Fixed point theorems in R-trees with applications to graph theory

By Rafa Esṕınola, W. A. Kirk. 2006

156 citations recorded by the source corpus.

Abstract It is proved that a commutative family of nonexpansive mappings of a complete R -tree X into itself always has a nonempty common fixed point set if X does not contain a geodesic ray. As a consequence of this, we show that any commuting family of edge preserving mappings of a connected reflexive graph G that contains no cycles or infinite paths always has at least one common fixed edge. This approach provides a new proof of the classical fixed edge theorem of Nowakowski and Rival. Several related results are also obtained.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3a742c5cafa270aabf9510a16fc99c4a5f7c172f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Naseer Shahzad
2. S. Aleomraninejad
3. W. A. Kirk
4. Sh. Rezapour
5. Nawab Hussain
6. G. Jungck
7. Poom Kumam
8. Wutiphol Sintunavarat

In Computational Theory and Mathematics

1. Naseer Shahzad
2. David Bryant
3. W. A. Kirk
4. Sh. Rezapour
5. Paul F. Tupper
6. Nawab Hussain
7. Bancha Panyanak
8. G. Jungck

In Geometry and Topology

1. Naseer Shahzad
2. S. Aleomraninejad
3. David Bryant
4. W. A. Kirk
5. Sh. Rezapour
6. Paul F. Tupper
7. Nawab Hussain
8. Poom Kumam

Citations in the corpus

• Some fixed point results on a metric space with a graph (2012, background): “In 2006, Espinola and Kirk provided useful results on combining fixed point theory and graph theory [6].”
• Some recent results in metric fixed point theory (2007, unknown): “Theorem 15 ([17]).”
• Hyperconvexity and tight-span theory for diversities (2010, background): “The closed, convex sets of a real tree satisfy the Helly property [34], so every finite subcollection of Γ has non-empty intersection.”
• Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces (2008, unknown): “B ( y, d(y,E) ) ∩ T (z) ∩ E = ∅, (see for example, [9]).”
• FIXED POINT THEOREMS IN CAT(0) SPACES AND R-TREES (2009, background): “In [ 5 ] the fixed edge theorem is extended as follows.”
• Fixed point results for multimaps in CAT(0) spaces (2009, background): “3 of Espinola and Kirk [9], Fix(F ) is nonempty closed convex and geodesically bounded.”