We start with the following definition.
Definition 3.1 A mapping
is said to be a probabilistic
Gcontraction if
f preserves edges and there exists
such that
Example 3.2 Let
be a metric space endowed with a graph G, and let the mapping
be a Banach Gcontraction. Then the induced Menger PM space
is a probabilistic Gcontraction.
To see this, let
, then
and there exists
such that
. Now, for
, we have
Thus f satisfies (3.1).
From Example 3.2 it is inferred that every Banach Gcontraction is a probabilistic Gcontraction with the same contraction constant.
Proposition 3.3
Let
be a probabilistic
G
contraction with contraction constant
.
Then
 (i)
f
is both a probabilistic
contraction and a probabilistic
contraction with the same contraction constant
α.
 (ii)
Proof
 (i)
It follows from the symmetry of
.
 (ii)
Suppose
. Then
since f is a probabilistic Gcontraction. But
is f invariant, so we conclude that
. Condition (3.1) is satisfied automatically, since
is a subgraph of G. □
Lemma 3.4
Let
be a Menger PMspace under a
tnorm Δ satisfying
. Assume that the mapping
is a probabilistic
Gcontraction. Let
, then
as
(
). Moreover, for
,
(
) if and only if
(
).
Proof Let
and
, then there exists a path
,
, in
from
x to
y with
,
and
. From Proposition 3.3,
f is a probabilistic
contraction. By induction, for
, we have
and
for all
and
. Thus we obtain
Let
and
be given. Since
, then there exists
such that
. Choose a natural number
such that for all
we have
and
. We get, for all
,
so that
as
(
). Continuing recursively, one can easily show that
Let
. Let
and
be given. Since
, then there exists
such that
. Choose a natural number
such that for all
we have
and
. So that for all
, we have
Hence,
as
. □
Every tnorm can be extended in a unique way to an nary as follows:
,
for
. Let
be a path between two vertices x and y in a graph G. Let us denote with
for all t. Clearly the function
is monotone nondecreasing.
Definition 3.5 Let
be a PMspace and
. Suppose that there exists a sequence
in
S such that
and
for
. We say that:
 (i)
 (ii)
Example 3.6 Let
be a Menger PMspace induced by the metric
on
, and let I be an identity map on S.
Consider the graph
consisting of
and
We note that
as
. Also, it is easy to see that
is a
graph. But since
, then
Thus
is not an
graph.
Example 3.7 Let
be a Menger PM space induced by the metric
on
, and let
I be an identity map on
S. Consider the graph
consisting of
and
Since
as
. Clearly,
is not a
graph. But
as
(
). Thus
is an
graph.
From the above examples, we note that the notions of
graph and
graph are independent even if f is an identity map.
The following lemma is essential to prove our fixed point results.
Lemma 3.8 (Miheţ [18])
Let
be a Menger PM
space under a
t
norm Δ
of ℋ
type.
Let
be a sequence in
S,
and let there exist
such that
Then
is a Cauchy sequence.
Theorem 3.9
Let
be a complete Menger PM
space under a
t
norm Δ
of ℋ
type.
Assume that the mapping
is a probabilistic
G
contraction and there exists
such that
,
then the following assertions hold.
 (i)
 (ii)
If
G
is a weakly connected
graph, then
f
is a Picard operator.
Proof Since
f is a probabilistic
Gcontraction and there exists
such that
. By induction
for all
and
(i) Since the
tnorm Δ is of ℋtype, then from Lemma 3.8 it can be inferred that
is a Cauchy sequence in
S. From completeness of the Menger PMspace
S, there exists
such that
Now we prove that
ϱ is a fixed point of
f. Let
G be a
graph. Then there exists a subsequence
of
and
such that
for all
. Note that
is a path in
G and so in
from
to
ϱ, thus
. Since
f is a probabilistic
Gcontraction and
for all
. For
and
, we get
Hence, we conclude that
. Now, let
, then from Lemma 3.4 we get
Next to prove the uniqueness of a fixed point, suppose
such that
. Then from Lemma 3.4, for
, we have
Hence,
. Moreover, if G is weakly connected, then f is a Picard operator as
.
(ii) Let
G be a weakly connected
graph. By using the same arguments as in the first part of the proof, we obtain
. For each
let
;
be a path in
from
to
ϱ with
,
and
.
where
.
Since
G is an
graph and
for
with
, then the sequence of functions
converges to
(
) uniformly. Let
and
be given. Since the family
is equicontinuous at point
, there exists
such that
for every
. Choose
such that for all
we have
and
. So that in view of (3.7), for all
, we have
Hence, we deduce
. Finally, let
be arbitrary, then from Lemma 3.4,
. □
Corollary 3.10
Let
be a complete Menger PM
space under a
t
norm Δ
of ℋ
type.
Assume that
S
is endowed with a graph
G
which is either

graph or

graph.
Then the following statements are equivalent:
 (i)
 (ii)
For every probabilistic
Gcontraction
f
on
S, if there exists
such that
, then
f
is a Picard operator.
Proof (i) ⇒ (ii): It is immediate from Theorem 3.9.
(ii) ⇒ (i): Suppose that
G is not weakly connected. Then
is disconnected,
i.e., there exists
such that
and
. Let
, we construct a selfmapping
f by
Let
, then
, which implies
. Hence
, since G contains all loops. Thus the mapping f preserves edges. Also, for
and
, we have
; thus (3.1) is trivially satisfied. But
and
are two fixed points of f contradicting the fact that f is a Picard operator. □
Remark 3.11 Taking
, Theorem 3.9 improves and extends the result of Sehgal [[12], Theorem 3] to all Menger PMspaces with tnorms of ℋtype. Theorem 3.9 generalizes claim 4^{0} of [[8], Theorem 3.2], and thus we have the following consequence.
Corollary 3.12 (Jachymski [[8], Theorem 3.2])
Let
be a complete metric space endowed with the graph
G. Assume that the mapping
is a Banach
Gcontraction and the following property is satisfied:
(
) For any sequence
in
S, if
in
S
and
for all
, then there exists a subsequence
with
for all
.
If there exists
with
, then
is a Picard operator. Furthermore, if
G
is weakly connected, then
f
is a Picard operator.
Proof Let
be the Menger PMspace induced by the metric d. Since the mapping f is a Banach Gcontraction, then it is a probabilistic Gcontraction (see Example 3.2) and property (
) invokes that G is a
graph. Hence the conclusion follows from Theorem 3.9(i). □
Example 3.13 Let
be a Menger PMspace where
and
for
. Then
is complete. Define a selfmapping
f on
X by
Further assume that X is endowed with a graph G consisting of
and
. It can be seen that f is a probabilistic Gcontraction with
and satisfies all the conditions of Theorem 3.9(i).
Note that for
and
and for each
, we can easily set
such that
Hence, one cannot invoke [[12], Theorem 3].
Definition 3.14 Let
be a Menger PMspace under a tnorm Δ of ℋtype. A mapping
is said to be: (i) continuous at point
whenever
in S implies
as
; (ii) orbitally continuous if for all
and any sequence
of positive integers,
implies
as
; (iii) orbitally Gcontinuous if for all
and any sequence
of positive integers,
and
imply
(see [8]).
Theorem 3.15
Let
be a complete Menger PMspace under a
tnorm Δ of ℋtype. Assume that the mapping
is a probabilistic
Gcontraction such that
f
is orbitally
Gcontinuous, and let there exist
such that
. Then
f
has a unique fixed point
and for every
,
. Moreover, if
G
is weakly connected, then
f
is a Picard operator.
Proof Let
, by induction
for all
. By using Lemma 3.8, it follows that
. Since f is orbitally Gcontinuous, then
. This gives
. From Lemma 3.4 for any
,
. □
Remark 3.16 We note that in Theorem 3.15 the assumption that f is orbitally Gcontinuous can be replaced by orbital continuity or continuity of f.
Remark 3.17 Theorem 3.15 generalizes and extends claims 2^{0} and 3^{0} [[8], Theorem 3.3] and claim 3^{0} [[8], Theorem 3.4].
As a consequence of Theorems 3.9 and 3.15, we obtain the following corollary, which is actually a probabilistic version of Theorem 1.1 and thus generalizes and extends the results of Nieto and RodríguezLópez [[4], Theorems 2.1 and 2.3], Petruşel and Rus [[3], Theorem 4.3] and Ran and Reurings [[2], Theorem 2.1].
Corollary 3.18
Let
be a partially ordered set,
and let
be a complete Menger PM
space under a
t
norm Δ
of ℋ
type.
Assume that the mapping
is nondecreasing (
nonincreasing)
with respect to the order ‘⪯
’ on
S
and there exists
such that
Also suppose that either
 (i)
 (ii)
If there exists
with
, then
f
has a fixed point. Furthermore, if
is such that every pair of elements of
S
has an upper or lower bound, then
f
is a Picard operator .
Proof Consider a graph
consisting of
and
. If f is nondecreasing, then it preserves edges w.r.t. graph
and condition (3.10) becomes equivalent to (3.1). Thus f is a probabilistic
contraction. In case f is nonincreasing, consider
with
and a vertex set coincides with S. Actually,
and from Proposition 3.3 if f is a probabilistic
contraction, then it is a probabilistic
contraction. Now if f is continuous, then the conclusion follows from Theorem 3.15. On the other hand, if (ii) holds, then
and
are
graphs and conclusions follow from the first part of Theorem 3.9. □
By relaxing ℋtype condition on a tnorm, our next result deals with a compact Menger PMspace using the following class of graphs as the fixed point property is closely related to the connectivity of a graph.
Definition 3.19 Let
be a PMspace endowed with a graph G and
. Assume the sequence
in S with
for
and
(
), we say that the graph G is
graph if for any subsequence
, there exists a natural number N such that
for all
.
Theorem 3.20
Let
be a compact Menger PMspace under a
tnorm Δ satisfying
. Assume that the mapping
is a probabilistic
Gcontraction, and let there exist
such that
. If
G
is an
graph, then
f
has a unique fixed point
.
Proof Since
, then
for
and
From compactness, let
be a subsequence such that
. Let
and
be given. Since
, then there exists
such that
choose
such that for all
we have
and
. Then we obtain
Thus,
.
Choose
such that for all
we have
and
. Since
G is an
graph, there exists
such that
for all
. Let
, then for
we get
Hence,
. Note that
is a path in
, so that
. □
So far it remains to investigate whether Theorem 3.20 can be extended to a complete PMspace?
Definition 3.21[12]
Let
be a PMspace, and let
and
be fixed real numbers. A mapping
is said to be
contraction if there exists a constant
such that for
and
we have
The PM space
is said to be
chainable if for each
there exists a finite sequence
of elements in S with
and
such that
for
.
It is important to note that every
contraction mapping is continuous. Let
in
S, then there exists a natural number
such that
for all
. Thus, for
and for all
, we obtain
Hence,
.
Theorem 3.22
Let
be a complete
chainable Menger PMspace under a
tnorm Δ of ℋtype. Let the mapping
be an
contraction. Then
f
is a Picard operator.
Proof Consider the graph
G consisting of
and
coinciding with
S. Let
. Since the PMspace
is
chainable, there exists a finite sequence
in
S with
and
such that
for
. Hence,
for
, which yields that
G is connected. Let
, then
. Since the mapping
f is an
contraction, thus (3.1) is satisfied. Finally we have
Thus,
. Hence, f is a probabilistic Gcontraction and the conclusion follows from Theorem 3.15. □
Remark 3.23 Theorem 3.22 has an advantage over Theorem 7 of Sehgal and BharuchaReid [12] which is only restricted to continuous tnorms satisfying
. Moreover, the proof of our result is rather simple and easy, which invokes the novelty of Theorem 3.22.
Definition 3.24 (Edelstein [19, 20])
The metric space
is εchainbale for some
if for every
, there exists a finite sequence
of elements in S with
,
and
for
.
Remark 3.25[12]
If
is an εchainable metric space, then the induced Menger PMspace
is an
chainable space.
Corollary 3.26 (Edelstein [19, 20])
Let
be a complete
ε
chainable metric space.
Let
and let there exist
such that
Then
f
is a Picard operator.
Proof Since the metric space
is
εchainable, then the induced Menger PMspace
is
chainable for each
. We only need to show that the selfmapping
f on
S is an
contraction. Let
be such that
,
i.e.,
or
. The definition of
implies
and thus
. Now, for
, we get
Hence the conclusion follows from Theorem 3.22. □
Kirk et al.[21] introduced the idea of cyclic contractions and established fixed point results for such mappings.
Let
S be a nonempty set, let
m be a positive integer, let
be nonempty closed subsets of
S, and let
be an operator. Then
is known as a cyclic representation of
S w.r.t.
f if
and the operator f is known as a cyclic operator [21].
In the following, we present the probabilistic version of the main result of [21], as a last consequence of Theorem 3.9.
Theorem 3.27
Let
be a complete Menger PM
space under a
t
norm Δ
of ℋ
type.
Let
m
be a positive integer,
let
be nonempty closed subsets of
S,
and
.
Assume that
 (i)
is a cyclic representation of
Y
w.r.t. f;
 (ii)
Then
f
has a unique fixed point
and
for any
.
Proof Since
is closed, then
is complete. Let us consider a graph G consisting of
and
. By (i) it follows that f preserves edges. Now, let
in Y such that
for all
. Then by (3.13) it is inferred that the sequence
has infinitely many terms in each
;
. So that one can easily identify a subsequence of
converging to
in each
; and since
’s are closed, then
. Thus, we can easily form a subsequence
in some
,
such that
for
. It elicits that G is a weakly connected
graph. Hence, by Theorem 3.9 conclusion follows. □