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A GENERALIZED COMMON FIXED POINT THEOREM FOR TWO FAMILIES OF SELF-MAPS

  • PHANEENDRA, T. (Applied Analysis Division School of Advanced Sciences VIT University)
  • Received : 2014.06.23
  • Published : 2015.11.30

Abstract

Brief developments in metrical fixed point theory are covered and a significant generalization of recent results obtained in [18], [27], [32] and [33] is established through an extension of the property (EA) to two sequences of self-maps using the notions of weak compatibility and implicit relation.

Keywords

property (EA);implicit relation;orbital completeness;weak compatibility;common fixed point

References

  1. M. A. Aamri and D. El. Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002), no. 1, 181-188. https://doi.org/10.1016/S0022-247X(02)00059-8
  2. A. Aliouche, Common fixed point theorems via an implicit relation and new properties, Sochow J. Math. 33 (2007), no. 4, 593-601.
  3. D. W. Boyd and J. S. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458-469. https://doi.org/10.1090/S0002-9939-1969-0239559-9
  4. Lj. B. Ciric, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc. 45 (1974), no. 2, 271-273.
  5. P. Collaco and J. Carvalho e Silva, A complete comparison of 25 contraction conditions, Nonlinear Anal. 30 (1997), no. 1, 471-476. https://doi.org/10.1016/S0362-546X(97)00353-2
  6. J. Danes, Two fixed point theorems in topological and metric spaces, Bull. Austral. Math. Soc. 14 (1976), no. 2, 259-265. https://doi.org/10.1017/S0004972700025077
  7. K. M. Das and K. V. Naik, Common fixed point theorems for commuting maps on a metric space, Proc. Amer. Math. Soc. 77 (1979), no. 3, 369-373.
  8. B. C. Dhage, On common fixed points of pairs of coincidentally commuting mappings in D-metric spaces, Indian J. Pure Appl. Math. 30 (1999), no. 4, 395-406.
  9. M. Edelstein, An extension of Banach's contraction principle, Proc. Amer. Math. Soc. 12 (1961), 7-10.
  10. B. Fisher, A fixed point theorem, Math. Mag. 48 (1975), no. 4, 223-225. https://doi.org/10.2307/2690350
  11. B. Fisher, Mappings with a common fixed point, Math. Sem. Notes Kobe Univ. 7 (1979), no. 1, 81-84.
  12. B. Fisher, Quasi-contractions on metric spaces, Proc. Amer. Math. Soc. 75 (1979), no. 2, 321-325.
  13. G. E. Hardy and T. D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull. 16 (1973), 201-206. https://doi.org/10.4153/CMB-1973-036-0
  14. G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976), no. 4, 261-263. https://doi.org/10.2307/2318216
  15. G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. 9 (1986), no. 4, 771-779. https://doi.org/10.1155/S0161171286000935
  16. G. Jungck, Common fixed points for non continuous and nonself mappings on nonmetric spaces, Far East J. Math. Sci. 4 (1996), no. 2, 199-215.
  17. R. Kannan, Some results on fixed points II, Amer. Math. Monthly 76 (1969), 405-408. https://doi.org/10.2307/2316437
  18. L. Kikina and K. Kikina, Fixed points for k mappings on a complete metric space, Demonstr. Math. 44 (2011), no. 2, 349-357.
  19. J. Kinces and V. Totok, Theorems and counterexamples on contractive mappings, Math. Balk. 4 (1990), no. 1, 69-90.
  20. N. Kosmatov, Countably many solutions of a fourth order boundary value problem, Electron. J. Qual. Theory Diff. Equ. 2004 (2004), no. 12, 1-15.
  21. Y. Liu, J. Wu, and Z. Li, Common fixed points of single-value and multivalued maps, Int. J. Math. Math. Sci. 19 (2005), no. 19, 3045-3055.
  22. A. Mohammad and P. Valeriu, Well-posedness of a common fixed point problem for three mappings under strict contractive conditions, Buletin. Univers. Petrol-Gaze din Ploiesti, Seria Math. Inform. Fiz. 61 (2009), 1-10.
  23. A. A. Mullin, Application of fixed point theory to number theory, Math. Sem. Notes Kobe Univ. 4 (1976), no. 1, 19-23.
  24. R. P. Pant, Common fixed points of two pairs of commuting mappings, Indian J. Pure Appl. Math. Sci. 17 (1986), no. 2, 187-192.
  25. R. P. Pant, Common fixed points of noncommuting mappings, J. Math. Anal. Appl. 188 (1994), no. 2, 436-440. https://doi.org/10.1006/jmaa.1994.1437
  26. R. P. Pant, A common fixed point theorem under a new condition, Indian J. Pure Appl. Math. 30 (1999), no. 2, 147-152.
  27. R. P. Pant, R. K. Bist, and D. Arora, Weak reciprocal continuity and fixed point theorems, Ann Univ. Ferrara Sez. VII Sci. Mat. 57 (2011), no. 1, 181-190. https://doi.org/10.1007/s11565-011-0119-3
  28. H. K. Pathak, Y. J. Cho, and S. M. Kang, Remarks on R-weakly commuting mappings and common fixed point theorems, Bull. Korean Math. Soc. 34 (1997), no. 2, 247-257.
  29. H. K. Pathak, Y. J. Cho, S. M. Kang, and B. S. Lee, Fixed point theorems for compatible mappings of type (P) and applications to dynamic programming, Matematiche (Catania) 50 (1995), no. 1, 15-33.
  30. H. K. Pathak and M. S. Khan, A comparison of various types of compatible maps and common fixed points, Indian. J. Pure Appl. Math. 28 (1997), no. 4, 477-485.
  31. H. K. Pathak and N. Shahzad, Gregus type fixed point results for tangential mappings satisfying contractive conditions of integral type, Bull. Beg. Math. Soc. Simon Stevin 16 (2009), no. 2, 277-288.
  32. T. Phaneendra and V. Sivarama Prasad, Two Generalized common fixed point theorems involving compatibility and property E.A., Demonstr. Math. 47 (2014), no. 2, 449-458.
  33. T. Phaneendra and Swatmaram, Contractive modulus and common fixed point for three asymptotically regular and weakly compatible self-maps, Malaya J. Mat. 4 (2013), no. 1, 76-80.
  34. V. Popa, Fixed point theorems for implicit contractive mappings, Stud. Cercet. Stiint. Ser. Mat. Univ. Bacau 7 (1997), 127-133.
  35. V. Popa, Some fixed point theorems for compatible mappings satisfying an implicit relation, Demonstr. Math. 32 (1999), no. 1, 157-163.
  36. B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977), 257-290. https://doi.org/10.1090/S0002-9947-1977-0433430-4
  37. B. E. Rhoades, Contractive definitions, Nonlinear analysis, 513-526, World Sci. Publishing, Singapore, 1987.
  38. K. R. R. Sastry and I. S. R. K. Murthy, Common fixed points of two partially commuting tangential self-maps on a metric space, J. Math. Anal. Appl. 250 (2000), no. 2, 731-734. https://doi.org/10.1006/jmaa.2000.7082
  39. S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math. Debre. 32 (1982), 149-153.
  40. P. K. Shrivastava, N. P. S. Bawa, and S. Pankaj, Coincidence theorems for hybrid contraction II, Soochow J. Math. 26 (2000), no. 4, 411-421.
  41. S. L. Singh and T. Anita, Weaker forms of commuting maps and existence of fixed points, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 10 (2003), no. 3, 145-161.
  42. S. L. Singh and S. N. Mishra, Remarks on Jachymski's fixed point theorems for compatible maps, Indian J. Pure Appl. Math. 28 (1997), no. 5, 611-615.
  43. S. L. Singh and S. N. Mishra, On a Ljubomir ciric fixed point theorem for nonexpansive type maps with applications, Indian J. Pure Appl. Math. 33 (2002), no. 4, 531-542.
  44. S. L. Singh and S. P. Singh, A fixed point theorem, Indian J. Pure Appl. Math. 11 (1980), no. 12, 1584-1586.
  45. P. V. Subrahmanyam, Completeness and fixed-points, Monatsh. Math. 80 (1975), no. 4, 325-330. https://doi.org/10.1007/BF01472580