Meir-Keeler Type n-tuplet Fixed Point Theorems in Partially Ordered Metric Spaces

Abstract

An n-tuplet fixed point is a generalization of the well-known concept of “coupled fixed point and tripled fixed point”. The intent of this paper is to introduce the concept of mixed strict monotone property and generalize Meir-Keeler contraction for mapping , where n is an arbitrary positive integer. Also establish an n-tuplet fixed point theorem for mappings under a generalized Meir-Keeler contraction in the setting of partially ordered metric spaces. Related examples are also given to support our main results. Our results are the generalizations of the results of B. Samet [8] and Hassen et al. [15]. Also as application, some results of integral type are given.

Meir-Keeler Type n-tuplet Fixed Point Theorems in Partially Ordered Metric Spaces

Abstract

An n-tuplet fixed point is a generalization of the well-known concept of “coupled fixed point and tripled fixed point”. The intent of this paper is to introduce the concept of mixed strict monotone property and generalize Meir-Keeler contraction for mapping , where n is an arbitrary positive integer. Also establish an n-tuplet fixed point theorem for mappings  under a generalized Meir-Keeler contraction in the setting of partially ordered metric spaces. Related examples are also given to support our main results. Our results are the generalizations of the results of B. Samet [8] and Hassen et al. [15]. Also as application, some results of integral type are given.

Keywords

• n-tuplet fixed point
• Meir-Keeler type contraction
• mixed strict monotone property
• partially ordered metric space

References

1. E. S. Wolk, “Continuous Convergence in partially ordered Sets,” General Topology and its Applications, vol. 5, no. 3, pp. 221-234, 1975.
2. B. Monjardet, “Metrics on partially ordered sets – a survey,” Discrete Mathematics, vol. 35, pp. 173-184, 1981.
3. A. C. M. Ram and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435-1443, 2004.
4. T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis: Theory, Method & Applications, vol. 65, no. 7, pp. 1379-1393, 2006.
5. V. Lakshmikantham and L. Ciric, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis: Theory, Method & Applications, vol. 70, no. 12, pp. 4341-4349, 2009.
6. V. Berinde and M. Borcut, “Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces,” Nonlinear Analysis: Theory, Method & Applications, vol. 74, no. 15, pp. 4889-4897, 2011.
7. B. Samet and C. Vetro, “Coupled fixed point, f- invariant set and fixed point of N-order,” Annals of Functional Analysis, Vol. 1, no. 2, pp. 46-56, 2010.
8. B. Samet, “Coupled fixed point theorems for a generalized Meir-Keeler Contraction in partially ordered metric spaces,” Nonlinear Analysis, 47:4508-4517, 2010. [9].E. Karapinar, “Quartet fixed point for nonlinear contraction,” http://arxivorg /abs/1106.5472.

9. E. Karapinar and N. V. Luong, “Quadruple fixed point theorems for nonlinear contractions,” Computers & Mathematics with Applications, Vol. 64, pp. 1839-1848, 2012.
10. E. Karapinar, “Quadruple fixed point theorems for weak ∅-contractions,” ISRN Mathematical Analysis, Vol. 2011, Article ID 989423, 15 pages, 2011.
11. E. Karapinar and V. Berinde, “Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Banach Journal of Mathematical Analysis, Vol. 6, no. 1, pp. 74-89, 2012.
12. E. Karapinar, “A new quartet fixed point theorem for nonlinear contractions,” Journal of Fixed point Theory, Appli, Vol. 6, no. 2, pp. 119-135, 2011.
13. E. Karapinar, W. Shatanwi, Z. Mustafa, “Quadruple fixed point theorems under nonlinear contractive conditions in partially ordered metric spaces,” Journal of Applied Mathematics, Vol. 2012, Article ID 951912, 17 pages, doi:10.1155/2012/951912.
14. Hassen Aydi, Erdal Karapinar, CalogeroVetro, “Meir-Keeler Type Contractions for Tripled fixed points” ActaMethematicaScientia 2012, 32B (6):L2119-2130.
15. T. Suzuki, “Meir-Keeler Contractions of Integral Type Are Still Meir Meir-Keeler Contractions,” Int. J. Math. Math. Sci., 2007, 2007: Article ID 39281.
16. Muzeyyen Erturk and Vatan Karakaya , “n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces.” Journal of Inequality and Applications, 2013:196, doi: 10.1186/1029-242X-2013- 196.