Fixed Points and Common Fixed Points of Quasi-Contractive Mappings on Partially Ordered-Cone Metric Spaces

ABSTRACT

In this paper, we use the mappings with quasi-contractive conditions, defined on a partially ordered set with cone metric structure, to construct convergent sequences and prove that the limits of the constructed sequences are the unique (common) fixed point of the mappings, and give their corollaries. The obtained results improve and generalize the corresponding conclusions in references.

In this paper, we use the mappings with quasi-contractive conditions, defined on a partially ordered set with cone metric structure, to construct convergent sequences and prove that the limits of the constructed sequences are the unique (common) fixed point of the mappings, and give their corollaries. The obtained results improve and generalize the corresponding conclusions in references.

Cite this paper

Jin, H. and Piao, Y. (2014) Fixed Points and Common Fixed Points of Quasi-Contractive Mappings on Partially Ordered-Cone Metric Spaces.*Applied Mathematics*, **5**, 3437-3444. doi: 10.4236/am.2014.521321.

Jin, H. and Piao, Y. (2014) Fixed Points and Common Fixed Points of Quasi-Contractive Mappings on Partially Ordered-Cone Metric Spaces.

References

[1] Huang, L. and Zhang, G.X. (2007) Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings. Journal of Mathematical Analysis and Applications, 332, 1468-1476.

http://dx.doi.org/10.1016/j.jmaa.2005.03.087

[2] Abbas, M. and Jungck, G. (2008) Common Fixed Point Results for Noncommuting Mappings without Continuity in Cone Metric Spaces. Journal of Mathematical Analysis and Applications, 341, 416-420.

http://dx.doi.org/10.1016/j.jmaa.2007.09.070

[3] Abbas, M. and Rhoades, B.E. (2009) Fixed and Periodic Point Results in Cone Metric Spaces. Applied Mathematics Letters, 22, 511-515.

http://dx.doi.org/10.1016/j.aml.2008.07.001

[4] Raja, P. and Vaezpour, S.M. (2008) Some Extensions of Banach’s Contraction Principle in Complete Cone Metric Spaces. Fixed Point Theory and Applications, 2008, Article ID: 768294.

[5] Piao, Y.J. (2014) Unique Common Fixed Points for Mixed Type Expansion Mappings on Cone Metric Spaces. Acta Mathematica Sinica, Chinese Series, 57, 1041-1046.

[6] Jungck, G., Radenovic, S., Radojevic, S. and Rakocevic, V. (2009) Common Fixed Point Theorems for Weakly Compatible Pairs on Cone Metric Spaces. Fixed Point Theory and Applications, 2009, Article ID: 643840.

[7] Ilic, D. and Rakocevic, V. (2009) Quasi-Contraction on a Cone Metric Space. Applied Mathematics Letters, 22, 728-731.

http://dx.doi.org/10.1016/j.aml.2008.08.011

[8] Ran, A.C.M. and Reuring, M.C.B. (2004) A Fixed Point Theorem in Partially Ordered Sets and Some Application to Matrix Equations. Proceedings of the American Mathematical Society, 132, 1435-1443.

http://dx.doi.org/10.1090/S0002-9939-03-07220-4

[9] Nieto, J.J. and Lopez, R.R. (2005) Contractive Mappings Theroem in Partially Ordered Sets and Applications to Ordinary Differential Equation. Order, 22, 223-239.

http://dx.doi.org/10.1007/s11083-005-9018-5

[10] Agarwal, R.P., El-Gebeily, M.A. and O’Regan, D. (2008) Generalized Contractions in Partially Ordered Metric Spaces. Applicable Analysis: An International Journal, 87, 109-116.

http://dx.doi.org/10.1080/00036810701556151

[11] Nieto, J.J. and Lopez, R.R. (2007) Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations. Acta Mathematica Sinica, English Series, 23, 2205-2212.

http://dx.doi.org/10.1007/s10114-005-0769-0

[12] Wang, C., Zhu, J.H., Damjanovic, B. and Hu, L.G. (2009) Approximating Fixed Points of a Pair of Contractive Mappings in Generalized Convex Metric Spaces. Applied Mathematics and Computation, 125, 1522-1525.

[13] Aitun, I., Damjanovic, B. and Djoric, D. (2010) Fixed Point and Common Fixed Point Theorems on Ordered Cone Metric Spaces. Applied Mathematics Letters, 23, 310-316.

http://dx.doi.org/10.1016/j.aml.2009.09.016

[14] Jankovic, S., Kadelburg, Z., Radenovic, S. and Rhoades, B.E. (2009) Assad-Kirk-Type Fixed Point Theorems for a Pair of Nonself Mappings on Cone Metric Spaces. Fixed Point Theory and Applications, 2009, Article ID: 7610386.

[15] Azam, A., Beg, I. and Arshad, M. (2010) Fixed Point in Topological Space Valued Cone Metric Spaces. Fixed Point Theory and Applications, 2010, Article ID: 604084.

[16] Zhang, X. (2010) Common Fixed Point Theorem of Lipschitz Type Mappings on Con Metric Spaces. Acta Mathematica Sinica, Chinese Series, 53, 1139-1148.

[1] Huang, L. and Zhang, G.X. (2007) Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings. Journal of Mathematical Analysis and Applications, 332, 1468-1476.

http://dx.doi.org/10.1016/j.jmaa.2005.03.087

[2] Abbas, M. and Jungck, G. (2008) Common Fixed Point Results for Noncommuting Mappings without Continuity in Cone Metric Spaces. Journal of Mathematical Analysis and Applications, 341, 416-420.

http://dx.doi.org/10.1016/j.jmaa.2007.09.070

[3] Abbas, M. and Rhoades, B.E. (2009) Fixed and Periodic Point Results in Cone Metric Spaces. Applied Mathematics Letters, 22, 511-515.

http://dx.doi.org/10.1016/j.aml.2008.07.001

[4] Raja, P. and Vaezpour, S.M. (2008) Some Extensions of Banach’s Contraction Principle in Complete Cone Metric Spaces. Fixed Point Theory and Applications, 2008, Article ID: 768294.

[5] Piao, Y.J. (2014) Unique Common Fixed Points for Mixed Type Expansion Mappings on Cone Metric Spaces. Acta Mathematica Sinica, Chinese Series, 57, 1041-1046.

[6] Jungck, G., Radenovic, S., Radojevic, S. and Rakocevic, V. (2009) Common Fixed Point Theorems for Weakly Compatible Pairs on Cone Metric Spaces. Fixed Point Theory and Applications, 2009, Article ID: 643840.

[7] Ilic, D. and Rakocevic, V. (2009) Quasi-Contraction on a Cone Metric Space. Applied Mathematics Letters, 22, 728-731.

http://dx.doi.org/10.1016/j.aml.2008.08.011

[8] Ran, A.C.M. and Reuring, M.C.B. (2004) A Fixed Point Theorem in Partially Ordered Sets and Some Application to Matrix Equations. Proceedings of the American Mathematical Society, 132, 1435-1443.

http://dx.doi.org/10.1090/S0002-9939-03-07220-4

[9] Nieto, J.J. and Lopez, R.R. (2005) Contractive Mappings Theroem in Partially Ordered Sets and Applications to Ordinary Differential Equation. Order, 22, 223-239.

http://dx.doi.org/10.1007/s11083-005-9018-5

[10] Agarwal, R.P., El-Gebeily, M.A. and O’Regan, D. (2008) Generalized Contractions in Partially Ordered Metric Spaces. Applicable Analysis: An International Journal, 87, 109-116.

http://dx.doi.org/10.1080/00036810701556151

[11] Nieto, J.J. and Lopez, R.R. (2007) Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations. Acta Mathematica Sinica, English Series, 23, 2205-2212.

http://dx.doi.org/10.1007/s10114-005-0769-0

[12] Wang, C., Zhu, J.H., Damjanovic, B. and Hu, L.G. (2009) Approximating Fixed Points of a Pair of Contractive Mappings in Generalized Convex Metric Spaces. Applied Mathematics and Computation, 125, 1522-1525.

[13] Aitun, I., Damjanovic, B. and Djoric, D. (2010) Fixed Point and Common Fixed Point Theorems on Ordered Cone Metric Spaces. Applied Mathematics Letters, 23, 310-316.

http://dx.doi.org/10.1016/j.aml.2009.09.016

[14] Jankovic, S., Kadelburg, Z., Radenovic, S. and Rhoades, B.E. (2009) Assad-Kirk-Type Fixed Point Theorems for a Pair of Nonself Mappings on Cone Metric Spaces. Fixed Point Theory and Applications, 2009, Article ID: 7610386.

[15] Azam, A., Beg, I. and Arshad, M. (2010) Fixed Point in Topological Space Valued Cone Metric Spaces. Fixed Point Theory and Applications, 2010, Article ID: 604084.

[16] Zhang, X. (2010) Common Fixed Point Theorem of Lipschitz Type Mappings on Con Metric Spaces. Acta Mathematica Sinica, Chinese Series, 53, 1139-1148.