Some Common Fixed Point Theorems in Menger Spaces

Sunny Chauhan; B. D. Pant

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2013)

  • Volume: 52, Issue: 1, page 57-69
  • ISSN: 0231-9721

Abstract

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In this paper, we prove some common fixed point theorems for occasionally weakly compatible mappings in Menger spaces. An example is also given to illustrate the main result. As applications to our results, we obtain the corresponding fixed point theorems in metric spaces. Our results improve and extend many known results existing in the literature.

How to cite

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Chauhan, Sunny, and Pant, B. D.. "Some Common Fixed Point Theorems in Menger Spaces." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 52.1 (2013): 57-69. <http://eudml.org/doc/260583>.

@article{Chauhan2013,
abstract = {In this paper, we prove some common fixed point theorems for occasionally weakly compatible mappings in Menger spaces. An example is also given to illustrate the main result. As applications to our results, we obtain the corresponding fixed point theorems in metric spaces. Our results improve and extend many known results existing in the literature.},
author = {Chauhan, Sunny, Pant, B. D.},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Menger space; weakly compatible mappings; occasionally weakly compatible mappings; fixed point; Menger space; weakly compatible mappings; occasionally weakly compatible mappings; fixed point},
language = {eng},
number = {1},
pages = {57-69},
publisher = {Palacký University Olomouc},
title = {Some Common Fixed Point Theorems in Menger Spaces},
url = {http://eudml.org/doc/260583},
volume = {52},
year = {2013},
}

TY - JOUR
AU - Chauhan, Sunny
AU - Pant, B. D.
TI - Some Common Fixed Point Theorems in Menger Spaces
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2013
PB - Palacký University Olomouc
VL - 52
IS - 1
SP - 57
EP - 69
AB - In this paper, we prove some common fixed point theorems for occasionally weakly compatible mappings in Menger spaces. An example is also given to illustrate the main result. As applications to our results, we obtain the corresponding fixed point theorems in metric spaces. Our results improve and extend many known results existing in the literature.
LA - eng
KW - Menger space; weakly compatible mappings; occasionally weakly compatible mappings; fixed point; Menger space; weakly compatible mappings; occasionally weakly compatible mappings; fixed point
UR - http://eudml.org/doc/260583
ER -

References

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  1. Abbas, M., Rhoades, B. E., Common fixed point theorems for occasionally weakly compatible mappings satisfying a generalized contractive condition, Math. Commun. 13, 2 (2008), 295–301. (2008) Zbl1175.47050MR2488678
  2. Ali, J., Imdad, M., Miheţ, D., Tanveer, M., 10.1007/s10474-011-0105-3, Acta Math. Hungar. 132, 4 (2011), 367–386. (2011) Zbl1249.54077MR2820123DOI10.1007/s10474-011-0105-3
  3. Al-Thagafi, M. A., Shahzad, N., 10.1007/s10114-007-5598-x, Acta Math. Sinica (English Series) 24, 5 (2008), 867–876. (2008) Zbl1175.41026MR2403120DOI10.1007/s10114-007-5598-x
  4. Al-Thagafi, M. A., Shahzad, N., A note on occasionally weakly compatible maps, Int. J. Math. Anal. (Ruse) 3, 2 (2009), 55–58. (2009) Zbl1181.54045MR2517562
  5. Bharucha-Reid, A. T., 10.1090/S0002-9904-1976-14091-8, Bull. Amer. Math. Soc. 82, 5 (1976), 641–657. (1976) Zbl0339.60061MR0413273DOI10.1090/S0002-9904-1976-14091-8
  6. Bhatt, A., Chandra, H., Sahu, D. R., 10.1016/j.na.2010.03.011, Nonlinear Analysis 73, 1 (2010), 176–182. (2010) Zbl1227.47034MR2645842DOI10.1016/j.na.2010.03.011
  7. Banach, S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133–181. (1922) 
  8. Chandra, H., Bhatt, A., Fixed point theorems for occasionally weakly compatible maps in probabilistic semi-metric space, Int. J. Math. Anal. 3, 12 (2009), 563–570. (2009) Zbl1176.54029MR2553969
  9. Chang, S. S., Cho, Y. J., Kang, S. M., Nonlinear Operator Theory in Probabilistic Metric Spaces, Nova Science Publishers, Inc., New York, 2001. (2001) Zbl1080.47054MR2018691
  10. Chauhan, S., Dhiman, N., 10.12785/msl/010201, Math. Sci. Lett. 1, 2 (2012), 89–96. (2012) DOI10.12785/msl/010201
  11. Chauhan, S., Kumar, S., Fixed points of occasionally weakly compatible mappings in fuzzy metric spaces, Sci. Magna 7, 2 (2011), 22–31. (2011) 
  12. Chauhan, S., Kumar, S., Fixed point theorems in Menger spaces and applications to metric spaces, J. Appl. Math. Inform. 30, 5-6 (2012), 729–740. (2012) MR2905814
  13. Chauhan, S., Kumar, S., Pant, B. D., 10.5373/jarpm.678.122010, J. Adv. Research Pure Math. 3, 4 (2011), 17–23. (2011) MR2859299DOI10.5373/jarpm.678.122010
  14. Chauhan, S., Pant, B. D., Common fixed point theorems for occasionally weakly compatible mappings using implicit relation, J. Indian Math. Soc. (N.S.) 77, 1-4 (2010), 13–21. (2010) Zbl1228.54035MR2724019
  15. Chauhan, S., Pant, B. D., Common fixed point theorems in fuzzy metric spaces, Bull. Allahabad Math. Soc. 27, 1 (2012), 27–43. (2012) Zbl1263.54048MR2986675
  16. Chauhan, S., Pant, B. D., Dhiman, N., A common fixed point theorem in Menger space, Sci. Magna 8, 1 (2012), 73–78. (2012) MR3171616
  17. Ćirić, Lj. B., On fixed points of generalized contractions on probabilistic metric spaces, Publ. Inst. Math. (Beograd) (N.S.) 18, 32 (1975), 71–78. (1975) Zbl0316.60008MR0383386
  18. Ćirić, Lj. B., Razani, A., Radenović, S., Ume, J. S, 10.1016/j.camwa.2007.10.009, Comput. Math. Appl. 55, 11 (2008), 2533–2543. (2008) Zbl1142.54363MR2416023DOI10.1016/j.camwa.2007.10.009
  19. Ćirić, Lj. B., Samet, B., Vetro, C., Common fixed point theorems for families of occasionally weakly compatible mappings, Math. Comp. Model. 53, 5-6 (2011), 631–636. (2011) Zbl1217.54042MR2769436
  20. Fang, J.-x., Common fixed point theorems of compatible and weakly compatible maps in Menger spaces, Nonlinear Anal. 71, 5-6 (2009), 1833–1843. (2009) Zbl1172.54027MR2524396
  21. Fang,J.-x., Gao, Y., 10.1016/j.na.2007.11.045, Nonlinear Anal. 70, 1 (2009), 184–193. (2009) Zbl1170.47061MR2468228DOI10.1016/j.na.2007.11.045
  22. Hicks, T. L., Fixed point theory in probabilistic metric spaces, Zbornik Radova, Prir.-Mat. Fak., Univerzitet u Novom Sadu 13 (1983), 63–72. (1983) Zbl0574.54044MR0786431
  23. Hicks, T. L., Sharma, P. L., Probabilistic metric structures: Topological classification, Zbornik Radova, Prir.-Mat. Fak., Univerzitet u Novom Sadu 14, 1 (1984), 35–42. (1984) Zbl0587.54049MR0786809
  24. Imdad, M., Ali, J., Tanveer, M., 10.1016/j.chaos.2009.04.017, Chaos, Solitons & Fractals 42, 5 (2009), 3121–3129. (2009) Zbl1198.54076MR2562820DOI10.1016/j.chaos.2009.04.017
  25. Jungck, G., 10.1090/S0002-9939-1988-0947693-2, Proc. Amer. Math. Soc. 103, 3 (1988), 977–983. (1988) Zbl0661.54043MR0947693DOI10.1090/S0002-9939-1988-0947693-2
  26. Jungck, G., Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J. Math. Sci. 4, 2 (1996), 199–215. (1996) Zbl0928.54043MR1426938
  27. Jungck, G., Rhoades, B. E., Fixed point theorems for occasionally weakly compatible mappings, Fixed Point Theory 7, 2 (2006), 286–296. (2006) Zbl1118.47045MR2284600
  28. Kasahara, S., Rhoades, B. E., Common fixed point theorems in compact metric spaces, Math. Japonica 23, 2 (1978/79), 227–229. (1978) Zbl0402.54048MR0517805
  29. Liu, Y., Wu, J., Li, Z., 10.1155/IJMMS.2005.3045, Int. J. Math. Math. Sci. 19 (2005), 3045–3055. (2005) MR2206083DOI10.1155/IJMMS.2005.3045
  30. Menger, K., 10.1073/pnas.28.12.535, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 535–537. (1942) Zbl0063.03886MR0007576DOI10.1073/pnas.28.12.535
  31. Menger, K., 10.1073/pnas.37.4.226, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 226–229. (1951) Zbl0042.37201MR0042081DOI10.1073/pnas.37.4.226
  32. Pant, R. P., 10.1006/jmaa.1994.1437, J. Math. Anal. Appl. 188, 2 (1994), 436–440. (1994) Zbl0830.54031MR1305460DOI10.1006/jmaa.1994.1437
  33. Pant, R. P., 10.1006/jmaa.1998.6029, J. Math. Anal. Appl. 226, 1 (1998), 251–258. (1998) Zbl0916.54027MR1646430DOI10.1006/jmaa.1998.6029
  34. Pant, R. P., R -weak commutativity and common fixed points, Soochow J. Math. 25, 1 (1999), 37–42. (1999) Zbl0918.54038MR1681606
  35. Pant, R. P., Pant, V., 10.1006/jmaa.2000.6871, J. Math. Anal. Appl. 248, 1 (2000), 327–332. (2000) Zbl0977.54038MR1772600DOI10.1006/jmaa.2000.6871
  36. Pant, B. D., Chauhan, S., Common fixed point theorem for occasionally weakly compatible mappings in Menger space, Surv. Math. Appl. 6 (2011), 1–7. (2011) MR2832553
  37. Pant, B. D., Chauhan, S., Common fixed point theorems for families of occasionally weakly compatible mappings in Menger spaces and application, Bull. Allahabad Math. Soc. 26, 2 (2011), 285–306. (2011) Zbl1256.54078MR2984889
  38. Pant, B. D., Chauhan, S., Common fixed point theorems for occasionally weakly compatible mappings in Menger probabilistic quasi-metric spaces, Adv. Nonlinear Var. Ineq. 14, 2 (2011), 55–63. (2011) Zbl1270.54049MR2857715
  39. Pant, B. D., Chauhan, S., 10.4134/CKMS.2012.27.3.513, Commun. Korean Math. Soc. 27, 3 (2012), 513–522. (2012) Zbl1250.54052MR2978718DOI10.4134/CKMS.2012.27.3.513
  40. Pant, B. D., Chauhan, S., Fixed point theorems for occasionally weakly compatible mappings in Menger spaces, Mat. Vesnik 64, 4 (2012), 267–274. (2012) Zbl1281.54036MR2965959
  41. Pant, B. D., Chauhan, S., Fisher, B., Fixed point theorems for families of occasionally weakly compatible mappings, J. Indian Math. Soc. (N.S.) 79, 1-4 (2012), 127–138. (2012) MR2976031
  42. Pant, B. D., Kumar, S., Chauhan, S., Fixed points of occasionally weakly compatible mappings in Menger spaces, Bull. Calcutta Math. Soc. 104, 1 (2012), 11–20. (2012) MR3088823
  43. Pant, B. D., Samet, B., Chauhan, S., 10.4134/CKMS.2012.27.4.733, Commun. Korean Math. Soc. 27, 4 (2012), 733–743. (2012) MR3025879DOI10.4134/CKMS.2012.27.4.733
  44. Razani, A., Shirdaryazdi, M., 10.1016/j.chaos.2005.10.096, Chaos, Solitons & Fractals 32, 1 (2007), 26–34. (2007) Zbl1134.54321MR2271101DOI10.1016/j.chaos.2005.10.096
  45. Schweizer, B., Sklar, A., 10.2140/pjm.1960.10.313, Pacific J. Math. 10 (1960), 313–334. (1960) Zbl0096.33203MR0115153DOI10.2140/pjm.1960.10.313
  46. Schweizer, B., Sklar, A., Probabilistic metric spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York, 1983. (1983) Zbl0546.60010MR0790314
  47. Sehgal, V. M., Some fixed point theorems in functional analysis and probability, Wayne State Univ., Detroit, 1966, Ph. D. Dissertation. (1966) MR2616264
  48. Sehgal, V. M., Bharucha-Reid, A. T., 10.1007/BF01706080, Math. Systems Theory 6 (1972), 97–102. (1972) Zbl0244.60004MR0310858DOI10.1007/BF01706080
  49. Singh, B., Jain, S., 10.1016/j.jmaa.2004.07.036, J. Math. Anal. Appl. 301, 2 (2005), 439–448. (2005) Zbl1068.54044MR2105684DOI10.1016/j.jmaa.2004.07.036
  50. Vetro, C., Some fixed point theorems for occasionally weakly compatible mappings in probabilistic semi-metric spaces, Int. J. Modern Math. 4, 3 (2009), 277–284. (2009) Zbl1221.54073MR2588994

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