@ Appl. Gen. Topol. 22, no. 2 (2021), 435-446doi:10.4995/agt.2021.15561
© AGT, UPV, 2021

On a probabilistic version of Meir-Keeler type

fixed point theorem for a family of

discontinuous operators

Ravindra K. Bisht
a
and Vladimir Rakočević

b

a
Department of Mathematics, National Defence Academy, Khadakwasla-411023, Pune, In-

dia (ravindra.bisht@yahoo.com)
b

University of Nǐs, Faculty of Sciences and Mathematics, Vǐsegradska 33, 18000 Nǐs, Ser-

bia. (vrakoc@sbb.rs)

Communicated by S. Romaguera

Abstract

A Meir-Keeler type fixed point theorem for a family of mappings is
proved in Menger probabilistic metric space (Menger PM-space). We
establish that completeness of the space is equivalent to fixed point
property for a larger class of mappings that includes continuous as well
as discontinuous mappings. In addition to it, a probabilistic fixed point
theorem satisfying (ǫ − δ) type non-expansive mappings is established.

2010 MSC: 47H09; 47H10.

Keywords: Menger PM-spaces; fixed point; almost orbital continuity; non-
expansive mapping.

1. Introduction and preliminaries

The idea of statistical metric space or probabilistic Menger space can be
traced back to Menger [10], who extended the concept of metric space (X, d),
by replacing the notion of distance d(x, y) (x, y ∈ X) by a distributive function
Fx,y : X × X → R, where Fx,y(t) represents the probability that the distance
between x and y is less than t. Schweizer and Sklar [22, 23] studied various
properties, e.g., topology, convergence of sequences, continuity of mappings,

Received 04 May 2021 – Accepted 15 June 2021

http://dx.doi.org/10.4995/agt.2021.15561


R. K. Bisht and V. Rakočević

completeness, etc., of these spaces. In 1972, Sehgal and Bharucha–Reid [24]
showed the role of distributive functions in metric fixed point theory and es-
tablished the probabilistic metric version of the classical Banach contraction
mapping principle. Since then the study of fixed point theorems in PM-space
has emerged as an active area of research.

Let g be a selfmapping which satisfy some contractive condition on a com-
plete Menger PM-space (X, F, T ). Then there exists a Cauchy sequence of
successive iterates {gnx}n∈N for each x in X which converges to some point,
say z ∈ X, and the limiting point z of the sequence of iterates is nothing but a
fixed point of g. However, there exist various contractive definitions which en-
sure the existence of the Cauchy sequence of iterates converging to some limit
point, but the limit point may not be a fixed point.

Pant et al. [17] (see also Bisht [2]) proved the following theorem where the
Meir-Keeler [9] type operator ensures the convergence of sequence of iterates
but does not ensure the existence of a fixed point.

Lemma 1.1. Let (X, F, T ) be a complete Menger PM-space, and let f be self-
mapping of X satisfying one of the following conditions

(i) for every ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that

ǫ − δ < min
{

Fx,gx(t), Fy,gy(t)
}

< ǫ ⇒ Fgx,gy(t) ≥ ǫ,

(ii) Fgx,gy(t) > min
{

Fx,gx(t), Fy,gy(t)
}

,

or

(i’) for every ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that

ǫ − δ ≤ min
{

Fx,gx(t), Fy,gy(t)
}

< ǫ ⇒ Fgx,gy(t) > ǫ,

for all x, y ∈ X. Then for any x in X the sequence of iterates {gnx}n∈N is
a Cauchy sequence and there exists a point z in X such that lim

n→∞
gnx = z for

each x in X.

The triple (X, F, Tmin) is a complete Menger PM-space, for X ⊆ R (see Re-
mark 2.3). The following example illustrates Lemma 1.1, but does not possess
a fixed point.

Example 1.2. Let X = [1, 2] ∪

{

1 −
1

3n
: n = 0, 1, 2, · · ·

}

and d be the usual

metric. Define g : X → X by

gx =

{

0 if 1 ≤ x ≤ 2.

1 −
1

3n+1
if x = 1 −

1

3n
, n = 1, 2, · · · .

Then

g(X) =

{

1 −
1

3n
: n = 1, 2, · · ·

}

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 436



On a probabilistic version of Meir-Keeler type fixed point theorem

and g is fixed point free. The mapping g satisfies the contractive condition
(i′) of Lemma 1.1 with

δ(ε) =

{

1
3n

− ε if 1
3n+1

≤ ε < 1
3n

, n = 1, 2, · · ·
ε if ε ≥ 1.

Therefore, to ensure the existence of a fixed point under such contractive
definitions, one needs to assume some additional hypotheses on the mappings.

Ćirić [5] introduced the notion of orbital continuity. If g is a self-mapping
of a metric space (X, d) then the set Og(x) = {g

nx | n = 0, 1, 2, . . .} is called
the orbit of g at x and g is called orbitally continuous if u = limi g

mix implies
gu = limi gg

mix. Every continuous self-mapping is orbitally continuous but not
conversely. In 1977, Jaggi [7] introduced the concept of x0-orbital continuity
which is weaker than orbital continuity of the mapping. A self-mapping g
of a metric space (X, d) is called x0-orbitally continuous for some x0 ∈ X

if its restriction to the set O(g, x0), is continuous, i.e., g : O(g, x0) → X,

is continuous, here O(g, x) represents closure of the orbit of g at x0. The
mapping g is said to be orbitally continuous if it is x0-orbitally continuous for
all x0 ∈ X. In 2011, Jungck [8] gave a generalized notion of orbital continuity,
namely, almost orbital continuity. A self-mapping g of a metric space (X, d)
is called almost orbitally continuous at x0 ∈ X if whenever limn g

in(x) = x0
for some x ∈ X and subsequence {gin(x)} of gn(x), there exists a subsequence
{gjn(x)} of gn(x) such that limn g

jn(x) = g(x0). Orbital continuity implies
almost orbital continuity, but the implication is not reversible. In 2017, Pant
and Pant [11] introduced the notion of k−continuity. A self-mapping g of a
metric space X is called k-continuous, k = 1, 2, 3, . . . , if gkxn → gt, whenever
{xn}n∈N is a sequence in X such that g

k−1xn → t. It may be observed that
1-continuity is equivalent to continuity and continuity implies 2-continuity, 2-
continuity implies 3-continuity and so on but not conversely. It is important
to note that k−continuity of the mapping implies orbital continuity but not
conversely. More recently, Pant et al. [12] introduced the notion of weak
orbital continuity, which is weaker than orbital continuity of the mapping. A
self-mapping g of a metric space (X, d) is called weakly orbitally continuous
[12] if the set {y ∈ X : limi g

miy = u implies limi gg
miy = gu} is nonempty,

whenever the set {x ∈ X : limi g
mix = u} is nonempty.

Example 1.3. Let X = [0, 2] and d be the usual metric. Define g : X → X
by

gx =
(1 + x)

2
if 0 ≤ x < 1, gx = 0 if 1 ≤ x < 2, g2 = 2.

Then [12]:

(i) g is not orbitally continuous. Since gn0 → 1 and g(gn0) → 1 6= g1.

(ii) g is weakly orbitally continuous. If we take x = 2 then gn2 → 2 and
g(gn2) → 2 = g2.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 437



R. K. Bisht and V. Rakočević

(iii) g is not k−continuous. If we consider the sequence {gn0}, then for any
positive integer k, we have gk−1(gn0) → 1 and gk(gn0) → 1 6= g1.

Example 1.4. Let X = [0, +∞) and d be the usual metric. Define g : X → X
by

gx = 1 if 0 ≤ x ≤ 1, gx =
x

5
if x > 1.

Then g is orbitally continuous. Let k ≥ 1 be any integer. Consider the sequence
{xn} given by xn = 5

k−1 + 1
n
. Then gk−1xn = 1+

1
n5k−1

, gkxn =
1
5
+ 1

n5k
. This

implies gk−1xn → 1, g
kxn →

1
5
6= g1 as n → +∞. Hence g is not k−continuous.

The above examples show that orbital continuity implies weak orbital con-
tinuity but the converse need not be true. Also, every k-continuous mapping
is orbitally continuous, but the converse is not true.

The question of continuity of contractive definitions at their fixed point in
metric space was studied by Rhoades [20] (see also, Hicks and Rhoades [6]). All
the contractive definitions studied by them forced the mappings to be contin-
uous at the fixed point. Rhoades [20] also listed the question of the existence
of a contractive condition that intromits discontinuity at the fixed point as an
open problem. Pant [15] gave the first affirmative answer to this problem in
the setting of metric space. Various other distinct answers to this problem and
their possible applications to neural networks having discontinuities in activa-
tion functions can be found in Bisht and Pant [1], Bisht and Rakočević [3],

Pant and Pant [11], Pant et al. [12, 16, 17, 18], Taş and Özgür [27].
Bisht and Rakočević [4] presented some new solutions to Rhoades’ open

problem on the existence of contractive mappings that admit discontinuity at
the fixed point. This was done via new fixed point theorems for a generalized
class of Meir-Keeler type mappings which were proved by the authors. Rhoade’s
question was related, in part, to the important problem of characterizing metric
completeness in terms of fixed point results; in this direction solutions to that
problem were deduced. In 2020. Romaguera [21] introduced and studied the
notion of w-distance for fuzzy metric spaces and he obtained a characterization
of complete fuzzy metric spaces via a suitable fixed point theorem.

In this paper, we prove a Meir-Keeler type fixed point theorem for a family
of mappings in Menger PM- space. A probabilistic fixed point theorem satis-
fying (ǫ − δ) type non-expansive mappings is also established. We assume the
notions of weak continuity which may imply discontinuity at the fixed point
but characterize completeness of the space.

2. Preliminaries

We start with some standard definitions and notations of a probabilistic
metric space.

Let D+ be the set of all distribution functions F : R → [0, 1] such that F is a
non-decreasing, left-continuous mapping satisfying F(0) = 0 and supx∈R F(x) =

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 438



On a probabilistic version of Meir-Keeler type fixed point theorem

1. The space D+ is partially ordered by the usual point-wise ordering of func-
tions, i.e., F ≤ G if and only if F(t) ≤ G(t) for all t ∈ R. The maximal element
for D+ in this order is the distribution function given by

ε0(t) =

{

0, t ≤ 0,
1, t > 0.

Definition 2.1 ([23]). A binary operation T : [0, 1] × [0, 1] → [0, 1] is a con-
tinuous t-norm if T satisfies the following conditions:

(a) T is commutative and associative;
(b) T is continuous;
(c) T (a, 1) = a for all a ∈ [0, 1];
(d) T (a, b) ≤ T (c, d), whenever a ≤ c and b ≤ d, and a, b, c, d ∈ [0, 1].

Some of the simple examples of t-norm are T (a, b) = max{a+b−1, 0}, T (a, b) =
min{a, b}, T (a, b) = ab and

T (a, b) =

{

ab
a+b−ab

, ab 6= 0,

0, ab = 0.

The t-norms are defined recursively by T 1 = T and

T n(x1, . . . , xn+1) = T (T
n−1(x1, . . . , xn), xn+1),

for n ≥ 2 and xi ∈ [0, 1] for all i ∈ {1, . . . , n + 1}.

Definition 2.2. A Menger probabilistic metric space (briefly, Menger PM-
space) is a triple (X, F, T ) where X is a non-void set, T is a continuous t-
norm, and F is a mapping from X × X into D+ such that, if Fx,y denotes the
value of F at the pair (x, y), then the following conditions hold:

(PM1) Fx,y(t) = ε0(t) if and only if x = y;
(PM2) Fx,y(t) = Fy,x(t);
(PM3) Fx,z(t + s) ≥ T (Fx,y(t), Fy,z(s)) for all x, y, z ∈ X and s, t ≥ 0.

Remark 2.3 ([24]). Every metric space is a PM-space. Let (X, d) be a metric
space and T (a, b) = min{a, b} is a continuous t-norm. Define Fx,y(t) = ε0(t −
d(x, y)) for all x, y ∈ X and t > 0. The triple (X, F, T ) is a PM-space induced
by the metric d.

Definition 2.4. Let (X, F, T ) be a Menger PM-space.

(1) A sequence {xn}n=1,2,... in X is said to be convergent to x in X if,
for every ε > 0 and λ > 0 there exists positive integer N such that
Fxn,x(ε) > 1 − λ whenever n ≥ N.

(2) A sequence {xn}n=1,2,... in X is called Cauchy sequence if, for every
ε > 0 and λ > 0 there exists positive integer N such that Fxn,xm(ε) >
1 − λ whenever n, m ≥ N.

(3) A Menger PM-space is said to be complete if every Cauchy sequence
in X is convergent to a point in X.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 439



R. K. Bisht and V. Rakočević

The following lemma was given in [22, 23].

Lemma 2.5 ([23]). Let (X, F, T ) be a Menger PM-space. Then the function
F is lower semi-continuous for every fixed t > 0, i.e., for every fixed t > 0 and
every two convergent sequences {xn}, {yn} ⊆ X such that xn → x, yn → y it
follows that

lim inf
n→+∞

Fxn,yn(t) = Fx,y(t).

3. Main results

3.1. Fixed points of a family of Meir-Keeler type mappings in Menger
PM-space. The Meir-Keeler type contractive condition employed in the next
theorem for a family of self-mappings ensures the convergence of sequence of
iterates as well as the existence of fixed points under some weaker notion of
continuity assumption.

Theorem 3.1. Let (X, F, T ) be a complete Menger PM-space, and let {fj :
0 ≤ j ≤ 1} be a family of self-mappings of X such that for any given fj the
following conditions are satisfied:

(i) Ffj x,fjy(t) ≥ min
{

Fx,fjx(t), Fy,fjy(t)
}

for all x, y ∈ X;
(ii) given ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that x, y ∈ X with

ǫ − δ ≤ min
{

Fx,fjx(t), Fy,fjy(t)
}

< ǫ implies Ffj x,fjy(t) > ǫ.

If fj is weakly orbitally continuous, then fj has a unique fixed point, say z, and
lim

n→+∞
fnj x0 = z for each x in X. Moreover, if every pair of mappings (fr, fs)

satisfies the condition

(iii) Ffj x,fsy(t) ≥ min
{

Fx,fjx(t), Fy,fsy(t)
}

;

then the mappings {fj} have a unique common fixed point which is also the
unique fixed point of each fr.

Proof. Consider any mapping fj. By virtue of (ii), it is obvious that fj satisfies
the following condition:

(3.1) Ffj x,fjy(t) > min
{

Fx,fjx(t), Fy,fjy(t)
}

.

Let x0 be any point in X. Define a sequence {xn} in X recursively by xn =
fjxn−1, n = 1, 2, . . .. If xp = xp+1 for some p ∈ N, then xp is a fixed point of
fj. Suppose xn 6= xn+1 for all n ≥ 0. Then using (3.1) we have

Fxn,xn+1(t) = Ffj xn−1,fjxn(t) > min
{

Fxn−1,fjxn−1(t), Fxn,fjxn(t)
}

= min
{

Fxn−1,xn(t), Fxn,xn+1(t)
}

= Fxn−1,xn(t).

Thus {Fxn,xn+1(t)} is a strictly increasing sequence of positive real numbers in
[0, 1] and, hence, tends to a limit r ≤ 1. Suppose r < 1. Then there exists a
positive integer N with n ≥ N such that

(3.2) r − δ(r) < Fxn,xn+1(t) < r.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 440



On a probabilistic version of Meir-Keeler type fixed point theorem

This further implies

r − δ(r) < min
{

Fxn,xn+1(t), Fxn+1,xn+2(t)
}

< r,

that is,

r − δ(r) < min
{

Fxn,fjxn(t), Fxn+1,fjxn+1(t)
}

< r.

By virtue of (ii), this yields Ffj xn,fjxn+1(t) = Fxn+1,xn+2(t) > r. This contra-
dicts (3.2). Hence lim inf

n→+∞
Fxn,xn+1(t) = 1. Further, if q is any positive integer

then for each t > 0, we have

Fxn,xn+q(t) = Ffj xn−1,fjxn+q−1(t) >

> min
{

Fxn−1,fjxn−1(t), Fxn+q−1,fjxn+q−1(t)
}

= min
{

Fxn−1,xn(t), Fxn+q−1,xn+q(t)
}

.

Since lim inf
n→+∞

Fxn,xn+1(t) = 1, making limit as n → +∞, the above inequality

yields

lim inf
n→+∞

Fxn,xn+q(t) = 1.

Therefore, {xn} is a Cauchy sequence. Since X is complete, there exists a
point z in X such that lim

n→+∞
xn = lim

n→+∞
fnj x0 = z. Moreover, if y0 is any

other point in X and yn = fjyn−1 = f
n
j y0, then (3.1) yields

Fxn,yn(t) = Ffj xn−1,fjyn−1(t) > min
{

Fxn−1,fjxn−1(t), Fyn−1,fyn−1(t)
}

= min
{

Fxn−1,xn(t), Fyn−1,yn(t)
}

.

Letting n → +∞, we get lim inf
n→+∞

Fz,yn(t) = 1 for each t > 0. Therefore,

lim
n→+∞

yn = lim
n→+∞

fnj y0 = z. Suppose that fj is weakly orbitally continuous.

Since fnj x0 → z for each x0, by virtue of weak orbital continuity of fj we get,

fnj y0 → z and f
n+1
j y0 → fjz for some y0 ∈ X. This implies that z = fjz

since fn+1j y0 → z. Therefore z is a fixed point of fj. Uniqueness of the fixed

point follows from (i). Moreover, if v and w are the fixed points of fj and fs
respectively, then by (iii) we have

Fv,w(t) = Ffj v,fsw(t) ≥ min
{

Fv,fj v(t), Fw,fsw(t)
}

.

In view of lim inf
n→+∞

Fv,w(t) = 1 for each t > 0, we get v = w and each mapping

{fj} has a unique fixed point which is also the unique common fixed point of
the family of mappings. �

The following result is an easy consequence of Theorem 3.1:

Corollary 3.2. Let (X, F, T ) be a complete Menger PM-space, and let {fj :
0 ≤ j ≤ 1} be a family of self-mappings of X such that for any given fj
satisfying conditions (i)-(ii) of Theorem 3.1. If fj is either k−continuous or
fkj is continuous for some positive integer k ≥ 1 or fj is orbitally continuous,
then fj has a unique fixed point Moreover, if every pair of mappings (fr, fs)

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 441



R. K. Bisht and V. Rakočević

satisfies the condition (iii) of Theorem 3.1, then the mappings {fj} have a
unique common fixed point which is also the unique fixed point of each fr.

The triple (X, F, Tmin) is a complete Menger PM-space, for X ⊆ R (see
Remark 2.3). The following example illustrates Theorem 3.1.

Example 3.3. Let X = [0, 2] and d be the usual metric. For each 0 ≤ j ≤ 1},
we define fj : X 7→ X by

fjx =

{

1, if 0 ≤ x ≤ 1,

j(x − 1), if 1 < x ≤ 2.

Then the mappings fj satisfy all the conditions of Theorem 3.1 and have a
unique common fixed point x = 1 which is also the unique fixed point of each
mapping. The mapping fj is discontinuous at the fixed point. The mapping fj
satisfies condition (ii) with δ(ǫ) = 1 − ǫ, if ǫ < 1, and δ(ǫ) = ǫ, for ǫ ≥ 1. It is
also easy to see that the mapping fj is orbitally continuous and, hence, weak
orbitally continuous [14].

Taking fj = g in Theorm 3.1, we get the following result as a corollary which
is a probabilistic version of Theorem 2.1 of Pant et al. [12]:

Theorem 3.4. Let (X, F, T ) be a complete Menger PM-space, and let g be a
self-mapping of X such that Fgx,gy(t) ≥ min

{

Fx,gx(t), Fy,gy(t)
}

for all x, y ∈
X;

(iv) given ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that x, y ∈ X with

ǫ − δ ≤ min
{

Fx,gx(t), Fy,gy(t)
}

< ǫ implies Fgx,gy(t) > ǫ.

Then

(a) g possesses a unique fixed point if and only if g is weakly orbitally
continuous.

(b) g possesses a unique fixed point provided g is either orbitally continuous
or k−continuous or gk is continuous for some positive integer k ≥ 1.

(c) g possesses a unique fixed point provided g is either x0-orbitally con-
tinuous or almost orbitally continuous.

In the next result, we show that Theorem 3.4 characterizes metric complete-
ness of X. Various workers have proved fixed point theorems that characterize
metric completeness [4, 17, 19, 25, 26]. In the next theorem, we show that
completeness of the space is equivalent to fixed point property for a large class
of mappings including both continuous and discontinuous mappings. In what
follows we use the notation a ≫ b (or a ≪ b) to show that the positive number
a is much greater (smaller) than the positive number b.

Theorem 3.5. Let (X, F, T ) be a Menger PM-space. If every k−continuous
or almost orbitally continuous self-mapping of X satisfying the condition (iv)
of Theorem 3.4 has a fixed point, then X is complete.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 442



On a probabilistic version of Meir-Keeler type fixed point theorem

Proof. Suppose that every k−continuous self-mapping of X satisfying condition
(iv) of Theorem 3.4 possesses a fixed point. We will prove that X is complete.
If possible, suppose X is not complete. Then there exists a Cauchy sequence
in X, say M = {u1, u2, u3, . . .}, consisting of distinct points which does not
converge. Let x ∈ X be given. Then, since x is not a limit point of the Cauchy
sequence M, there exists a least positive integer N(x) such that x 6= uN(x) and
for each m ≥ N(x) and t > 0 we have

(3.3) 1 − Fx,uN(x)(t) ≫ 1 − FuN(x),um(t).

Consider a mapping g : X 7→ X by g(x) = uN(x). Then, g(x) 6= x for each x
and, using (3.3), for any x, y in X and t > 0 we get

1 − Fgx,gy(t) = 1 − FuN(x),uN(y)(t) ≪ 1 − Fx,uN(x)(t) = 1 − Fx,gx(t)

if N(x) ≤ N(y), or

1 − Fgx,gy(t) = 1 − FuN(x),uN(y)(t) ≪ 1 − Fy,uN(y)(t) = 1 − Fy,gy(t)

if N(x) > N(y).
This implies that

(3.4) Fgx,gy(t) > min
{

Fx,gx(t), Fy,gy(t)
}

.

In other words, given ǫ > 0 we can select δ(ǫ) = ǫ such that

(3.5) ǫ − δ ≤ min
{

Fx,gx(t), Fy,gy(t)
}

< ǫ implies Fgx,gy(t) > ǫ.

It is clear from (3.4) and (3.5) that the mapping g satisfies condition (iv)
of Theorem 3.4. Moreover, g is a fixed point free mapping whose range is
contained in the non-convergent Cauchy sequence M = {un}n∈N. Hence, there
exists no sequence {xn}n∈N in X for which {gxn}n∈N converges, that is, there
exists no sequence {xn}n∈N in X for which the condition gxn → t implies
g2xn → gt is violated. Therefore, g is a 2-continuous mapping. In a similar
manner it follows that g is almost orbitally continuous. Thus, we have a self-
mapping g of X which satisfies condition (iv) of Theorems 3.4 but does not
possess a fixed point. This contradicts the hypothesis of the theorem. Hence
X is complete. �

We now give a weaker version of Theorem 3.4 which extends Theorem 3.2
of [17].

Theorem 3.6. Let (X, F, T ) be a complete Menger PM-space, and let g be a
self-mapping of X such that

(v) Fgx,gy(t) > min
{

Fx,gx(t), Fy,gy(t)
}

for all x, y ∈ X;
(vi) given ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that x, y ∈ X with

ǫ − δ < min
{

Fx,gx(t), Fy,gy(t)
}

< ǫ implies Fgx,gy(t) ≥ ǫ.

Then g possesses a unique fixed point if g is either weakly orbitally continuous
or x0-orbitally continuous or almost orbitally continuous.

Proof. The proof follows on the similar lines as the proof of Theorem 3.5. �

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 443



R. K. Bisht and V. Rakočević

3.2. Fixed points of a family of (ǫ − δ) non-expansive mappings in
Menger PM-space. We now prove a fixed point theorem for a family of
(ǫ − δ) non-expansive mappings in Menger PM-space.

Theorem 3.7. Let (X, F, T ) be a complete Menger PM-space, and let {fj :
0 ≤ j ≤ 1} be a family of self-mappings of X such that for any given fj the
following conditions are satisfied:

(i’) Ffj x,fjy(t) ≥ min
{

Fx,fjx(t), Fy,fjy(t)
}

for all x, y ∈ X;
(ii’) given ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that x, y ∈ X with

ǫ − δ < min
{

Fx,fjx(t), Fy,fjy(t)
}

< ǫ implies Ffj x,fjy(t) ≤ ǫ;

If fj is continuous, then fj has a unique fixed point, say z, Moreover, if every
pair of mappings (fr, fs) satisfies the condition

(iii’) Ffj x,fsy(t) ≥ min
{

Fx,fjx(t), Fy,fsy(t)
}

;

then the mappings {fj} have a unique common fixed point which is also the
unique fixed point of each fr.

Proof. Let x0 be any point in X. Define a sequence {xn} in X recursively by
xn = fjxn−1, n = 1, 2, . . .. Then following the lines of Theorem 3.1, it can
be shown that {xn} is a Cauchy sequence. Continuity of fj now implies that
fjz = z and z is a fixed point of fj. Rest of the proof follows from Theorem
3.1. �

Taking fj = g in Theorm 3.7, we get the following result as a corollary:

Theorem 3.8. Let (X, F, T ) be a complete Menger PM-space, and let g be a
continuous self-mapping of X such that

(iv’) Fgx,gy(t) ≥ min
{

Fx,gx(t), Fy,gy(t)
}

for all x, y ∈ X;
(v’) given ǫ ∈ (0, 1) there exists δ ∈ (0, ǫ] such that x, y ∈ X with

ǫ − δ < min
{

Fx,gx(t), Fy,gy(t)
}

< ǫ implies Fgx,gy(t) ≤ ǫ.

Then g possesses a unique fixed point.

The triple (X, F, Tmin) is a complete Menger PM-space, for X ⊆ R (Remark
2.3). The following example [13] illustrates Theorem 3.8.

Example 3.9. Let X = [−1, 1] and d be the usual metric. Define g : X 7→ X
by

gx = −|x|x, for each x ∈ X.

Then the mapping g satisfies all the conditions of Theorem 3.8 and has a unique
fixed point x = 0. Also, g possesses two periodic points x = 1 and x = −1.
The mapping g satisfies condition (v′) with δ(ǫ) = (

√

(ǫ/2)) − (ǫ/2), if ǫ < 2,
and δ(2) = 2.

Remark 3.10. It is pertinent to mention here that uniqueness of the fixed point
in Theorem 3.8 is because of the particular form (iv′). If we change (iv′) by
the following

Fgx,gy(t) ≥ min
{

Fx,y(t), Fx,gx(t), Fy,gy(t)
}

for all x, y ∈ X,

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 444



On a probabilistic version of Meir-Keeler type fixed point theorem

then the fixed point need not be unique.

Remark 3.11. Theorem 3.1 provides a new answer to the once open question
(see Rhoades [20], p. 242) on the existence of contractive mappings which
admit discontinuity at the fixed point in the setting of Menger PM-space.

References

[1] R. K. Bisht and R. P. Pant, A remark on discontinuity at fixed point, J. Math. Anal.
Appl. 445 (2017), 1239–1242.

[2] R. K. Bisht, A probabilistic Meir-Keeler type fixed point theorem which characterizes
metric completeness, Carpathain J. Math. 36, no. 2 (2020), 215–222.

[3] R. K. Bisht and V. Rakočević, Generalized Meir-Keeler type contractions and disconti-
nuity at fixed point, Fixed Point Theory 19, no. 1 (2018), 57–64.

[4] R. K. Bisht and V. Rakočević, Discontinuity at fixed point and metric completeness,
Appl. Gen. Topol. 21, no. 2 (2020), 349–362.

[5] Lj. B. Ćirić, On contraction type mappings, Math. Balkanica 1 (1971), 52–57.
[6] T. Hicks and B. E. Rhoades, Fixed points and continuity for multivalued mappings,

International J. Math. Math. Sci. 15 (1992), 15–30.
[7] D. S. Jaggi, Fixed point theorems for orbitally continuous functions, Indian J. Math.

19, no. 2 (1977), 113–119.
[8] G. F. Jungck, Generalizations of continuity in the context of proper orbits and fixed

pont theory, Topol. Proc. 37 (2011), 1–15.
[9] A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28

(1969), 326–329.
[10] K. Menger, Statistical metric, Proc. Nat. Acad. Sci. USA 28 (1942), 535–537.
[11] A. Pant and R. P. Pant, Fixed points and continuity of contractive maps, Filomat 31,

no. 11 (2017), 3501–3506.
[12] A. Pant, R. P. Pant and M. C. Joshi, Caristi type and Meir-Keeler type fixed point

theorems, Filomat 33, no. 12 (2019), 3711–3721.
[13] A. Pant and R. P. Pant, Fixed points and continuity of contractive maps, Filomat 31,

no. 11 (2017), 3501–3506.
[14] A. Pant, R. P. Pant and W. Sintunavarat, Analytical Meir-Keeler type contraction

mappings and equivalent characterizations, RACSAM 37 (2021), 115.

[15] R. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl. 240 (1999), 284–289.

[16] R. P. Pant, N. Y. Özgür and N. Taş, On discontinuity problem at fixed point, Bull.
Malays. Math. Sci. Soc. 43, no. 1 (2020), 499–517.

[17] R. P. Pant, A. Pant, R. M. Nikolić and S. N. Ješić, A characterization of completeness
of Menger PM-spaces, J. Fixed Point Theory Appl. 21, (2019) 90.

[18] R. P. Pant, N. Y. Özgür and N. Taş, Discontinuity at fixed points with applications,
Bulletin of the Belgian Mathematical Society-Simon Stevin 25, no. 4 (2019), 571–589.

[19] O. Popescu, A new type of contractions that characterize metric completeness,
Carpathian J. Math. 31, no. 3 (2015), 381–387.

[20] B. E. Rhoades, Contractive definitions and continuity, Contemporary Mathematics 72
(1988), 233–245.

[21] S. Romaguera, w-distances on fuzzy metric spaces and fixed points, Mathematics 8, no.
11 (2020), 1909.

[22] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 415–417.
[23] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, New York, El-

sevier 1983.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 445



R. K. Bisht and V. Rakočević

[24] V. M. Sehgal and A. T. Bharucha-Reid, Fixed points of contraction mappings in PM-
spaces, Math. System Theory 6 (1972), 97–102.

[25] P. V. Subrahmanyam, Completeness and fixed points, Monatsh. Math. 80 (1975), 325–
330.

[26] T. Suzuki, A generalized Banach contraction principle that characterizes metric com-
pleteness, Proc. Amer. Math. Soc. 136, no. 5 (2008), 1861–1869.

[27] N. Taş and N. Y. Özgür, A new contribution to discontinuity at fixed point, Fixed Point
Theory 20, no. 2 (2019), 715–728.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 446