DataPDF Available
The 2nd Regional conference on Mathematics And Applications Payame Noor university
Mazandaran,Tonekabon , 2 October 2014
GRAPH AND FIXED POINT
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:XXpreserves edges of Gwhenever (x, y)E(G)
implies (f x, fy)E(G) for all x, y X. Also, a mapping f:XX
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.
1
2 F. AUTHOR, S. AUTHOR
α(0,1) such that d(fx, fy)αd(x, y) for all (x, y)E(G) (). If
fis a G-contraction, then fis
G-contraction().
Example 1.1. () Each Banach contraction is a G0-contraction, where
the graph G0is deﬁned by E(G0) = X×X.
Example 1.2. () Let be a partial order on X. Deﬁne the graph
G1by
E(G1) = {(x, y)X×X:xy}.
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(xi1, 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 xG.
We say that Gis a (C)-graph whenever for any {xn}n0in X,
if xnxand (xn, xn+1)E(G) for n0, there is a subsequence
{xkn}n0with (xkn, x)E(G) for n0.
We say that Gis a (P)-graph whenever {xn}n1is 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:nN}N{0}with the Euclidean
metric. Deﬁne the undirected graph G2:V(G) := Xand
E(G) = {(1
n, n) : nN}{(n, 0) : nN}.
Let {xn=1
n}
n=1. We see xn0, xnG2and
r(xn,0) = d(1
n, n) + d(n, 0) = 2n1
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:nN}{1
2+n:nN}{0}with the
Euclidean metric. Deﬁne the undirected graph G3:V(G) := Xand
E(G) = {(1
n,1
n+ 1) : nN}{(1
n,1
2 + n) : nN}{(1
2 + n,0) : nN}.
Let {xn=1
n}
n=1. We see xn0 and (xn, xn+1)E(G3), But there is
not subsequence {xkn}n0with (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 Gbe a subgraph of the directed graph G. We
say that bis upper bound for G, if for any gin G,g[b]Gand cis
supremum of Gwhenever 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 Gand Gis 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 xG
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 x0X
such that φ(x0) = infxXφ(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={xG:x[T x]G} =and Ghas 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:XXbe monotone. Assume that there
exists aXwith af(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.