Fixed Point and Common Fixed Point Theorems for Cyclic Quasi-Contractions in Metric and Ultrametric Spaces

Affiliation(s)

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok, Thailand.

Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, Korea.

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok, Thailand.

Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, Korea.

ABSTRACT

In this paper, we prove introduce some fixed point theorems for quasi-contraction under the cyclical conditions. Then, we point out that a common fixed point extension is also applicable via our earlier results equipped together with a weaker cyclical properties, namely a co-cyclic representation. Examples are as well provided along this paper.

In this paper, we prove introduce some fixed point theorems for quasi-contraction under the cyclical conditions. Then, we point out that a common fixed point extension is also applicable via our earlier results equipped together with a weaker cyclical properties, namely a co-cyclic representation. Examples are as well provided along this paper.

Cite this paper

P. Chaipunya, Y. Cho, W. Sintunavarat and P. Kumam, "Fixed Point and Common Fixed Point Theorems for Cyclic Quasi-Contractions in Metric and Ultrametric Spaces,"*Advances in Pure Mathematics*, Vol. 2 No. 6, 2012, pp. 401-407. doi: 10.4236/apm.2012.26060.

P. Chaipunya, Y. Cho, W. Sintunavarat and P. Kumam, "Fixed Point and Common Fixed Point Theorems for Cyclic Quasi-Contractions in Metric and Ultrametric Spaces,"

References

[1] S. Banach, “Sur les Opérations Dans les Ensembles Abstraits Et Leur Applications Aux équations Intégrales,” Fundamenta Mathematicae, Vol. 3, 1922, p. 160.

[2] Lj. Ciric, “Generalized Contractions and Fixed-Point Theorems,” Publications de l’Institut Mathématique, Vol. 12, No. 26, 1971, pp. 19-26.

[3] Lj. B. Ciric, “A Generalization of Banach’s Contraction Principle,” Proceedings of the American Mathematical Society, Vol. 45, No. 2, 1974, pp. 267-273. doi:10.2307/2040075

[4] I. A. Rus, “Cyclic Representations and Fixed Points,” Ann. T. Popoviciu Seminar Funct. Eq. Approx. Convexity, Vol. 3, 2005, pp. 171-178.

[5] M. Pacurar and I. A. Rus, “Fixed Point Theory for Cyclic φ-Contractions,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 72, No. 3-4, 2010, pp. 1181-1187. doi:10.1016/j.na.2009.08.002

[6] E. Karapinar, “Fixed Point Theory for Cyclic Weakφ-Contraction,” Applied Mathematics Letters, Vol. 24, No. 6, 2011, pp. 822-825. doi:10.1016/j.aml.2010.12.016

[7] W. Sintunavarat and P. Kumam, “Common Fixed Point Theorem for Cyclic Generalized Multi-Valued Contraction Mappings,” Applied Mathematics Letters, Vol. 25, No. 11, 2012, pp. 1849-1855. doi:10.1016/j.aml.2012.02.045

[8] R. H. Haghi, Sh. Rezapour and N. Shahzad, “Some Fixed Point Generalizations Are Not Real Generalizations,” Non-linear Analysis: Theory, Methods & Applications, Vol. 74, No. 5, 2011, pp. 1799-1803. doi:10.1016/j.na.2010.10.052

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

[1] S. Banach, “Sur les Opérations Dans les Ensembles Abstraits Et Leur Applications Aux équations Intégrales,” Fundamenta Mathematicae, Vol. 3, 1922, p. 160.

[2] Lj. Ciric, “Generalized Contractions and Fixed-Point Theorems,” Publications de l’Institut Mathématique, Vol. 12, No. 26, 1971, pp. 19-26.

[3] Lj. B. Ciric, “A Generalization of Banach’s Contraction Principle,” Proceedings of the American Mathematical Society, Vol. 45, No. 2, 1974, pp. 267-273. doi:10.2307/2040075

[4] I. A. Rus, “Cyclic Representations and Fixed Points,” Ann. T. Popoviciu Seminar Funct. Eq. Approx. Convexity, Vol. 3, 2005, pp. 171-178.

[5] M. Pacurar and I. A. Rus, “Fixed Point Theory for Cyclic φ-Contractions,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 72, No. 3-4, 2010, pp. 1181-1187. doi:10.1016/j.na.2009.08.002

[6] E. Karapinar, “Fixed Point Theory for Cyclic Weakφ-Contraction,” Applied Mathematics Letters, Vol. 24, No. 6, 2011, pp. 822-825. doi:10.1016/j.aml.2010.12.016

[7] W. Sintunavarat and P. Kumam, “Common Fixed Point Theorem for Cyclic Generalized Multi-Valued Contraction Mappings,” Applied Mathematics Letters, Vol. 25, No. 11, 2012, pp. 1849-1855. doi:10.1016/j.aml.2012.02.045

[8] R. H. Haghi, Sh. Rezapour and N. Shahzad, “Some Fixed Point Generalizations Are Not Real Generalizations,” Non-linear Analysis: Theory, Methods & Applications, Vol. 74, No. 5, 2011, pp. 1799-1803. doi:10.1016/j.na.2010.10.052

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