@
Appl. Gen. Topol. 21, no. 2 (2020), 349-362

doi:10.4995/agt.2020.13943

c© AGT, UPV, 2020

Discontinuity at fixed point and metric

completeness

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

In this paper, we prove some new fixed point theorems for a generalized
class of Meir-Keeler type mappings, which give some new solutions
to the Rhoades open problem regarding the existence of contractive
mappings that admit discontinuity at the fixed point. In addition to
it, we prove that our theorems characterize completeness of the metric
space as well as Cantor’s intersection property.

2010 MSC: 47H09; 47H10.

Keywords: fixed point; completeness; discontinuity; Cantor’s intersection
property.

1. Introduction and Preliminaries

If f is a self-mapping on a complete metric space (X, d) satisfying a contrac-
tive condition, then the contractive mapping, in general, ascertains that: For
a point x ∈ X, the sequence of iterates is a Cauchy sequence, {fnx} → z, and
z is the fixed point of f. However, there exists Meir-Keeler type contractive
mapping which ensures the existence of sequence of iterates which is Cauchy
and {fnx} → z, but z may not be a fixed point of f. Meanwhile, a more com-
plete study (data dependence, well-posedness, Ulam-Hyers stability, Ostrowski

Received 30 June 2020 – Accepted 30 July 2020

http://dx.doi.org/10.4995/agt.2020.13943


R. K. Bisht and V. Rakočević

property) was recently proposed in [34].

Fixed point theorems studied by Kannan are considered as the genesis of the
question on continuity of contractive mappings at the fixed point (see [15, 16]).
Subsequently, the question whether there exists a contractive definition which
is strong enough to generate a fixed point but which does not force the mapping
to be continuous at the fixed point was ingeminated as an open problem by
Rhoades[33]. In 1999, Pant [27] proved two fixed point theorems in which
the considered mappings were discontinuous at the fixed points, hence gave
affirmative solutions to the Rhoades problem for both the single and a pair of
self-mappings. Some new solutions to this problem with applications to neural
networks have been reported in [2, 3, 4, 5, 6, 12, 23, 24, 25, 26, 29, 30, 31, 32, 37,
39]. Fixed point theorems for discontinuous mappings have found a variety of
applications, e.g., neural networks are generally used in character recognition,
image compression, stock market prediction and to solve non-negative sparse
approximation problems ([10, 11, 20, 21, 22, 38]).
Here, we list various classes of Meir-Keeler (M-K) type conditions which ensure
convergence of the successive approximations but the limiting point may or may
not be a fixed point for the given mappings. For i ∈ {1, ..., 11}, we consider:

[M-K] for a given ! > 0 there exists a δ(!) > 0 such that, for any x, y ∈ X,

! ≤ mi(x, y) < ! + δ implies d(fx, fy) < !.

It is obvious that [M-K] satisfies the following contractive condition:

d(Tx, Ty) < mi(x, y), for any x, y ∈ X with mi(x, y) > 0, where

m1(x, y) = d(x, y), (Meir-Keeler [18])

m2(x, y) =
d(x, fx) + d(y, fy)

2
, (Kannan [15])

m3(x, y) = max{d(x, fx), d(y, fy)}, (Bianchini [1])

m4(x, y) =
d(x, fy) + d(y, fx)

2
, (Chatterjea [7])

m5(x, y) = max

!
ad(x, fx) + (1 − a)d(y, fy), (1 − a)d(x, fx) + ad(y, fy)

"
,

0 ≤ a < 1, (Pant [28])
m6(x, y) = max{d(x, y), d(x, fx), d(y, fy)}, (Maiti and Pal [17])

m7(x, y) = max

!
d(x, y), d(x, fx), d(y, fy), d(x, fy), d(y, fx)

"
, (Ćirić [9])

m8(x, y) = max

!
d(x, y), d(x, fx), d(y, fy),

d(x, fy) + d(y, fx)

2

"
, (Jachymski[14])

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 350



Discontinuity at fixed point and metric completeness

m9(x, y) = max

!
d(x, y),

k[d(x, fx) + d(y, fy)]

2
,
k[d(x, fy) + d(y, fx)]

2

"
, 0 ≤ k < 1,

m10(x, y) = max

!
d(x, y), ad(x, fx) + (1 − a)d(y, fy), (1 − a)d(x, fx) + ad(y, fy),

b[d(x, fy) + d(y, fx)]

2

"
, 0 ≤ a < 1 and 0 ≤ b < 1,

( Bisht and Rakočević [5])

m11(x, y) = max

!
d(x, y), ad(x, fx) + (1 − a)d(y, fy), (1 − a)d(x, fx) + ad(y, fy),

[d(x, fy) + d(y, fx)]

2

"
, 0 ≤ a < 1.

Now, we recall some notions of weaker forms of continuity conditions.

Definition 1.1 ([8, 9]). If f is a self-mapping of a metric space (X, d), then
the set O (x, f) = {fnx : n = 1, 2, ...} is called the orbit of f at x and f is called
orbitally continuous if u = limi f

mix implies fu = limi ff
mix.

Definition 1.2 ([25]). A self-mapping f of a metric space X is called k-
continuous, k = 1, 2, 3, ..., if fkxn → ft whenever, {xn} is a sequence in X
such that fk−1xn → t.

Definition 1.3 ([26]). A self-mapping f of a metric space (X, d) is called
weakly orbitally continuous if the set {y ∈ X : limifmiy = u =⇒ limiffmiy =
fu} is nonempty, whenever the set {x ∈ X : limifmix = u} is nonempty.

Remark 1.4. The following observations are now well-established (see [25, 26]):
(i) Continuity implies orbital continuity but not conversely.
(ii) 1-continuity is equivalent to continuity and
continuity =⇒ 2 − continuity =⇒ 3 − continuity =⇒ . . . , but not con-
versely.
(iii) Orbital continuity implies weak orbital continuity but the converse need
not be true.
(iv) k-continuous mappings are orbitally continuous but the converse need not
be true.

The notion of f-orbitally lower semi-continuity was given by Hicks and
Rhoades [13].

Definition 1.5. Let (X, d) be a metric space and f : X → X. A mapping
g : X → R is said to be f-orbitally lower semi-continuous at a point z ∈ X
if {xn} is a sequence in O(x, f) for some x ∈ X, limn→∞ xn = z implies
lim infn→∞ g(xn) ≥ g(z).

In [19], the author has shown that the f-orbital lower semi-continuity of
x → d(x, fx) is weaker than orbital continuity and k-continuity of f. The
following example illustrates this fact:

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 351



R. K. Bisht and V. Rakočević

Example 1.6. Let X = {0, 1}∪
!

1

3n
: n = 1, 2, · · ·

"
∪
!
1 +

1

3n
: n = 1, 2, · · ·

"

and d be the usual metric. Define f : X → X by

fx =

#
$$$$$$%

$$$$$$&

0 if x = 0
4

3
if x = 1

1 +
1

3n+1
if x =

1

3n
, n = 1, 2, · · ·

1

3n
if x = 1 +

1

3n
, n = 1, 2, · · ·

Then

O(1, f) =

!
1, 1 +

1

3
,
1

3
, 1 +

1

32
,
1

32
, · · · , 1 +

1

3n
,
1

3n
, · · ·

"
.

Let {xn} be a sequence in O(1, f) with xn =
1

3n
for n ≥ 1. Then, xn → 0

as n → ∞. However, fxn = 1 + 13n+1 → 1 ∕= f0. Thus, f is not orbitally

continuous. Now, let {zn} be a sequence in X with zn =
1

3n
for n ≥ 1. Then

we have fzn = 1 +
1

3n+1
and f2zn =

1

3n+1
. Since limn→∞ fzn = 1 and

limn→∞ f
2zn = 0 ∕= 4/3 = f1, the mapping f is not 2-continuous.

Also,

g(x) = d(x, fx) =

#
$$$$$$%

$$$$$$&

0 if x = 0
1

3
if x = 1

1 −
2

3n+1
if x =

1

3n
, n = 1, 2, · · ·

1 if x = 1 +
1

3n
, n = 1, 2, · · ·

Let x ∈ X and {xn} ⊂ O(x, f). If {xn} converges, then it converges to 0 or 1.
If limn→∞ xn = 0, then

lim inf
n→∞

g(xn) =

#
%

&

0
or
1

≥ g(0) = 0

. If limn→∞ xn = 1, then lim infn→∞ g(xn) = 1 > 1/3 = g(1). Thus, x →
d(x, fx) is f-orbitally lower semi-continuous.

In this paper, we prove some new fixed point theorems for a generalized
class of Meir-Keeler type mappings, which give some new solutions to Rhoades
open problem regarding the existence of contractive mappings that admit dis-
continuity at the fixed point. Further, we prove that our theorems characterize
completeness of the metric space as well as Cantor’s intersection property.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 352



Discontinuity at fixed point and metric completeness

2. Sequence of successive approximations

Here we show that there exists a large class of contractive mappings which
ensure that the sequences of iterates are Cauchy and {fnx} → z, but z may
not be a fixed point of f. We begin with the following result:

Proposition 2.1. Let f be a self-mapping of a complete metric space (X, d)
such that for all x, y in X we have

(i) Given ε > 0 there exists a δ > 0 such that ε ≤ m10(x, y) < ε + δ ⇒
d(fx, fy) < ε. Then given x in X, the sequence of iterates {fnx} is a Cauchy
sequence and lim

n→∞
fnx = z for some z in X.

Proof. Condition (i) implies that if m10(x, y) > 0, then

(2.1) d(fx, fy) < m10(x, y).

Let x0 be any point in X. Define a sequence {xn} in X recursively by xn+1 =
fxn = f

nx0 and γn = d(xn, xn+1) for each n ∈ N ∪ {0}.
We assume that xn ∕= xn+1 for each n. Using (2.1), we get

γn = d(xn, xn+1) = d(fxn−1, fxn)

< max

'
d(xn−1, xn), ad(xn−1, fxn−1) + (1 − a) d(xn, fxn),

(1 − a)d(xn−1, fxn−1) + ad(xn, fxn),
b[d(xn−1, fxn) + d(xn, fxn−1)]

2

(

= max

'
γn−1, aγn−1 + (1 − a) γn,

(1 − a)γn−1 + aγn,
b[γn−1 + γn]

2

(

= γn−1.

This shows that {γn} is a strictly decreasing sequence of positive real num-
bers and, hence, tends to a limit γ ≥ 0. Suppose γ > 0. Then there exists a
positive integer N such that

(2.2) n ≥ N ⇒ γ < γn < γ + δ(γ).
By virtue of (i) the above inequality yields d(fxn, fxn+1) = d(xn+1, xn+2) <

r. This contradicts with (2.2). Hence, γn → 0 as n → ∞.
We now prove that {xn} is a Cauchy sequence. Arguing by contradiction,

suppose that {xn} is not a Cauchy sequence. Then there exist an ε > 0 and a
sub-sequence {xni} of {xn} such that
(2.3) d(xni, xni+1) > 2ε.

Select δ in (i) such a way that 0 < δ ≤ ε. Since lim
n→∞

γn = 0, there exists a

positive integer N such that

(2.4) γn <
δ

6
,

whenever n ≥ N. Let ni > N. Then there exist integers mi satisfying ni <
mi < ni+1 such that

(2.5) d(xni, xmi) ≥ ε +
δ

3
.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 353



R. K. Bisht and V. Rakočević

If not, then using (2.4) and (2.5), we have

d(xni, xni+1) ≤ d(xni, xni+1−1) + d(xni+1−1, xni+1)

< ε +
δ

3
+

δ

6
= ε +

δ

2
< 2ε,

a contradiction with (2.3). Let m∗i be the smallest integer such that ni < m
∗
i <

ni+1 and

(2.6) d(xni, xm∗i ) ≥ ε +
δ

3
.

Then d(xni, xm∗i −1) < ε +
δ
3
. In view of (2.1) and (2.2), we get

ε < ε +
δ

3
≤ d(xni, xm∗i ) = d(fxni−1, fxm∗i −1) < m10(xni−1, xm∗i −1)

= max

#
$%

$&

d
)
xni−1, xm∗i −1

*
, ad(xni−1, fxni−1) + (1 − a)d(xm∗i −1, fxm∗i −1),

(1 − a)d(xni−1, fxni−1) + ad(xm∗i −1, fxm∗i −1),
b[d(xni−1, fxm∗i −1) + d(xm

∗
i
−1, fxni−1)]

2

+
$,

$-

≤ max
.
d(xni−1, xm∗i −1), d(xni−1, xni), d(xm

∗
i
−1, xni) + d(xni−1, xni)

/

≤ d(xni, xm∗i −1) + d(xni−1, xni)

< ε +
δ

3
+

δ

6
= ε +

δ

2
,

that is,

ε +
δ

3
≤ m(xni−1, xm∗i −1) < ε +

δ

2
.

By virtue of (i), the last inequality yields d(fxni−1, fxm∗i −1) < ε, i.e.,
d(xni, xm∗i ) < ε. This contradicts (2.6) and, hence, {xn} is a Cauchy sequence.
Since X is complete, there exists a point z ∈ X such that lim

n→∞
xn = lim

n→∞
fnx =

z. □

We now present two examples which satisfy the above proposition but f is
fixed point free.

Example 2.2. Let X = [1, 2] ∪
!
1 −

1

3n
: n = 0, 1, 2, · · ·

"
and d be the usual

metric. Define f : X → X by

fx =

'
0 if 1 ≤ x ≤ 2.

1 −
1

3n+1
if x = 1 −

1

3n
, n = 0, 1, 2, · · · .

Then

f(X) =

!
1 −

1

3n
: n = 0, 1, 2, · · ·

"

and f is fixed point free. The mapping f satisfies the contractive condition
(i) with

δ(ε) =

!
1
3n

− ε if 1
3n+1

≤ ε < 1
3n

, n = 0, 1, 2, · · ·
ε if ε ≥ 1.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 354



Discontinuity at fixed point and metric completeness

Example 2.3. Let X = [0, 2] equipped with the Euclidean metric d. Define
f : X → X by

fx =

!
1+3x

4
if 0 ≤ x < 1

0 if x ≥ 1 .

Then X is a complete metric space and f satisfies the contractive condition (i)
with

δ(ε) =

#
%

&

ε
3

if 0 ≤ ε ≤ 3
4

1 − ε if 3
4
< ε < 1

ε if ε ≥ 1
but does not posses a fixed point. It is easy to verify that for each x in X, the
sequence of iterates {fnx} is a Cauchy sequence and fnx → 1 (see [29]).

Replacing m10(x, y) given in the condition (i) of Proposition 2.1 by m11(x, y)
and using similar arguments, proof of the following proposition follows easily.

Proposition 2.4. Let (X, d) be a complete metric space and f : X → X be a
self-mapping such that for all x, y ∈ X we have

(i)′ Given ε > 0 there exists a δ > 0 such that

ε ≤ m11(x, y) < ε + δ =⇒ d(fx, fy) < ε.
Then the sequence of iterates {fnx} is a Cauchy sequence for a given x ∈ X

and lim
n→∞

fnx = z for some z ∈ X.

3. Discontinuity at fixed point

We give a new solution to the problem of continuity at the fixed point for
the number m10(x, y).

Theorem 3.1. Let f be a self-mapping of a complete metric space (X, d) such
that

(i) Given ε > 0 there exists a δ = δ(ε) > 0 such that
ε ≤ m10(x, y) < ε + δ ⇒ d(fx, fy) < ε,

for all x, y in X. If x → d(x, fx) is f-orbitally lower semi-continuous, then
f has a unique fixed point z ∈ X and fnx → z as n → ∞. Moreover, f is
continuous at z if and only if lim

x→z
max {d (x, fx) , d (z, fz)} = 0 or, equivalently,

lim
x→z

sup d (fz, fx) = 0.

Proof. Let x be any point in X. Define a sequence {xn} in X recursively by
xn = f

nx, n = 0, 1, 2, 3.... Then following the proof of Proposition 2.1 above,
we get that {xn} is a Cauchy sequence. Since X is complete, there exists a
point z ∈ X such that xn → z.

Since xn → z satisfying d(xn, fxn) = d(fnx, fn+1x) → 0 as n → ∞, by
f-orbital lower semi-continuity of x → d(x, fx), one gets

d(z, fz) ≤ lim inf
n→∞

d(xn, fxn) = 0,

which implies that z = fz, i.e., z is a fixed point of f. Uniqueness of the fixed
point follows easily.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 355



R. K. Bisht and V. Rakočević

Let f be continuous at the fixed point z. For lim
x→z

max {d (x, fx) , d (z, fz)} =
0; let yn → z. Then fyn → fz = z, d (yn, fyn) = 0 and

lim
n→∞

max {d (yn, fyn) , d (z, fz)} = 0.

Conversely, let lim
x→z

max {d (x, fx) , d (z, fz)} = 0. To show that f is contin-
uous at the fixed point z, let yn → z. Then, we have

lim
n→∞

max {d (yn, fyn) , d (z, fz)} = 0.

This implies that lim
n→∞

d (yn, fyn) = 0 and, hence, lim
n→∞

fyn = lim
n→∞

yn = z = fz.

Therefore, f is continuous at the fixed point z. This completes the proof. □

Theorem 3.2. Let f be a self-mapping of a complete metric space (X, d) such
that

(i) Given ε > 0 there exists a δ = δ(ε) > 0 such that
ε ≤ m10(x, y) < ε + δ ⇒ d(fx, fy) < ε,

for all x, y in X. Then f possesses a fixed point if and only if f is weakly
orbitally continuous. Moreover, the fixed point is unique and f is contin-
uous at z if and only if lim

x→z
max {d (x, fx) , d (z, fz)} = 0 or, equivalently,

lim
x→z

sup d (fz, fx) = 0.

Proof. Let x be any point in X. Define a sequence {xn} in X recursively by
xn = f

nx, n = 0, 1, 2, 3.... Then following the proof of Theorem 3.1 above, we
get that {xn} is a Cauchy sequence. Since X is complete, there exists a point
z ∈ X such that xn → z.

Suppose that f is weakly orbitally continuous. Since fnx0 → z for each x0,
by virtue of weak orbital continuity of f we get, fny0 → z and fn+1y0 → fz
for some y0 ∈ X. This implies that z = fz since fn+1y0 → z. Therefore z is a
fixed point of f.

Conversely, suppose that the mapping f has a fixed point, say z. Then
{fnz = z} is a constant sequence such that limn fnz = z and limn fn+1z =
z = fz. Hence, f is weak orbitally continuous. Rest of the the proof follows
from the proof of Theorem 3.1. □

Remark 3.3. The last part of Theorem 3.1 or Theorem 3.2 can alternatively be
stated as: f is discontinuous at z if and only if lim

x→z
max {d (x, fx) , d (z, fz)} ∕= 0

or equivalently, lim
x→z

sup d (fz, fx) > 0.

Remark 3.4. Theorems 3.1 and 3.2 hold true if we replace m10(x, y) given in
(i) by m11(x, y).

The following theorem is a consequence of above theorems.

Theorem 3.5. Let f be a self-mapping of a complete metric space (X, d) such
that

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 356



Discontinuity at fixed point and metric completeness

(i) Given ε > 0 there exists a δ = δ(ε) > 0 such that
ε ≤ m3(x, y) < ε + δ ⇒ d(fx, fy) < ε,

for all x, y in X. Then f possesses a fixed point if and only if f is weakly
orbitally continuous or if x → d(x, fx) is f-orbitally lower semi-continuous,
then f has a fixed point z ∈ X and fnx → z as n → ∞. Moreover, the fixed
point is unique and f is continuous at z if and only if lim

x→z
sup d (fz, fx) = 0.

The following result is an easy consequence of Theorem 3.1.

Corollary 3.6. Let f be a self-mapping of a complete metric space (X, d)
satisfying (i) of Theorem 3.1 for all x, y in X. If f is k-continuous for some
k ≥ 1 or if f is orbitally continuous then f has a unique fixed point, say z.
Moreover, f is continuous at z if and only if lim

x→z
sup d (fz, fx) = 0.

W now give an example to illustrate the above result.

Example 3.7. Let X = [0, 2] and d be the usual metric. Define f : X → X
by

fx =

!
1+x
2

if 0 ≤ x ≤ 1
x−1
2

if 1 < x ≤ 2 .

Then f satisfies all the conditions of Theorem 3.5 and has a unique fixed point
z = 1 at which f is discontinuous [29]. One can compute that

g(x) = d(x, fx) =

!
x−1
2

if 0 ≤ x ≤ 1
1+x
2

if 1 < x ≤ 2 .

The mapping g(x) = x → d(x, fx) is f-orbitally lower semi-continuous and
satisfies the condition (i) with

δ(ε) =

#
%

&

ε if ε ≤ 1
2

1 − ε if 1
2
< ε < 1

ε if ε ≥ 1
.

Here, lim
x→z

max {d (x, fx) , d (z, fz)} does not exist. Also, lim
x→z

sup d (fz, fx) =

1.

Theorem 3.8. Let f be a self-mapping of a complete metric space (X, d) such
that

(ii) Given ε > 0 there exists a δ = δ(ε) > 0 such that
ε ≤ m10(x, y) < ε + δ ⇒ d(fx, fy) < ε,

for all x, y in X. If a = b = 0, then f has a unique fixed point whenever x →
d(x, fx) is f-orbitally lower semi-continuous or f is weakly orbitally continuous.
If a, b > 0, then f possesses a unique fixed point at which f is continuous.

Proof. The proof follows on the same lines of the proof of Theorem 3.1 above.
As seen in Theorem 3.1 above, f need not be continuous at the fixed point if
a = 0 = b. We now show that f is continuous at the fixed point when a, b > 0.
Suppose a > 0 and z is the fixed point of f. Let {xn} be any sequence in X

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 357



R. K. Bisht and V. Rakočević

such that xn → z as n → ∞. Then using (ii), for sufficiently large values of n,
we get

d(z, fxn) = d(fz, fxn)

< max

!
d(z, xn), ad(z, fz) + (1 − a) d(xn, fxn),

(1 − a)d(z, fz) + ad(xn, fxn),
b[d(z, fxn) + d(xn, fz)]

2

"

= max

!
d(z, xn), (1 − a) d(xn, fxn), ad(xn, fxn),

b[d(z, fxn) + d(xn, fz)]

2

"

≤ max
!
!1, !2 + (1 − a) d(z, fxn), !3 + ad(z, fxn),

b[d(z, fxn) + !4]

2

"
,

where !i(i = 1, 2, 3, 4) → 0 as n → ∞. This yields d(z, fxn) < !1, ad(z, fxn) <

!2, (1 − a)d(z, fxn) < !3 or
bd(z, fxn)

2
< !4. On letting n → ∞, we get

limn→∞ fxn = z = fz. Hence f is continuous at the fixed point. □

4. Characterization of metric completeness

In a complete metric space, the following well-known variant of Cantor’s
intersection theorem holds.

Theorem 4.1. Suppose that X is a complete metric space, and Cn is a se-
quence of non-empty closed nested subsets of X whose diameters tend to zero:

limn→∞ diam(Cn) = 0, where diam(Cn) is defined by diam(Cn) = sup{d(x, y)|
x, y ∈ Cn}. Then the intersection of the Cn contains exactly one point:0∞

n=1 Cn = {x} for some x ∈ X.
A converse to this theorem is also true: if X is a metric space with the

property that the intersection of any nested family of non-empty closed subsets
whose diameters tend to zero is non-empty, then X is a complete metric space.

In the next result, we show that Theorem 3.5 characterizes metric complete-
ness of X. However, there is a substantive difference between the next theorem
and similar theorems (e. g., Subrahmanyam [35], Suzuki [36]) giving charac-
terization of completeness in terms of fixed point property for contractive type
mappings. Subrahmanyam [35] and Suzuki [36] have shown that the contrac-
tive condition implies continuity at the fixed point; and completeness of the
metric space X is equivalent to the existence of fixed point. In the next theorem
motivated by ([25, 26]) we prove that completeness of the space is equivalent
to fixed point property for a larger class of mappings including continuous as
well as discontinuous mappings.

Theorem 4.2. If every x → g(x) = d(x, fx) is f-orbitally lower semi-continuous
or weak orbitally continuous self-mapping f of a metric space (X, d) satisfying
the condition (i) of Theorem 3.5 has a fixed point, then X is complete.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 358



Discontinuity at fixed point and metric completeness

Proof. Suppose that every x → d(x, fx) is f-orbitally lower semi-continuous
or weak orbitally continuous self-mapping f of a metric space (X, d) satisfying
the condition (i) of Theorem 3.5 possesses a fixed point. We will prove that X
is complete. Arguing by contradiction, suppose that X is not complete. Then
there exists a Cauchy sequence in X, say M = {un}n∈N having distinct points
which does not converge in X. Let x ∈ X be any arbitrary point. Then, since x
is not a limit point of the Cauchy sequence M, and we have d(x, M − {x}) > 0
and there exists an integer nx ∈ N such that x ∕= unx and for each m ≥ nx

(4.1) d(unx, um) <
1

2
d(x, unx).

Consider a mapping f : X +→ X by f(x) = unx. Then, f(x) ∕= x for each x
and, using (4.1), for any x, y ∈ X we get

d(fx, fy) = d(unx, unx=y) <
1

2
d(x, unx) = d(x, fx), if nx ≤ ny,

or

d(fx, fy) = d(unx, uny ) <
1

2
d(y, uny ) = d(y, fy), if nx > ny.

This implies that

(4.2) d(fx, fy) <
1

2
max{d(x, fx), d(y, fy)}.

Or equivalently, given ! > 0 we can select δ(!) = ! such that

(4.3) ! ≤ max{d(x, fx), d(y, fy)} < ! + δ ⇒ d(fx, fy) < !.
It is clear from (4.2) and (4.3) that the mapping f satisfies condition (i) of The-
orem 3.5. Moreover, f is a fixed point free mapping whose range is contained
in the non-convergent Cauchy sequence M. Hence, there exists no sequence
{xn}n∈N in X for which {fxn}n∈N converges, i.e., there exists no sequence
{xn}n∈N in X for which the condition fk−1xn → t ⇒ fkxn → ft for k > 1 is
violated. Therefore, f is k-continuous mapping. Since xn ∈ O(x, f), it is of
the form xn = f

inx with in ∈ N. Set yn = fin+1−kx when in ≥ k − 1. Then,
fk−1yn = xn → t as n → ∞. By k-continuity of f, we get fxn = fkyn → ft
as n → ∞. Thus, limn→∞ g(xn) = limn→∞ d(xn, fxn) = d(t, ft) = g(t), which
implies that g is f-orbitally lower semi-continuous at t. In a similar manner
it follows that f is weak orbitally continuous. Thus, we have a self-mapping
f of X which satisfies all the conditions of Theorems 3.5 but does not possess
a fixed point. This contradicts the hypothesis of the theorem. Hence X is
complete. □

We now show that Theorem 3.5 characterizes Cantor’s intersection property.

Theorem 4.3. Let (X, d) be a metric space and f a self-mapping of X satis-
fying condition (i) of Theorem 3.5. Suppose X satisfies Cantor’s intersection
property and x → d(x, fx) is f-orbitally lower semi-continuous or f is weakly
orbitally continuous. Then f has a fixed point and f is continuous at z if and
only if limx→z max{d(x, fx), d(z, fz)} = 0.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 359



R. K. Bisht and V. Rakočević

Proof. Let x be any point in X. Define a sequence {xn} in X recursively by
xn = f

nx, n = 0, 1, 2, 3.... Then following the proof of Proposition 2.1 above,
we get that {xn} is a Cauchy sequence. Define a sequence {Tn : n = 1, 2, 3, ...}
of nonempty subsets of X by Tn = {xi : i ≥ n}. Let Cn denote the closure of
Tn ∈ X. Then for each n it is obvious that Cn is a nonempty closed subset of X,
Cn+1 ⊆ Cn and limn→∞ diam(Cn) = 0. Since Cantor’s intersection property
holds in X, ∩{Cn} consists of exactly one point, say z, which is nothing but
the limit of the Cauchy sequence {xn}. Rest of the proof follows from Theorem
3.5. □

Theorem 4.4. Let (X, d) be a metric space. If every x → d(x, fx) is f-
orbitally lower semi-continuous or weak orbitally continuous self-mapping f of
a metric space (X, d) satisfying the condition (i) of Theorem 3.5 has a fixed
point, then X satisfies Cantor’s intersection property.

Proof. Suppose that every x → d(x, fx) is f-orbitally lower semi-continuous
or weak orbitally continuous self-mapping f of a metric space (X, d) satisfying
the condition (i) of Theorem 3.5 possesses a fixed point. We assert that X
satisfies Cantor’s intersection property. If possible, suppose X does not satisfy
Cantor’s intersection property, then there exists a sequence {Cn}of nonempty
closed subsets of X satisfying Cn+1 ⊆ Cn and limn→∞ diam(Cn) = 0 and
having empty intersection. Construct a sequence T = {xn} ∈ X such that
xi ∈ Ai. limn→∞ diam(Cn) = 0, given ! > 0 there exists a positive integer
N such that n, m ≥ N implies d(xn, xm) < !. Therefore {xn} is a Cauchy
sequence. However, T = {xn} is a non-convergent Cauchy sequence since the
sequence {Cn} has empty intersection. As done in the proof of Theorem 4.2 we
can now define f-orbitally lower semi-continuous or weakly orbitally continuous
self-mapping on a metric space (X, d) satisfying condition (i) which does not
possess a fixed point. This contradicts our hypothesis. Therefore, Cantor’s
intersection property holds in X. □

Combining Theorem 4.2 and Theorem 4.4, we get the following theorem:

Theorem 4.5. For a metric space (X, d), the following are equivalent:
(a) X is complete.
(b) X satisfies Cantor’s intersection property.
(c) every x → d(x, fx) is f-orbitally lower semi-continuous or weak orbitally
continuous self-mapping f of a metric space (X, d) satisfying the condition:

Given ε > 0 there exists a δ = δ(ε) > 0 such that
ε ≤ m3(x, y) < ε + δ ⇒ d(fx, fy) < ε,

for all x, y in X, has a fixed point.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 360



Discontinuity at fixed point and metric completeness

References

[1] R. M. T. Bianchini, Su un problema di S. Reich riguardante la teoria dei puntifissi, Boll.
Un. Mat. Ital. 5 (1972), 103–108.

[2] R. K. Bisht and N. Özgür, Geometric properties of discontinuous fixed point set of (!−δ)
contractions and applications to neural networks, Aequat. Math. 94 (2020), 847–863.

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

[4] R. K. Bisht and R. P. Pant, Contractive definitions and discontinuity at fixed point,
Appl. Gen. Topol. 18, no. 1 (2017), 173–182.

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

[6] R. K. Bisht and V. Rakočević , Fixed points of convex and generalized convex contrac-
tions, Rend. Circ. Mat. Palermo, II. Ser., 69, no. 1 (2020), 21–28

[7] S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci. 25 (1972), 15–18.

[8] Lj. B. Ćirić, On contraction type mapping, Math. Balkanica 1 (1971), 52–57.

[9] Lj. B. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc.
45, no. 2 (1974), 267–273.

[10] X. Ding, J. Cao, X. Zhao and F. E. Alsaadi, Mittag-Leffler synchronization of delayed
fractional-order bidirectional associative memory neural networks with discontinuous
activations: state feedback control and impulsive control schemes, Proc. Royal Soc. A:
Math. Eng. Phys. Sci. 473 (2017), 20170322.

[11] M. Forti and P. Nistri, Global convergence of neural networks with discontinuous neuron
activations, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50, no. 11 (2003) 1421–
1435.

[12] H. Garai, L. K. Dey and Y. J. Cho, On contractive mappings and discontinuity at fixed
points, Appl. Anal. Discrete Math. 14 (2020), 33–54.

[13] T. L. Hicks and B. E. Rhoades, A Banach type fixed-point theorem, Math. Japon. 24,
(1979/80), 327–330.

[14] J. Jachymski, Equivalent conditions and Meir-Keeler type theorems, J. Math. Anal.
Appl. 194 (1995), 293–303.

[15] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71–76.
[16] R. Kannan, Some results on fixed points-II, Amer. Math. Monthly 76 (1969), 405–408.
[17] M. Maiti and T. K. Pal, Generalizations of two fixed point theorems, Bull. Calcutta

Math. Soc. 70 (1978), 57–61.
[18] A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28

(1969), 326–329.
[19] L. V. Nguyen, On fixed points of asymptotically regular mappings, Rend. Circ. Mat.

Palermo, II. Ser., to appear.
[20] X. Nie and W. X. Zheng, On multistability of competitive neural networks with dis-

continuous activation functions. In: 4th Australian Control Conference (AUCC), (2014)
245–250.

[21] X. Nie and W. X. Zheng, Multistability of neural networks with discontinuous non-
monotonic piecewise linear activation functions and time-varying delays, Neural Net-
works 65 (2015), 65–79.

[22] X. Nie and W. X. Zheng, Dynamical behaviors of multiple equilibria in competitive
neural networks with discontinuous nonmonotonic piecewise linear activation functions,
IEEE Transactions On Cybernatics 46, no. 3 (2015), 679–693.

[23] N. Y. Özgür and N. Taş, Some fixed-circle theorems and discontinuity at fixed circle,
AIP Conference Proceedings 1926 (2018), 020048 .

[24] N. Y. Özgür and N. Taş, Some fixed-circle theorems on metric spaces, Bull. Malays.
Math. Sci. Soc. 42, no. 4 (2019), 1433–1449.

[25] A. Pant and R. P. Pant, Fixed points and continuity of contractive maps, Filomat 31,
no. 11 (2017), 3501–3506.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 361



R. K. Bisht and V. Rakočević

[26] 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.

[27] R. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl. 240 (1999), 284–289.
[28] R. P. Pant, Fixed points of Lipschitz type mappings, preprint.

[29] R. P. Pant, N. Özgür, N. Taş, A. Pant and M. C. Joshi, New results on discontinuity at
fixed point, J. Fixed Point Theory Appl. (2020) 22:39.

[30] R. P. Pant, N. Y. Özgür and N. Taş, On discontinuity problem at fixed point, Bull.
Malays. Math. Sci. Soc. 43 (2020), 499–517.

[31] 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.

[32] M. Rashid, I. Batool and N. Mehmood, Discontinuous mappings at their fixed points
and common fixed points with applications, J. Math. Anal. 9, no. 1 (2018), 90–104.

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

[34] I. A. Rus, Some variants of contraction principle, generalizations and applications, Stud.
Univ. Babes-Bolyai Math. 61, no. 3 (2016), 343–358.

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

[36] T. Suzuki, A generalized Banach contraction principle that characterizes metric com-
pleteness, Proc. Amer. Math. Soc. 136, no. 5 (2008), 186–1869.

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

[38] H. Wu and C. Shan, Stability analysis for periodic solution of BAM neural networks
with discontinuous neuron activations and impulses, Appl. Math. Modelling 33, no. 6
(2017), 2564–2574.

[39] D. Zheng and P. Wang, Weak θ-φ-contractions and discontinuity, J. Nonlinear Sci. Appl.
10 (2017), 2318–2323.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 362