Common Fixed Points for a Countable Family of Set-Valued Mappings with Quasi-Contractive Conditions on Metrically Convex Spaces

ABSTRACT

In this paper, we consider a countable family of set-valued mappings satisfying some quasi-contractive conditions. We also construct a sequence by the quasi-contractive conditions of mappings and the boundary condition of a closed subset of a metrically convex space, and then prove that the unique limit of the sequence is the unique common fixed point of the mappings. Finally, we give more generalized common fixed point theorems for a countable family of single-valued mappings. The main results generalize and improve many common fixed point theorems for a finite or countable family of single valued or set-valued mappings with quasi-contractive conditions.

Cite this paper

Y. Jin, A. Jin and Y. Piao, "Common Fixed Points for a Countable Family of Set-Valued Mappings with Quasi-Contractive Conditions on Metrically Convex Spaces,"*Advances in Pure Mathematics*, Vol. 4 No. 1, 2014, pp. 17-24. doi: 10.4236/apm.2014.41003.

Y. Jin, A. Jin and Y. Piao, "Common Fixed Points for a Countable Family of Set-Valued Mappings with Quasi-Contractive Conditions on Metrically Convex Spaces,"

References

[1] N. A. Assad, “On Fixed Point Theorem of Kannan in Banach 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,” Bolletino della Unione Matematica Italiana, 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, No. 3, 1972, pp. 553-562. http://dx.doi.org/10.2140/pjm.1972.43.553

[4] X. Zhang, “Common Fixed Point Theorem of Lipschitz Type Mappings on Convex Cone Metric Spaces,” Acta Mathematica Sinica (Chinese Series), Vol. 53, No. 6, 2010, pp. 1139-1148.

[5] M. Abbas, B. E. Rhoades, et al., “Common Fixed Points of Generalized Contractive Multivalued Mappings in Cone Metric Spaces,” Mathematical Communications, Vol. 14, No. 2, 2009, pp. 365-378.

[6] S. L. Singh and B. Ram, “Common Fixed Points of Commuting Mappings in 2-Metric Spaces,” Mathematical Semester Notes, Vol. 10, 1982, pp. 197-207.

[7] Y. J. Piao and Y. F. Jin, “New Unique Common Fixed Point Results for Four Mappings with -Contractive Type Theorems in 2-Metric Spaces,” Applied Mathematics, Vol. 3, No. 7, 2012, pp. 734-737.

http://dx.doi.org/10.4236/am.2012.37108

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

[9] S. K. Chatterjea, “Fixed Point Theorems,” Comptes rendus de l'Académie des Sciences, Vol. 25, 1972, pp. 727-730.

[10] O. Hadzic, “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.

[11] Y. J. Piao, “Unique Common Fixed Point Theorems for a Family of Non-Self Maps in Metrically Convex Spaces,” Applied Mathematics, Vol. 22, No. 4, 2009, pp. 852-857.

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

[13] J. R. Wu and H. Y. Liu, “Common Fixed Point Theorems for Sequences of -Type Contraction Set-Valued Mappings,” Chinese Quarterly Journal of Mathematics, Vol. 24, No. 4, 2009, pp. 504-510.

[1] N. A. Assad, “On Fixed Point Theorem of Kannan in Banach 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,” Bolletino della Unione Matematica Italiana, 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, No. 3, 1972, pp. 553-562. http://dx.doi.org/10.2140/pjm.1972.43.553

[4] X. Zhang, “Common Fixed Point Theorem of Lipschitz Type Mappings on Convex Cone Metric Spaces,” Acta Mathematica Sinica (Chinese Series), Vol. 53, No. 6, 2010, pp. 1139-1148.

[5] M. Abbas, B. E. Rhoades, et al., “Common Fixed Points of Generalized Contractive Multivalued Mappings in Cone Metric Spaces,” Mathematical Communications, Vol. 14, No. 2, 2009, pp. 365-378.

[6] S. L. Singh and B. Ram, “Common Fixed Points of Commuting Mappings in 2-Metric Spaces,” Mathematical Semester Notes, Vol. 10, 1982, pp. 197-207.

[7] Y. J. Piao and Y. F. Jin, “New Unique Common Fixed Point Results for Four Mappings with -Contractive Type Theorems in 2-Metric Spaces,” Applied Mathematics, Vol. 3, No. 7, 2012, pp. 734-737.

http://dx.doi.org/10.4236/am.2012.37108

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

[9] S. K. Chatterjea, “Fixed Point Theorems,” Comptes rendus de l'Académie des Sciences, Vol. 25, 1972, pp. 727-730.

[10] O. Hadzic, “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.

[11] Y. J. Piao, “Unique Common Fixed Point Theorems for a Family of Non-Self Maps in Metrically Convex Spaces,” Applied Mathematics, Vol. 22, No. 4, 2009, pp. 852-857.

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

[13] J. R. Wu and H. Y. Liu, “Common Fixed Point Theorems for Sequences of -Type Contraction Set-Valued Mappings,” Chinese Quarterly Journal of Mathematics, Vol. 24, No. 4, 2009, pp. 504-510.