Common Fixed Point Result of Multivalued and Singlevalued Mappings in Partially Ordered Metric Space

Affiliation(s)

Department of Mathematical Sciences and Computer Applications, Bundelkhand University, Jhansi, India.

Department of Mathematical Sciences and Computer Applications, Bundelkhand University, Jhansi, India.

ABSTRACT

In recent times the fixed point results in partially ordered metric spaces has greatly developed. In this paper we prove common fixed point results for multivalued and singlevalued mappings in partially ordered metric space. Our theorems generalized the theorem in [1] and extends the many more recent results in such spaces.

KEYWORDS

Multi-Valued Mapping; Single-Valued Mapping; Partial Ordering; Control Function; Fixed Point Theorem

Multi-Valued Mapping; Single-Valued Mapping; Partial Ordering; Control Function; Fixed Point Theorem

Cite this paper

R. Saini and A. Sharma, "Common Fixed Point Result of Multivalued and Singlevalued Mappings in Partially Ordered Metric Space,"*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 142-148. doi: 10.4236/apm.2013.31019.

R. Saini and A. Sharma, "Common Fixed Point Result of Multivalued and Singlevalued Mappings in Partially Ordered Metric Space,"

References

[1] R. A. Rashwan and M. A. Ahmed, “Common Fixed Points for δ-Compatible Mappings,” Southwest Journal of Pure and Applied Mathematics, Vol. 1, 1996, pp. 1-6.

[2] M. A. Ahmed, “Common Fixed Point Theorems for Weakly Compatible Mappings,” Rocky Mountain Journal of Mathematics, Vol. 33, No. 4, 2003, pp. 1189-1203. doi:10.1216/rmjm/1181075457

[3] B. Fisher, “Common Fixed Point of Four Mappings,” Bulletin of the Institute of Mathematics, Academia Sinica, Vol. 11, 1983, pp. 103-113.

[4] B. Fisher, “Common Fixed Point Theorems for Commuting Mappings,” Bulletin of the Institute of Mathematics, Academia Sinica, Vol. 9, 1981, pp. 399-406.

[5] B. Fisher, “Common Fixed Points of Mappings and Set Valued Mappings,” Rostocker Mathematisches Kolloquium, Vol. 18, 1981, pp. 69-77.

[6] B. Fisher and S. Sessa, “Two Common Fixed Point Theorems for Weakly Commuting Mappings,” Periodica Mathematica Hungarica, Vol. 20, No. 3, 1989, pp. 207-218. doi:10.1007/BF01848124

[7] S. V. R. Naidu, “Some Fixed Point Theorems in Metric Spaces by Altering Distances,” Czechoslovak Mathematical Journal, Vol. 53, No. 1, 2003, pp. 205-212. doi:10.1023/A:1022991929004

[8] R. A. Rashwan and M. A. Ahmed, “Fixed Points of Single and Set Valued Mappings,” Kyungpongk Mathematica Journal, Vol. 38, No. 1, 1998, pp. 29-37.

[9] G. Jungck and B. E. Rhoades, “Fixed Points for Set Valued Functions without Continuity,” Indian Journal of Pure Applied Mathematics, Vol. 29, No. 3, 1998, pp. 227-238.

[10] G. Jungck and B. E. Rhoades, “Some Fixed Point Theorems for Compatible Maps,” International Journal of Mathematics and Mathematical Sciences, Vol. 16, No. 3, 1993, pp. 417-428. doi:10.1155/S0161171293000535

[11] I. Beg and A. R. Butt, “Common Fixed Point for Generalized Set Valued Contractions Satisfying an Implicit Relation in Partially Ordered Metric Spaces,” Mathematical Communications, Vol. 15, No. 1, 2010, pp. 65-76.

[12] M. S. Khan, M. Swaleh and S. Sessa, “Fixed Points Theorems by Altering Distances between the Points,” Bulletin Australian Mathematical Society, Vol. 30, No. 1, 1984, pp. 1-9. doi:10.1017/S0004972700001659

[13] S. V. R. Naidu, “Some Fixed Point Theorems in Metric Spaces by Altering Distances,” Czechoslovak Mathematical Journal, Vol. 53, No. 1, 2003, pp. 205-212. doi:10.1023/A:1022991929004

[14] B. S. Choudhury, “A Common Unique Fixed Point Result in Metric Spaces Involving Generalised Altering Distances,” Mathematical Communications, Vol. 10, No. 2, 2005, pp. 105-110.

[15] B. S. Choudhury and P. N. Dutta, “Common Fixed Points for Fuzzy Mappings Using Generalized Altering Distances,” Soochow Journal of Mathematics, Vol. 31, No. 1, 2005, pp. 71-81.

[16] K. P. R. Sastry and G. V. R Babu, “Some Fixed Point Theorems by Altering Distances between the Points,” Indian Journal of Pure and Applied Mathematics, Vol. 30, No. 6, 1999, pp. 641-647.

[17] B. S. Choudhury and N. Metiya, “Multivalued and Singlevalued Fixed Point Results in Partially Ordered Metric Spaces,” Arab Journal of Mathematical Sciences, Vol. 17, No. 2, 2011, pp. 135-151. doi:10.1016/j.ajmsc.2011.03.001

[1] R. A. Rashwan and M. A. Ahmed, “Common Fixed Points for δ-Compatible Mappings,” Southwest Journal of Pure and Applied Mathematics, Vol. 1, 1996, pp. 1-6.

[2] M. A. Ahmed, “Common Fixed Point Theorems for Weakly Compatible Mappings,” Rocky Mountain Journal of Mathematics, Vol. 33, No. 4, 2003, pp. 1189-1203. doi:10.1216/rmjm/1181075457

[3] B. Fisher, “Common Fixed Point of Four Mappings,” Bulletin of the Institute of Mathematics, Academia Sinica, Vol. 11, 1983, pp. 103-113.

[4] B. Fisher, “Common Fixed Point Theorems for Commuting Mappings,” Bulletin of the Institute of Mathematics, Academia Sinica, Vol. 9, 1981, pp. 399-406.

[5] B. Fisher, “Common Fixed Points of Mappings and Set Valued Mappings,” Rostocker Mathematisches Kolloquium, Vol. 18, 1981, pp. 69-77.

[6] B. Fisher and S. Sessa, “Two Common Fixed Point Theorems for Weakly Commuting Mappings,” Periodica Mathematica Hungarica, Vol. 20, No. 3, 1989, pp. 207-218. doi:10.1007/BF01848124

[7] S. V. R. Naidu, “Some Fixed Point Theorems in Metric Spaces by Altering Distances,” Czechoslovak Mathematical Journal, Vol. 53, No. 1, 2003, pp. 205-212. doi:10.1023/A:1022991929004

[8] R. A. Rashwan and M. A. Ahmed, “Fixed Points of Single and Set Valued Mappings,” Kyungpongk Mathematica Journal, Vol. 38, No. 1, 1998, pp. 29-37.

[9] G. Jungck and B. E. Rhoades, “Fixed Points for Set Valued Functions without Continuity,” Indian Journal of Pure Applied Mathematics, Vol. 29, No. 3, 1998, pp. 227-238.

[10] G. Jungck and B. E. Rhoades, “Some Fixed Point Theorems for Compatible Maps,” International Journal of Mathematics and Mathematical Sciences, Vol. 16, No. 3, 1993, pp. 417-428. doi:10.1155/S0161171293000535

[11] I. Beg and A. R. Butt, “Common Fixed Point for Generalized Set Valued Contractions Satisfying an Implicit Relation in Partially Ordered Metric Spaces,” Mathematical Communications, Vol. 15, No. 1, 2010, pp. 65-76.

[12] M. S. Khan, M. Swaleh and S. Sessa, “Fixed Points Theorems by Altering Distances between the Points,” Bulletin Australian Mathematical Society, Vol. 30, No. 1, 1984, pp. 1-9. doi:10.1017/S0004972700001659

[13] S. V. R. Naidu, “Some Fixed Point Theorems in Metric Spaces by Altering Distances,” Czechoslovak Mathematical Journal, Vol. 53, No. 1, 2003, pp. 205-212. doi:10.1023/A:1022991929004

[14] B. S. Choudhury, “A Common Unique Fixed Point Result in Metric Spaces Involving Generalised Altering Distances,” Mathematical Communications, Vol. 10, No. 2, 2005, pp. 105-110.

[15] B. S. Choudhury and P. N. Dutta, “Common Fixed Points for Fuzzy Mappings Using Generalized Altering Distances,” Soochow Journal of Mathematics, Vol. 31, No. 1, 2005, pp. 71-81.

[16] K. P. R. Sastry and G. V. R Babu, “Some Fixed Point Theorems by Altering Distances between the Points,” Indian Journal of Pure and Applied Mathematics, Vol. 30, No. 6, 1999, pp. 641-647.

[17] B. S. Choudhury and N. Metiya, “Multivalued and Singlevalued Fixed Point Results in Partially Ordered Metric Spaces,” Arab Journal of Mathematical Sciences, Vol. 17, No. 2, 2011, pp. 135-151. doi:10.1016/j.ajmsc.2011.03.001