Appl. Sci. Technol. 4 (2011), 1-9. (2011) MR3001859
  11. Connell, E. H., 10.1090/S0002-9939-1959-0110093-3, Proc. Am. Math. Soc. 10 (1959), 974-979. (1959) Zbl0163.17705MR0110093DOI10.1090/S0002-9939-1959-0110093-3
  12. Dutta, P. N., Choudhury, B. S., 10.1007/s10496-010-0110-3, Anal. Theory Appl. 26 (2010), 110-121. (2010) Zbl1224.54091MR2653705DOI10.1007/s10496-010-0110-3
  13. G{ä}hler, S., 10.1002/mana.19640280309, Math. Nachr. 28 German (1965), 235-244. (1965) Zbl0142.39804MR0178452DOI10.1002/mana.19640280309
  14. G{ä}hler, S., 10.1002/mana.19630260109, Math. Nachr. 26 German (1963), 115-148. (1963) Zbl0117.16003MR0162224DOI10.1002/mana.19630260109
  15. Gole{ţ}, I., A fixed point theorem in probabilistic 2-metric spaces, Inst. Politehn. Traian Vuia Timişoara Lucrăr. Sem. Mat. Fiz. (1988), 21-26. (1988) MR1221431
  16. Had{ž}i{ć}, O., A fixed point theorem for multivalued mappings in 2 -Menger spaces, Zb. Rad., Prir.-Mat. Fak., Univ. Novom Sadu, Ser. Mat. 24 (1994), 1-7. (1994) Zbl0897.54036
  17. Hadži{ć}, O., Pap, E., Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications 536 Kluwer Academic Publishers, Dordrecht (2001). (2001) MR1896451
  18. Janos, L., 10.1090/S0002-9939-1976-0425936-3, Proc. Am. Math. Soc. 61 (1976), 171-175. (1976) Zbl0364.54022MR0425936DOI10.1090/S0002-9939-1976-0425936-3
  19. Kada, O., Suzuki, T., Takahashi, W., Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Jap. 44 (1996), 381-391. (1996) Zbl0897.54029MR1416281
  20. Kannan, R., 10.2307/2316437, Am. Math. Mon. 76 (1969), 405-408. (1969) Zbl0179.28203MR0257838DOI10.2307/2316437
  21. Kannan, R., Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), Article No. 11, 71-76. (1968) Zbl0209.27104MR0257837
  22. Karpagam, S., Agrawal, S., 10.1016/j.na.2010.07.026, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 1040-1046. (2011) Zbl1206.54047MR2746787DOI10.1016/j.na.2010.07.026
  23. Khan, M. S., On the convergence of sequences of fixed points in 2-metric spaces, Indian J. Pure Appl. Math. 10 (1979), 1062-1067. (1979) Zbl0417.54020MR0547888
  24. Khan, M. S., Swaleh, M., Sessa, S., 10.1017/S0004972700001659, Bull. Aust. Math. Soc. 30 (1984), 1-9. (1984) Zbl0553.54023MR0753555DOI10.1017/S0004972700001659
  25. Kikkawa, M., Suzuki, T., Some similarity between contractions and Kannan mappings. {II}, Bull. Kyushu Inst. Technol., Pure Appl. Math. 55 (2008), 1-13. (2008) Zbl1163.54022MR2455257
  26. Kikkawa, M., Suzuki, T., Some similarity between contractions and Kannan mappings, Fixed Point Theory Appl. (electronic only) 2008 (2008), Article No. 649749, 8 pages. (2008) Zbl1163.54022MR2395313
  27. Kirk, W. A., Srinivasan, P. S., Veeramani, P., Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (2003), 79-89. (2003) Zbl1052.54032MR2031823
  28. Lal, S. N., Singh, A. K., 10.1017/S0004972700007887, Bull. Aust. Math. Soc. 18 (1978), 137-143. (1978) Zbl0385.54028MR0645161DOI10.1017/S0004972700007887
  29. Mihe{ţ}, D., 10.1016/j.na.2009.01.107, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 2734-2738. (2009) Zbl1176.54034MR2532798DOI10.1016/j.na.2009.01.107
  30. Naidu, S. V. R., 10.1155/S016117120101064X, Int. J. Math. Math. Sci. 28 (2001), 625-636. (2001) Zbl1001.47037MR1892319DOI10.1155/S016117120101064X
  31. Saha, P. K., Tiwari, R., An alternative proof of Kannan's fixed point theorem in a generalized metric space, News Bull. Calcutta Math. Soc. 31 (2008), 15-18. (2008) Zbl1218.54049MR2682190
  32. Sastry, K. P. R., Babu, G. V. R., Some fixed point theorems by altering distances between the points, Indian J. Pure Appl. Math. 30 (1999), 641-647. (1999) Zbl0938.47044MR1701042
  33. Sastry, K. P. R., Naidu, S. V. R., Babu, G. V. R., Naidu, G. A., Generalization of common fixed point theorems for weakly commuting map by altering distances, Tamkang J. Math. 31 (2000), 243-250. (2000) MR1778222
  34. Schweizer, B., Sklar, A., Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics North-Holland Publishing, New York (1983). (1983) Zbl0546.60010MR0790314
  35. Sehgal, V. M., Bharucha-Reid, A. T., 10.1007/BF01706080, Math. Syst. Theory 6 (1972), 97-102. (1972) Zbl0244.60004MR0310858DOI10.1007/BF01706080
  36. Shi, Y., Ren, L., Wang, X., 10.1007/BF02936092, J. Appl. Math. Comput. 13 (2003), 277-286. (2003) Zbl1060.47057MR2000215DOI10.1007/BF02936092
  37. Shioji, N., Suzuki, T., Takahashi, W., 10.1090/S0002-9939-98-04605-X, Proc. Am. Math. Soc. 126 (1998), 3117-3124. (1998) Zbl0955.54009MR1469434DOI10.1090/S0002-9939-98-04605-X
  38. Subrahmanyam, P. V., 10.1007/BF01472580, Monatsh. Math. 80 (1975), 325-330. (1975) Zbl0312.54048MR0391065DOI10.1007/BF01472580
  39. W{ł}odarczyk, K., Plebaniak, R., Banach, A., 10.1016/j.na.2008.04.037, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70 (2009), 3332-3341 erratum ibid. 71 3585-3586 (2009). (2009) Zbl1171.54311MR2503079DOI10.1016/j.na.2008.04.037
  40. W{ł}odarczyk, K., Plebaniak, R., Obczy{ń}ski, C., 10.1016/j.na.2009.07.024, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 794-805. (2010) Zbl1185.54020MR2579346DOI10.1016/j.na.2009.07.024
  41. Zeng, W. Z., Probabilistic 2-metric spaces, J. Math. Res. Exposition 7 (1987), 241-245. (1987) MR0929343

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.