Graph and fixed point

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The 2nd Regional conference on Mathematics And Applications Payame Noor university
Mazandaran,Tonekabon , 2 October 2014
GRAPH AND FIXED POINT
S. M. A. ALEOMRANINEJAD1∗
Abstract. Over the past few decades, there have been a lot of
activity in fixed point theory and another branches in mathemat-
ics such differential equations, geometry and algebraic topology. In
2006, Espinola and Kirk started combining fixed point theory and
graph theory. Recently, Reich and Zaslavski have studied a new
inexact iterative scheme for fixed points of contractive and non-
expansive multifunctions. In this paper,In this paper, we obtain
some fixed point results on subgraphs of directed graphs.
1. Introduction and Preliminaries
Let (X, d) be a metric space and ∆ = X×X. Consider a directed
graph Gsuch that the set V(G) of its vertices coincides with Xand
the set E(G) of its edges contains all loops, that is, ∆ ⊂E(G). We
assume that Ghas no parallel edges. By 
Gwe denote the undirected
graph obtained from Gby ignoring direction of the edges. We say that
a mapping f:X→Xpreserves edges of Gwhenever (x, y)∈E(G)
implies (f x, fy)∈E(G) for all x, y ∈X. Also, a mapping f:X→X
is called a G-contraction whenever fpreserves edges of Gand there is
2010 Mathematics Subject Classification. Primary 47H10; Secondary 05C40.
Key words and phrases. Fixed point, Connected graph, G-contraction, G-
nonexpansive mapping.
∗Speaker.
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2 F. AUTHOR, S. AUTHOR
α∈(0,1) such that d(fx, fy)≤αd(x, y) for all (x, y)∈E(G) ([2]). If
fis a G-contraction, then fis 
G-contraction([2]).
Example 1.1. ([2]) Each Banach contraction is a G0-contraction, where
the graph G0is defined by E(G0) = X×X.
Example 1.2. ([2]) Let ⪯be a partial order on X. Define the graph
G1by
E(G1) = {(x, y)∈X×X:x⪯y}.
The preserving edges of G1means fis nondecreasing with respect to
this order.
Definition 1.3. Let xand ybe two vertices in a graph G. A path in
Gfrom xto yof length nis a sequence {xi}n
i=0 of n+1 distinct vertices
such that x0=x,xn=yand (xi, xi+1)∈E(G) for all i= 0,1, ..., n −1.
We denote by r(x, y), the sum of edges distance between x and y,
that is, r(x, y) = n
i=1 d(xi−1, xi).[x]Gis the set of all vertices in G
that there is a path from those to x. Let Gbe a directed graph and T
a selfmap on G. We say that Tis a self-path map whenever x∈[T x]G
for all x∈G.
We say that Gis a (C)-graph[1] whenever for any {xn}n≥0in X,
if xn→xand (xn, xn+1)∈E(G) for n≥0, there is a subsequence
{xkn}n≥0with (xkn, x)∈E(G) for n≥0.
We say that Gis a (P)-graph[1] whenever {xn}n≥1is a convergent
sequence to a point xand xn∈[x]G, we have r(xn, x)→0. Now,
by providing next examples, we show that the notions (C)-graph and
(P)-graph are independent.
Example 1.4. Set X={1
n:n∈N}N{0}with the Euclidean
metric. Define the undirected graph G2:V(G) := Xand
E(G) = {(1
n, n) : n∈N}{(n, 0) : n∈N}.
Let {xn=1
n}∞
n=1. We see xn→0, xn∈[0]G2and
r(xn,0) = d(1
n, n) + d(n, 0) = 2n−1
n.

SHORT TITLE 3
So r(xn,0) →0 and then G2is not (P)-graph. It is easy to see that G2
is (C)-graph.
Example 1.5. Set X={1
n:n∈N}{1
√2+n:n∈N}{0}with the
Euclidean metric. Define the undirected graph G3:V(G) := Xand
E(G) = {(1
n,1
n+ 1) : n∈N}{(1
n,1
√2 + n) : n∈N}{(1
√2 + n,0) : n∈N}.
Let {xn=1
n}∞
n=1. We see xn→0 and (xn, xn+1)∈E(G3), But there is
not subsequence {xkn}n≥0with (xkn, x)∈E(G). So G3is not (c)-graph.
It is easy to see that G3is (P)-graph.
2. Main results
Definition 2.1. Let G′be a subgraph of the directed graph G. We
say that bis upper bound for G′, if for any g′in G′,g′∈[b]Gand cis
supremum of G′whenever for any upper bound b, we have c∈[b]G.
Example 2.2. Let V(G) = {1
n}{0},E(G) = {(1
n,1
n+1 )}{(1
n,0)}
and E(G′) = {(1
n,1
n+1 )}. It is clear that 0 is a supremum of G′, not
belong to G′and G′is infinity.
Supremum don’t need to be unique.
Example 2.3. Let V(G) = {a, b, c, d},E(G) = {(a, b),(b, c),(c, d),(d, a)}
and E(G′) = {(a, b),(b, c)}. It is clear that a, b, c, d are upper bounds
and supremum of G′.
Definition 2.4. The element x0in Gis end point of Gif for any x∈G
with (x0, x)∈G, we have x0=x.
Example 2.5. Let φ:X→(−∞,∞) and Gbe a directed graph with
V(G) = X,E(G) = {(x, y) : d(x, y)≤φ(x)−φ(y)}. If there is x0∈X
such that φ(x0) = infx∈Xφ(x) ,then Ghas an end point.
Proof. It is clear that x0is the end point otherwise there is y=x0in
Gsuch that (x0, y)∈E(G). We see that
d(x0, y)≤φ(x0)−φ(y)≤φ(x0)−φ(x0) = 0
and then x0=y. □

4 F. AUTHOR, S. AUTHOR
Let Gbe the directed graph andMthe set of all paths in G. Then
⊆is a partial order on M. By using Hausdorffsmaximum principle, M
has a maximal element. This means that Ghas a maximal path. We
use this subject in our results.
Theorem 2.6. Let Gbe a directed graph with the property that every
path in Ghas an upper bound. Then Ghas an end point or cycle.
Theorem 2.7. Let Gbe a directed graph. Then Ghas an end point if
and only if each self-path map on Ghas a fixed point.
Theorem 2.8. Let Gbe a directed graph such that every path has a
supremum and Ta selfmap on Gsuch that T x ∈[T y]Gfor all x∈[y]G,
G′={x∈G:x∈[T x]G} =∅and G′has no cycle. Then Thas a fixed
point.
Knaster-Tarski fixed point theorem is a consequence of Theorem 2.8.
Theorem 2.9. Let (X, ≥X)be a partially ordered such that each chain
in Xhas a supremum and f:X→Xbe monotone. Assume that there
exists a∈Xwith a≤f(a). Then fhas a fixed point.
Acknowledgements: Research of the author was supported by
Qom University of technology .
References
1. S. M. A. Aleomraninejad, Sh. Rezapour, N. Shahzad, Convergence of an iter-
ative scheme for multifunctions,J. Fixed Point Thoery Appl. (2011).
2. J. Jachymski, The contraction principle for mappings on a metric space with a
graph,Proc. Amer. Math. Soc. 136 (2008) 1359-1373.
1Department of Mathematics, Qom University of Technology, Qom,
Iran.
E-mail address:aleomran63@yahoo.com