Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

FIXED POINT THEOREMS IN MENGER SPACES AND APPLICATIONS TO METRIC SPACES

  • Received : 2011.08.15
  • Accepted : 2011.12.06
  • Published : 2012.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we prove common fixed point theorems for occasionally weakly compatible maps in Menger spaces with a continuous t-norm of H-type. As application to our results, we obtain the corresponding fixed point theorems in metric spaces. Our results improve and generalize many known results in Menger spaces as well as in metric spaces.

Keywords

References

  1. J. Ali, M. Imdad and D. Bahuguna, Common fixed point theorems in Menger spaces with common property (E.A), Comput. Math. Appl. 60(12) (2010), 3152-3159. MR2739482 (2011g:47124) https://doi.org/10.1016/j.camwa.2010.10.020
  2. M.A. Al-Thagafi and N. Shahzad, Generalized I-nonexpansive selfmaps and invariant approximations, Acta Math. Sinica (English Series) 24(5) (2008), 867-876. MR2403120 https://doi.org/10.1007/s10114-007-5598-x
  3. A. Bhatt, H. Chandra and D.R. Sahu, Common fixed point theorems for occasionally weakly compatible mappings under relaxed conditions, Nonlinear Analysis 73(1) (2010), 176-182. MR2645842 https://doi.org/10.1016/j.na.2010.03.011
  4. S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math. 3 (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181
  5. S. Chauhan, S. Kumar and B.D. Pant, Common fixed point theorems for occasionally weakly compatible mappings in Menger spaces, J. Adv. Research Pure Math. 3(4) (2011), 17-23. DOI:10.5373/jarpm.678.122010
  6. Lj.B. Ciric, Generalized contractions and fixed-point theorems, Publ. Inst. Math. (Beograd) (N.S.) 12(26) (1971), 19-26. MR0309092 (46 #8203)
  7. Lj.B. Ciric, On a family of contractive maps and fixed points, Publ. Inst. Math. (Beograd) (N.S.) 17(31) (1974), 45-51. MR0370546 (51 #6773)
  8. Lj.B. Ciric, A. Razani, S. Radenovic and J.S. Ume, Common fixed point theorems for families of weakly compatible maps, Comput. Math. Appl. 55(11) (2008), 2533-2543. MR2416023 (2009e:54090) https://doi.org/10.1016/j.camwa.2007.10.009
  9. Lj.B. Ciric, B. Samet and C. Vetro, Common fixed point theorems for families of occasionally weakly compatible mappings, Math. Comp. Model. 53(5-6) (2011), 631-636. MR2769436 https://doi.org/10.1016/j.mcm.2010.09.015
  10. J.-x. Fang, Common fixed point theorems of compatible and weakly compatible maps in Menger spaces, Nonlinear Analysis 71(5-6) (2009), 1833-1843. MR2524396 (2010g:54045) https://doi.org/10.1016/j.na.2009.01.018
  11. O. Hadzic, Fixed point theorems for multi-valued mappings in probabilistic metric spaces, Mat. Vesnik 3(16)(31)(2) (1979), 125-133. MR0613901 (82g:54068)
  12. O. Hadzic and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and its Applications, 536. Kluwer Academic Publishers, Dordrecht, 2001. ISBN: 1-4020-0129-0 MR1896451 (2003a:47113)
  13. M. Imdad, J. Ali and M. Tanveer, Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces, Chaos, Solitons & Fractals 42(5) (2009), 3121-3129. MR2562820 (2010j:54064) https://doi.org/10.1016/j.chaos.2009.04.017
  14. G. Jungck and B.E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math. 29(3) (1998), 227-238. MR1617919
  15. G. Jungck and B.E. Rhoades, Fixed point theorems for occasionally weakly compatible mappings, Fixed Point Theory. 7(2) (2006), 287-296. MR2284600
  16. K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 535-537. MR0007576 https://doi.org/10.1073/pnas.28.12.535
  17. D. Mihet, A note on a common fixed point theorem in probabilistic metric spaces, Acta Math. Hungar. 125(1-2) (2009), 127-130. MR2564425 https://doi.org/10.1007/s10474-009-8238-3
  18. S.N. Mishra, Common fixed points of compatible mappings in PM-spaces, Math. Japon. 36(2) (1991), 283-289. MR1095742
  19. D. O'Regan and R. Saadati, Nonlinear contraction theorems in probabilistic spaces, Appl. Math. Comput. 195(1) (2008), 86-93. MR2379198 https://doi.org/10.1016/j.amc.2007.04.070
  20. B.D. Pant and S. Chauhan, Common fixed point theorem for occasionally weakly compatible mappings in Menger space. Surv. Math. Appl. 6 (2011), 1-7.
  21. B.D. Pant and S. Chauhan, Fixed points of occasionally weakly compatible mappings using implicit relation, Commun. Korean Math. Soc. (2012), Article in press.
  22. B.D. Pant and S. Chauhan, Fixed point theorems for occasionally weakly compatible mappings in Menger spaces, Mat. Vesnik 64 (2012), Article in press.
  23. B.D. Pant, S. Chauhan and B. Fisher, Fixed point theorems for families of occasionally weakly compatible mappings, J. Indian Math. Soc. (N.S.) 79(1-4) (2012), Article in press.
  24. A. Razani and M. Shirdaryazdi, A common fixed point theorem of compatible maps in Menger space, Chaos, Solitons & Fractals 32(1) (2007), 26-34. MR2271101 (2007g:54067) https://doi.org/10.1016/j.chaos.2005.10.096
  25. R. Saadati, D. O'Regan, S.M. Vaezpour and J.K. Kim, Generalized distance and common fixed point theorems in Menger probabilistic metric spaces, Bull. Iranian Math. Soc. 35(2) (2009), 97-117. MR2642929
  26. B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 313-334. MR0115153 https://doi.org/10.2140/pjm.1960.10.313
  27. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics. North-Holland Publishing Co., New York, 1983. ISBN: 0-444-00666-4 MR0790314
  28. V.M. Sehgal, Some fixed point theorems in functional analysis and probability, Ph. D. Dissertation, Wayne State Univ. 1966.
  29. V.M. Sehgal and A.T. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Math. Systems Theory 6 (1972), 97-102. MR0310858 (46 #9956) https://doi.org/10.1007/BF01706080
  30. B. Singh and S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl. 301(2) (2005), 439-448. MR2105684 https://doi.org/10.1016/j.jmaa.2004.07.036