A Family of Non-Self Maps Satisfying Φi-Contractive Condition and Having Unique Common Fixed Point in Metrically Convex Spaces

Yongjie  Piao; Dongzhe  Piao

doi:10.4236/apm.2012.24036

APM Vol.2 No.4 , July 2012

A Family of Non-Self Maps Satisfying Φi-Contractive Condition and Having Unique Common Fixed Point in Metrically Convex Spaces

Yongjie Piao^*, Dongzhe Piao

Abstract: Class of 5-dimensional functions Φ was introduced and a convergent sequence determined by non-self mappings satisfying certain Φi-contractive condition was constructed, and then that the limit of the sequence is the unique com-mon fixed point of the mappings was proved. Finally, several more general forms were given. Our main results gener-alize and unify many same type fixed point theorems in references.

Keywords: Metrically Convex Space; 5-Dimensional Functions Φ ;Φi-Contractive Condition; Common Fixed Point; Complete

Cite this paper: Y. Piao and D. Piao, "A Family of Non-Self Maps Satisfying Φ_i-Contractive Condition and Having Unique Common Fixed Point in Metrically Convex Spaces," Advances in Pure Mathematics, Vol. 2 No. 4, 2012, pp. 280-284. doi: 10.4236/apm.2012.24036.

References

[1] N. A. Assad, “On Fixed Point Theorem of Kannan in Ba-nach Spaces,” TamKang Journal of Mathematics, Vol. 7, 1976, pp. 91-94.

[2] N. A. Assad, “Fixed Point Theorems for Set-Valued Transformations on Compact Sets,” Boll. Un. Math Ital, Vol. 7, No. 4, 1973, pp. 1-7.

[3] N. A. Assad and W. A Kirk, “Fixed Point Theorems for Set-Valued Mappings of Contractive Type,” Pacific Journal of Mathematics, Vol. 43, 1972, pp. 553-562.

[4] M. S. Khan, H. K. Pathak and M. D. Khan, “Some Fixed Point Theorems in Metrically Convex Spaces,” Georgian Journal of Mathematics, Vol.7, No.3, 2000, pp. 523-530.

[5] Y. J. Piao, “A New Generalized Fixed Point Theorem in Metrically Convex Metric Spaces,” Journal of Yanbian University (Natural Science Edition), Vol. 29, No. 1, 2003, pp. 12-16.

[6] S. K. Chatterjea, “Fixed Point Theorems,” C.R. ACad., Bulgare Sci, Vol. 25, 1972, pp. 727-730.

[7] Y. J. Piao, “A Fixed Point Theroem for Non-Self-Mapping in Metrically Convex Metric Spaces,” Journal of Jilin Normal University (Natural Science Edition), Vol. 24, No. 3, 2003, pp. 15-18.

[8] O. Hadzic, “A Common Fixed Point Theorem for a Family of Mappings in Convex Metric Spaces,” Univ. U. Novom Sadu, Zb. Rad. Prirod. Mat. Fak. Ser. Mat, Vol. 20, No. 1, 1990, pp. 89-95.

[9] Y. J. Piao and D. Z. Piao, “Unique Common Fixed Point Theorems for a Family of Non-Self Maps in Metrically Convex Spaces,” Mathematica Appicata, Vol.22, No.4, 2009, pp. 852-857.

[10] Y. J. Piao, “Unique Common Fixed Point for a Family of Quasi-Contractive Type Maps in Metrically Convex Spaces,” Acta Mathematica Scientia, Vol. 30A, No. 2, 2010, pp. 487-493 (In Chinese).

[11] Y. J. Piao, “Unique Common Fixed Point for a Family of Self-Maps with Same Type Contractive Condition in 2-Metric Spaces,” Analysis in Theory and Applications, Vol. 24, No. 4, 2008, pp. 316-320. doi:10.1007/s10496-008-0316-9

[12] Y. J. Piao, “Unique Common Fixed Point for a Family of Self-Maps with Same Quasi-Contractive Type Condition in 2-Metric Space,” Journal of Nanjing University Mathematical Biquarterly, Vol. 27, No. 1, 2010, pp. 82-87 (In Chinese).

[13] Y. J. Piao, “Uniqueness of Common Fixed Points for a Family of Mappings with Φ-Contractive Condition in 2-Metric Spaces,” Applied Mathematics, Vol. 3, No. 1, 2012, pp. 73-77. doi:10.4236/am.2012.31012