J. Aust. Math. Soc. 83 (2007), no. 1, pp. 17–29.
On two pairs of non-self hybrid mappings
Ljubomir B. Ćirić Jeong Sheok Ume Nebojša T. Nikolić
Faculty of Mechanical Engineering
Al. Rudara 12-35
11 070 Belgrade
Serbia and Montenegro
lciric@afrodita.rcub.bg.ac.yu
Department of Applied Mathematics
Changwon National University
Changwon 641-773
Korea
jsume@changwon.ac.kr
Faculty of Organizational Science
Jove Ilica 154
11 000 Belgrade
Serbia
sigma@fon.bg.ac.yu
Received 2 November 2005; revised 31 May 2006
Communicated by A. Pryde

Abstract

In this paper we obtain some results on coincidence and common fixed points for two pairs of multi-valued and single-valued non-self mappings in complete convex metric spaces. We improve on previously used methods of proof and obtain results for mappings which are not necessarily compatible and not necessarily continuous, generalizing some known results. In particular, a theorem by Rhoades [19] and a theorem by Ahmed and Rhoades [2] are generalized and improved.

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2000 Mathematics Subject Classification: primary 54H25, 47H10
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2354??? Z'blatt-MATH: pre05231330
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References

  1. A. Ahmed and A. R. Khan, ‘Some fixed point theorems for non-self hybrid contractions’, J. Math. Anal. Appl. 213 (1997), 275–280. MR1469374
  2. A. Ahmed and B. E. Rhoades, ‘Some common fixed point theorems for compatible mappings’, Indian J. Pure Appl. Math. 32. (2001), 1247–1254. MR1853906
  3. N. A. Assad, ‘Fixed point theorems for set valued transformations on compact sets’, Boll. Un. Math. Ital. (4) 8 (1973), 1–7. MR326700
  4. N. A. Assad and W. A. Kirk., ‘Fixed point theorems for set valued mappings of contractive type’, Pacific J. Math. 43 (1972), 553–562. MR341459
  5. Lj. B. Ćirić, ‘Fixed points for generalized multi-valued mappings’, Mat. Vesnik 9 (1972), 265–272. MR341460
  6. Lj. B. Ćirić, ‘A remark on rhoades fixed point theorem for non-self mappings’, Internat. J.Math. Math. Sci. 16 (1993), 397–400. MR1210850
  7. Lj. B. Ćirić, ‘Quasi-contraction non-self mappings on banach spaces’, Bull. Acad. Serbe Sci. Arts 23 (1998), 25–31. MR1744096
  8. Lj. B. Ćirić, J.S.Ume, M.S. Khan and H. K. Pathak, ‘On some non-self mappings’, Math. Nachr. 251 (2003), 28–33. MR1960802
  9. Lj. B. Ćirić and J. S. Ume, ‘Some common fixed point theorems for weakly compatible mappings’, J. Math. Anal. Appl. 314 (2006), 488–499. MR2185245
  10. O. Hadzić, ‘A theorem on coincidence points for multi-valued mappings in convex metric spaces’, Zb. Radova Univ. u Novom Sadu 19 (1989), 233–240. MR1100008
  11. S. Itoh, ‘Multi-valued generalized contractions and fixed point theorems’, Comment. Math. Univ. Carolin. 18 (1977), 247–258. MR451227
  12. G. Jungck, ‘Compatible mappings and common fixed points’, Internat. J. Math. Math. Sci. 9 (1986), 771–779. MR870534
  13. H. Kaneko and S. Sessa, ‘Fixed point theorems for compatible multi-valued and single-valued mappings’, Internat J. Math. Math. Sci. 12 (1989), 257–262. MR994907
  14. M. S. Khan, ‘Common fixed point theorems for multi-valued mappings’, Pacific J. Math. 95 (1981), 337–347. MR632191
  15. J. T. Markin, ‘A fixed point theorem for set-valued mappings’, Bull. Amer. Math. Soc. 74 (1968), 639–640. MR227825
  16. S. B. Nadler, ‘Multi-valued contraction mappings’, Pacific J. Math. 30 (1969), 475–488. MR254828
  17. B. K. Ray, ‘On Ćirić's fixed point theorem’, Fund. Math. 94 (1977), 221–229. MR428317
  18. B. E. Rhoades, ‘A. fixed point theorem for some non-self mappings’, Math. Japon. 23 (1978), 457–459. MR524997
  19. B. E. Rhoades, ‘A fixed point theorem for non-self set-valued mappings’, Internat. J.Math. Math. Sci. 20 (1997), 9–12. MR1431417
  20. B. E. Rhoades, ‘Common fixed points for multi-maps in a metric space’, Nonlinear Anal. 13 (1989), 221–229. MR986445
  21. T. Tsachev and V. G. Angelov, ‘Fixed points of non-self mappings and applications’, Nonlinear Anal. 21 (1993), 9–16. MR1231524