Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T14:37:06.754Z Has data issue: false hasContentIssue false
  • English
  • Français

Coincidence and common fixed points of hybrid contractions

Part of: Connections with other structures, applications

Published online by Cambridge University Press:  09 April 2009

M. S. Khan ,
Y. J. Cho ,
W. T. Park  and
T. Mumtaz
M. S. Khan
Affiliation:
Sultan Qaboos University, Department of Mathematics and Computing, College of Science, P.O. Box 32486, Al-Khod, Muscat, Sultanate of, Oman
Y. J. Cho
Affiliation:
Gyeongsang National University, Jinju 660-701, Korea
W. T. Park
Affiliation:
Gyeongsang National University, Jinju 660-701, Korea
T. Mumtaz
Affiliation:
Aligarh Muslim University, Aligarh 202002, India
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we show the existence of solutions of functional equations fix ∈ Sx ∪ Tx and x = fix ∈ Sx ∪ Tx under certain nonlinear hybrid contraction and asymptotic regularity conditions, generalize and improve a recent result due to Kaneko concerning common fixed points of multivalued mappings weakly commuting with a single-valued mapping and satisfying a generalized contraction type. Some related results are also obtained.

Keywords

Coincidence and common fixed pointsHausdorff metricorbital completeproximinal subsets and weakly commuting mappings.

MSC classification

Secondary: 54H25: Fixed-point and coincidence theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 3 , December 1993 , pp. 369 - 385
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Hadzic, O., ‘Common fixed point theorems for family of mappings in complete metric spaces’, Math. Japon. 29 (1984), 127–134.Google Scholar
[2]Itoh, S. and Takahashi, W., ‘Single-valued mappings, multi-valued mappings and fixed point theorems’, J. Math. Anal. Appl. 59 (1977), 261–263.CrossRefGoogle Scholar
[3]Jungck, G., ‘Common fixed points for commuting and compatible maps on compacts’, to appear.Google Scholar
[4]Jungck, G., ‘Commuting mapping and fixed points’, Amer. Math. Monthly 83 (1976), 261–263.CrossRefGoogle Scholar
[5]Jungck, G., ‘Periodic and fixed points, and commuting mappings’, Proc. Amer. Math. Soc. 76 (1979), 333–338.CrossRefGoogle Scholar
[6]Kaneko, H., ‘Single-valued and multi-valued f-contractions’, Boll. Un. Mat. Ital. A 4 (1985), 29–33.Google Scholar
[7]Kaneko, H., ‘A comparison of contractive conditions for multi-valued mappings’, Kobe J. Math. 3 (1986), 37–45.Google Scholar
[8]Khan, M. S., ‘Common fixed point theorems for multi-valued mappings’, Pacific J. Math. 95 (1981), 337–347.CrossRefGoogle Scholar
[9]Kubiak, T., ‘Fixed point theorems for contractive type multi-valued mapping’, Math. Japon. 30 (1985), 89–101.Google Scholar
[10]Meade, B. A. and Singh, S. P., ‘On common fixed point theorems’, Bull. Austral. Math. Soc. 16 (1977), 49–53.Google Scholar
[11]Mukherjee, R. N., ‘On fixed points of single-valued and set-valued mappings’, J. Indian Acad. Math. 4 (1982), 101–103.Google Scholar
[12]Nadler, S. B., ‘Multivalued contraction mappings’, Pacfic J. Math. 30 (1969), 475–488.CrossRefGoogle Scholar
[13]Naimpally, S. A., Singh, S. L. and Whitfield, J. H. M., ‘Coincidence theorems for hybrid contractions’, Math. Nachr. 127 (1986), 177–180.CrossRefGoogle Scholar
[14]Rhoades, B. E. and Sessa, S., ‘Common fixed point theorems for three mappings under a weak commutativity condition’, Indian J. Pure Appl. Math. 17 (1986), 47–57.Google Scholar
[15]Rhoades, B. E., Sessa, S., Khan, M. S. and Swaleh, M., ‘On fixed points of asymptotically regular mappings’, J. Austral. Math. Soc. (Series A) 43 (1987), 328–346.CrossRefGoogle Scholar
[16]Rhoades, B. E., Singh, S. L. and Kulshrestha, C., ‘Coincidence theorems for some multi-valued mappings’, Internat. J. Math. Sci. 7 (1984), 429–434.CrossRefGoogle Scholar
[17]Sessa, S., ‘On a weak commutativity condition in fixed point considerations’, Publ. Inst. Math. 32 (1982), 149–153.Google Scholar
[18]Sessa, S. and Fisher, B., ‘On common fixed points of weakly commuting mappings and set-valued mappings’, Internat. J. Math. Soc. 9 (1986), 323–329.CrossRefGoogle Scholar
[19]Sessa, S., Khan, M. S. and Imada, M., ‘A common fixed point theorem with a weak commutativity condition’, Glas. Mat. Ser. III 21 (1986), 225–235.Google Scholar
[20]Sessa, S., Mukherjee, R. N. and Som, T., ‘A common fixed point theorem for weakly commuting mappings’, Math. Japon. 31 (1986), 235–245.Google Scholar
[21]Singer, I., ‘The theory of best approximation and functional analysis’, in: CBMS-NSF Regional Conf Ser. in Appl. Math. 13 (SIAM, Philadelphia, 1974).Google Scholar
[22]Smithon, R. E., ‘Fixed points for contractive multifunctions’, Proc. Amer. Math. Soc. 27 (1971), 192–194.CrossRefGoogle Scholar
[23]Čirič, Lj. B., ‘Fixed points for generalized multivalued contractions’, Math. Vesnik 9 (1972), 265–272.Google Scholar
[24]Wegrzyk, R., ‘Fixed point theorems for multi-valued functions and their applications to functional equations’, Dissertatione Math. (Rozprawy Mat.) 201 (1982), 1–28.Google Scholar