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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.

1

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