Advances in Analysis
Order Theoretic Common n-tuple Fixed Point
Download PDF (563.2 KB) PP. 129 - 134 Pub. Date: March 9, 2017
Author(s)
- Yaé Ulrich Gaba*
University of Cape Town, Rondebosch 7701, South Africa.
Abstract
Abstract In this article, we solve an open problem initially suggested in [2], namely:
Let (X,d) be a Hausdorff left K-complete T0-quasi-pseudometric space, Φ: X→ℜ be a bounded from above function and ≤ the preorder induced by Φ . Let F : X × X → X and Gi : X → X; i = 1, 2, · · · ,N for N > 2 be N + 1 d-sequentially continuous mapping on X such that the pairs {F;Gi}; i = 1, 2, · · · ,N are weakly left-related.
Problem:
1. Can we prove that F,G1, · · · ,GN have a common coupled fixed point in X?
2. Alternatively, what could be a correct formulation of the statement, using the induced preorder and the weakly left-related property that guarantees a positive answer?
We answer this question by the affirmative. In fact we prove that a more general result holds when F : Xn → X for a natural number n > 2.
Let (X,d) be a Hausdorff left K-complete T0-quasi-pseudometric space, Φ: X→ℜ be a bounded from above function and ≤ the preorder induced by Φ . Let F : X × X → X and Gi : X → X; i = 1, 2, · · · ,N for N > 2 be N + 1 d-sequentially continuous mapping on X such that the pairs {F;Gi}; i = 1, 2, · · · ,N are weakly left-related.
Problem:
1. Can we prove that F,G1, · · · ,GN have a common coupled fixed point in X?
2. Alternatively, what could be a correct formulation of the statement, using the induced preorder and the weakly left-related property that guarantees a positive answer?
We answer this question by the affirmative. In fact we prove that a more general result holds when F : Xn → X for a natural number n > 2.
Keywords
Quasi-pseudometric space, left K-complete, preordered space, left-weakly related,
common couple fixed point, common n-tuple fixed
References
[1] Ertürk, M, Karakaya, V; n-tuple fixed point theorems for contractive type mappings in partially ordered metric spaces. J. Inequal. Appl. 2013, 196 (2013).
[2] Y. U. Gaba; "An order theoretic approach in fixed point theory." Mathematical Sciences (2014): 1-7. DOI 10.1007/s40096-014-0133-6.
[3] V. Lakshmikantham and Lj. B. Ciric; Coupled fixed point theorems for nonlinear contractions in partially ordered metric space, Nonlinear Anal. 70, no. 12, 4341-4349, 2009.
[4] Y. U. Gaba; Startpoints and ( , )-contractions in quasi-pseudometric spaces, Journal of Mathematics, Volume 2014 (2014), Article ID 709253, 8 pages http://dx.doi.org/10.1155/2014/709253.