@
Appl. Gen. Topol. 18, no. 1 (2017), 173-182

doi:10.4995/agt.2017.6713

c© AGT, UPV, 2017

Contractive definitions and discontinuity at

fixed point

Ravindra K. Bishta and R. P. Pant b

a Department of Mathematics, National Defence Academy, Khadakwasla, Pune, India

(ravindra.bisht@yahoo.com)
b Department of Mathematics, Kumaun University, Nainital, Uttarakhand, India.

(pant rp@rediffmail.com)

Communicated by I. Altun

Abstract

In this paper, we investigate some contractive definitions which are
strong enough to generate a fixed point but do not force the mapping to
be continuous at the fixed point. We also obtain a fixed point theorem
for generalized nonexpansive mappings in metric spaces by employing
Meir-Keeler type conditions.

2010 MSC: Primary: 47H09; Secondary: 47H10.

Keywords: Fixed point; (�−δ) contractions; power contraction; orbital con-
tinuity.

1. Introduction

The well-known Banach-Picard-Caccioppoli contraction principle states that:

Theorem 1.1. If a self-mapping T of a complete metric space (X,d) satisfies
the condition; d(Tx,Ty) ≤ ad(x,y), 0 ≤ a < 1, for each x,y ∈ X, then T has
a unique fixed point. The Picard iteration {xn} defined by xn+1 = Txn, (n =
0, 1, 2, ...) converges to x∗ for any initial value x0 ∈ X.

It is known that the mapping T of Banach-Picard-Caccioppoli contraction
is continuous in the entire domain of X.

In an interesting development, Kannan [9] proved the following theorem:

Received 11 October 2016 – Accepted 09 January 2017

http://dx.doi.org/10.4995/agt.2017.6713


R. K. Bisht and R. P. Pant

Theorem 1.2 ([9]). If a self-mapping T of a complete metric space (X,d)
satisfies the condition:

d(Tx,Ty) ≤ b[d(x,Tx) + d(y,Ty)], 0 ≤ b < 1/2,
for each x,y ∈ X, then T has a unique fixed point.

The Kannan fixed point theorem gave rise to the famous question of conti-
nuity of contractive mappings at their fixed points. It may be observed that
Kannan contractive condition does not require the continuity of the mapping T
for the existence of the fixed point. However, a mapping T satisfying Kannan
contractive condition turns out to be continuous at the fixed point. To see this,
suppose that z = Tz is a fixed point of T and xn → z. Then
d(Txn,z) = d(Txn,Tz) ≤ b[d(xn,Txn) + d(z,Tz)] ≤ b[d(xn,z) + d(z,Txn)],

that is, (1 − b)d(Txn,z) ≤ bd(xn,z). This implies that Txn → z = Tz and T
is continuous at the fixed point z.

Kannan’s paper generated a far-flung interest in the study of fixed points of
generalized contractive mappings and soon these were followed by a flood of pa-
pers involving contractive definitions, many of which did not require continuity
of the mapping. Also, Kannan contractive condition contained the geometri-
cally elegant idea of defining generalized contractions (generally referred to as
contractive definitions in the literature) by replacing d(x,y) in Theorem 1.1
above, by a convex combination of distances between the four points x,y,Tx
and Ty. As a result of this, a large number of contractive definitions were
soon introduced and studied by various researchers (for various contractive
conditions see [3, 4, 15, 16, 17]).

One of the most interesting generalizations of the Banach-Picard-Caccioppoli
contraction principle consists of replacing the Lipschitz constant k by some
real valued function whose functional values are less than 1. In 1969, Boyd and
Wang [2] initiated the work along these lines and proved the following theorem:

Theorem 1.3 ([2]). Let T be a mapping of a complete metric space (X,d) into
itself. Suppose there exists a function φ, upper semicontinuous from right from
R+ into itself such that d(Tx,Ty) ≤ φ(d(x,y)), for all x,y ∈ X. If φ(t) < t
for each t > 0, then T has a unique fixed point.

Another noteworthy generalizations of both Banach-Picard-Caccioppoli con-
traction principle and Boyd and Wang fixed point theorem was obtained by
Meir and Keeler [12] in 1969. They proved the following theorem:

Theorem 1.4 ([12]). If a self-mapping T of a complete metric space (X,d)
satisfies the condition:

(i) for a given � > 0 there exists a δ(�) > 0 such that
� ≤ d(x,y) < � + δ implies d(Tx,Ty) < �

then T has a unique fixed point.

A mapping satisfying Boyd and Wong or Meir-Keeler type condition is also
continuous in the entire domain of X.

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Contractive definitions and discontinuity at fixed point

The following theorem was established by J. Matkowski [11] in 1975 as a
generalization of Meir and Keeler fixed point theorem (see also [6]):

Theorem 1.5 ([11]). If a self-mapping T of a complete metric space (X,d)
satisfy the conditions:

(i) d(Tx,Ty) < d(x,y), for all x,y ∈ X,x 6= y;
(ii) for a given � > 0 there exists a δ(�) > 0 such that

� < d(x,y) < � + δ implies d(Tx,Ty) ≤ �
then there exists exactly one fixed point of T ; moreover, its domain of attraction
coincides with the whole of X.

In [8] Jachymski listed some Meir-Keeler type conditions and established
relations between them. Further he gave some new Meir-Keeler type conditions
ensuring a convergence of the successive approximations (see also [5]).

In a survey paper of contractive definitions, Rhoades [17] compared 250 con-
tractive definitions and showed that majority of the contractive definitions do
not require the mapping to be continuous in the entire domain. However, in
all the cases the mapping is continuous at the fixed point. He further demon-
strated that the contractive definitions force the mapping to be continuous at
the fixed point though continuity was neither assumed nor implied by the con-
tractive definitions. The question whether there exists a contractive definition
which is strong enough to generate a fixed point but which does not force the
map to be continuous at the fixed point was reiterated by Rhoades in [18] as
an existing open problem.

The question of the existence of contractive mappings which are discontin-
uous at their fixed points was settled in the affirmative by Pant [13]. Recently,
Bisht and Pant[1] also gave a contractive definition which does not force the
map to be continuity at the fixed point.

In this note we provide more solutions to the open question of the existence
of contractive definitions which are strong enough to generate a fixed point but
which do not force the mapping to be continuous at the fixed point.

Recall that the set O(x; T) = {Tnx : n = 0, 1, 2, ...} is called the orbit of the
self-mapping T at the point x ∈ X.

Definition 1.6. A self-mapping T of a metric space (X,d) is called orbitally
continuous at a point z ∈ X if for any sequence {xn} ⊂ O(x; T) (for some
x ∈ X) xn → z implies Txn → Tz as n →∞.

It is easy to check that every continuous self-mapping of a metric space is
orbitally continuous, but converse need not be true.

2. Main results

In what follows we shall denote

M(x,y) = max{d(x,y),d(x,Tx),d(y,Ty), [d(x,Ty) + d(y,Tx)]/2};
N(x,y) = max{d(x,y),d(x,Tx),d(y,Ty),a[d(x,Ty) + d(y,Tx)]/2}, 0 ≤ a < 1.

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R. K. Bisht and R. P. Pant

Theorem 2.1. Let (X,d) be a complete metric space. Let T be a self-mapping
on X such that for any x,y ∈ X;

(i) d(Tx,Ty) ≤ φ(N(x,y)), where φ : R+ → R+ is such that φ(t) < t for
each t > 0;

(ii) for a given � > 0 there exists a δ(�) > 0 such that � < M(x,y) < � + δ
implies d(Tx,Ty) ≤ �.

Suppose T is orbitally continuous. Then T has a unique fixed point, say z,
and Tnx → z for each x ∈ X. Moreover, T is continuous at z iff limx→zM(x,z) =
0.

Proof. Let x0 be any point in X. Define a sequence {xn} in X given by the
rule xn+1 = T

nx0 = Txn and qn = d(xn,xn+1) for all n ∈ N
⋃
{0}. Then by

(i)

qn = d(xn,xn+1) = d(Txn−1,Txn) ≤ φ(N(xn−1,xn)) < N(xn−1,xn)
= max{qn,qn−1} = qn−1.

Thus {qn} is a strictly decreasing sequence of positive real numbers and,
hence, tends to a limit q ≥ 0. If possible, suppose q > 0. Then there exists a
positive integer k ∈ N such that n ≥ k implies

(2.1) q < qn < q + δ(q).

It follows from (ii) and qn < qn−1 that qn ≤ q, for n ≥ k, which contradicts
the above inequality. Thus we have q = 0.

We shall show that {xn} is a Cauchy sequence. Fix an � > 0. Without loss
of generality, we may assume that δ(�) < �. Since qn → 0, there exists k ∈ N
such that qn <

1
2
δ, for n ≥ k.

Following Jachymski [7, 8] we shall use induction to show that, for any n ∈ N,

(2.2) d(xk,xk+n) < � +
1

2
δ.

Inequality (2.2) holds for n = 1. Assuming (2.2) is true for some n we shall
prove it for n + 1. By the triangle inequlaity, we have

(2.3) d(xk,xk+n+1) ≤ d(xk,xk+1) + d(xk+1,xk+n+1).

Observe that it suffices to show that

(2.4) d(xk+1,xk+n+1) ≤ �.

To show it we shall prove that M(xk,xk+n) ≤ � + δ, where
M(xk,xk+n) =max{d(xk,xk+n),d(xk,Txk),d(xk+n,Txk+n),

[d(xk,Txk+n) + d(xk+n,Txk)]/2}.
(2.5)

By the induction hypothesis, we get

(2.6) d(xk,xk+n) < � +
1

2
δ,d(xk,xk+1) <

1

2
δ,d(xk+n,xk+n+1) <

1

2
δ.

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Contractive definitions and discontinuity at fixed point

Also,

1

2
[d(xk,xk+n+1) + d(xk+1,xk+n)] ≤

1

2
[d(xk,xk+n) + d(xk+n+1,xk+n)

+ d(xk,xk+1) + d(xk,xk+n)] < � + δ.

Thus M(xk,xk+n) < � + δ so by (ii) d(xk+1,xk+n+1) ≤ �, completing the
induction. Hence (2.2) implies that {xn} is a Cauchy sequence. Since X is
complete, there exists a point z ∈ X such that xn → z as n → ∞. Also
Txn → z. Orbital continuity of T implies that limn→∞Txn = Tz. This yields
Tz = z, that is, z is a fixed point of T. Uniqueness of the fixed point follows
from (i).

Now, let T be continuous at the fixed point z and xn → z. Then Txn →
Tz = z. Hence

lim
n
M(xn,z) = lim

n
max{d(xn,z),d(xn,Txn),d(z,Tz), [d(xn,Tz) + d(z,Txn)]/2}

= 0.

On the other hand, if limxn→z M(xn,z) = 0, then d(xn,Txn) → 0 as xn → z.
This implies that Txn → z = Tz, i.e., T is continuous at z. This concludes the
theorem. �

In the next theorem, we replace the orbital continuity of the mapping T by
continuity condition on Tp, where p is a positive integer ≥ 2.

Theorem 2.2. Let (X,d) be a complete metric space. Let T be a self-mapping
on X such that Tp is continuous and for any x,y ∈ X;

(i) d(Tx,Ty) ≤ φ(N(x,y)), where φ : R+ → R+ is such that φ(t) < t for
each t > 0;

(ii) for a given � > 0 there exists a δ(�) > 0 such that � < M(x,y) < � + δ
implies d(Tx,Ty) ≤ �.

Then T has a unique fixed point, say z, and Tnx → z for each x ∈ X. More-
over, T is continuous at z iff limx→zM(x,z) = 0.

Proof. Let x0 be any point in X. Define a sequence {xn} in X given by the
rule xn+1 = T

nx0 = Txn. Then following the proof of above theorem we
conclude that {xn} is a Cauchy sequence. Since X is complete, there exists a
point z ∈ X such that xn → z as n → ∞. Also Txn → z and Tpxn → z. By
continuity of Tp, we have Tpxn → Tpz. This implies Tpz = z. We claim that
Tz = z. For if Tz 6= z, we get
d(Tz,z) = d(Tz,Tpz) ≤ φ(N(z,Tp−1z)) < N(z,Tp−1z) = d(Tpz,Tp−1z);
d(Tpz,Tp−1z) ≤ φ(N(Tp−1z,Tp−2z)) < N(Tp−1z,Tp−2z) = d(Tp−1z,Tp−2z);
. . .
d(T 2z,Tz) ≤ φ(N(Tz,z)) < N(Tz,z) = d(Tz,z),

that is z = Tz and z is a fixed point of T. Uniqueness of the fixed point follows
from (i). �

Taking M(x,y) = N(x,y) = max{d(x,y),d(x,Tx),d(y,Ty),a[d(x,Ty) +
d(y,Tx)]/2}, 0 ≤ a < 1 we now state the following theorems:

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R. K. Bisht and R. P. Pant

Theorem 2.3. Let (X,d) be a complete metric space. Let T be a self-mapping
on X such that for any x,y ∈ X;

(i) d(Tx,Ty) ≤ φ(M(x,y)), where φ : R+ → R+ is such that φ(t) < t for
each t > 0;

(ii) for a given � > 0 there exists a δ(�) > 0 such that � < M(x,y) < � + δ
implies d(Tx,Ty) ≤ �.

Then T has a unique fixed point, say z, and Tnx → z for each x ∈ X.
Moreover, T is continuous at z iff limx→zM(x,z) = 0.

Proof. It may be completed on the lines of the proof of Theorem 2.1 above. �

Theorem 2.4. Let (X,d) be a complete metric space. Let T be a self-mapping
on X such that Tp is continuous and for any x,y ∈ X;

(i) d(Tx,Ty) ≤ φ(M(x,y)), where φ : R+ → R+ is such that φ(t) < t for
each t > 0;

(ii) for a given � > 0 there exists a δ(�) > 0 such that � < M(x,y) < � + δ
implies d(Tx,Ty) ≤ �.

Then T has a unique fixed point, say z, and Tnx → z for each x ∈ X.
Moreover, T is continuous at z iff limx→zM(x,z) = 0.

Proof. It may be completed on the lines of the proof of Theorem 2.2 above. �

Remark 2.5. The last part of Theorems 2.1 and 2.2 can alternatively be stated
as: T is discontinuous at z iff limx→zM(x,z) 6= 0.

The following example illustrates the above theorems:

Example 2.6. Let X = [0, 2] and d be the usual metric on X. Define T :
X → X by

T(x) = 1 if x ∈ [0, 1], T(x) = 0 if x ∈ (1, 2].

Then T satisfies the conditions of Theorems 2.1 and 2.2 and has a unique
fixed point x = 1 at which T is discontinuous. The mapping T satisfies the
contractive condition (i) with φ(t) = 1 for t > 1 and φ(t) = t

2
for t ≤ 1. Also, T

satisfies condition (ii) with δ(�) = 1 for � ≥ 1 and δ(�) = 1−� for � < 1. It can
also be easily seen that limx→1M(x, 1) 6= 0 and T is discontinuous at the fixed
point x = 1. However, Tp is continuous, since Tp(x) = 1 for all x ∈ X(p ≥ 2).

Theorem 2.7. Let (X,d) be a complete metric space. Let T be a self-mapping
on X such that for any x,y ∈ X;

(i) d(Tx,Ty) ≤ φ(N(x,y)), where φ : R+ → R+ is such that φ(t) < t for
each t > 0;

(ii) for a given � > 0 there exists a δ(�) > 0 such that � < M(x,y) < � + δ
implies d(Tx,Ty) ≤ �.

Then T has a unique fixed point. Moreover, T is continuous at z iff limx→zM(x,z)
= 0.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 178



Contractive definitions and discontinuity at fixed point

Proof. Let x0 be any point in X and let x 6= Tx. Define a sequence {xn} in X
given by the rule xn+1 = T

nx0 = Txn. Then following the proof of Theorem
2.1, we conclude that {xn} is a Cauchy sequence. Since X is complete, there
exists a point z ∈ X such that xn → z as n → ∞. Also Txn → z. We claim
that Tz = z. For if Tz 6= z, we get

d(Tz,Txn) ≤ φ(max{d(z,xn),d(z,Tz),d(xn,Txn),a[d(z,Txn)+d(xn,Tz)]/2}).

On letting n → ∞ this yields, d(Tz,z) ≤ φ(d(Tz,z)) < d(Tz,z), a contradic-
tion. Thus z is a fixed point of T . Uniqueness of the fixed point follows from
(i). �

The following theorem shows that power contraction allows the possibility
of discontinuity at the fixed point. In the next theorem we denote:

M′(x,y) =max{d(x,y),d(x,Tmx),d(y,Tmy), [d(x,Tmy) + d(y,Tmx)]/2},
N′(x,y) =max{d(x,y),d(x,Tmx),d(y,Tmy),a[d(x,Tmy) + d(y,Tmx)]/2},

0 ≤ a < 1

where m ∈ N.

Theorem 2.8. Let (X,d) be a complete metric space. Let T be a self-mapping
on X such that for any x,y ∈ X:

(i) d(Tmx,Tmy) ≤ φ(N′(x,y)), where φ : R+ → R+ is such that φ(t) < t
for each t > 0;

(ii) for a given � > 0 there exists a δ(�) > 0 such that � < M′(x,y) < � + δ
implies d(Tmx,Tmy) ≤ �.

Then T has a unique fixed point.

Proof. By Theorem 2.7, Tm has a unique fixed point z ∈ X; i.e., Tm(z) = z.
Then T(z) = T(Tm(z)) = Tm(T(z)) and so T(z) is a fixed point of Tm. Since
the fixed point of Tm is unique, Tz = z. �

Taking M′(x,y) = N′(x,y) = max{d(x,y),d(x,Tmx),d(y,Tmy),a[d(x,Tmy)+
d(y,Tmx)]/2}, 0 ≤ a < 1 we get the following result:

Theorem 2.9. Let (X,d) be a complete metric space. Let T be a self-mapping
on X such that for any x,y ∈ X:

(i) d(Tmx,Tmy) ≤ φ(M′(x,y)), where φ : R+ → R+ is such that φ(t) < t
for each t > 0;

(ii) for a given � > 0 there exists a δ(�) > 0 such that � < M′(x,y) < � + δ
implies d(Tmx,Tmy) ≤ �.

Then T has a unique fixed point.

Proof. It may be completed following Theorem 2.7 above. �

Remark 2.10. Theorems 2.1, 2.2, and 2.3 unify and improve the results due to
Bisht and Pant [1], Ćirić [5, 6], Jachymski [8], Kuczma et al. [10], Matkowski
[11], and Pant [13].

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R. K. Bisht and R. P. Pant

Some consequences of the above proved theorems are the following corollaries
which also generalize and extend the results of Jachymski [8], Kuczma et al.
[10], Matkowski [11], and Pant [13].

Corollary 2.11. Let (X,d) be a complete metric space. Let T be a self-mapping
on X such that:

(i) d(Tx,Ty) < N(x,y), for any x,y ∈ X with M(x,y) > 0;
(ii) for a given � > 0 there exists a δ(�) > 0 such that � < M(x,y) < � + δ

implies d(Tx,Ty) ≤ �.
Suppose T is orbitally continuous. Then T has a unique fixed point, say z,

and Tnx → z for each x ∈ X. Moreover, T is continuous at z iff limx→zM(x,z) =
0.

Corollary 2.12. Let (X,d) be a complete metric space. Let T be a self-mapping
on X such that Tp is continuous:

(i) d(Tx,Ty) < N(x,y), for any x,y ∈ X with M(x,y) > 0;
(ii) for a given � > 0 there exists a δ(�) > 0 such that � < M(x,y) < � + δ

implies d(Tx,Ty) ≤ �.
Then T has a unique fixed point, say z, and Tnx → z for each x ∈ X.

Moreover, T is continuous at z iff limx→zM(x,z) = 0.

3. Fixed points of nonexpansive mappings

In what follows we shall denote

P(x,y) = max{d(x,y),b[d(x,Tx) + d(y,Ty)]/2,c[d(x,Ty) + d(y,Tx)]/2}, 0 ≤
b,c < 1.

Theorem 3.1. Let (X,d) be a complete metric space. Let T be a self-mapping
on X such that for any x,y ∈ X;

(i) for a given � > 0 there exists a δ(�) > 0 such that � < M(x,y) < � + δ
implies d(Tx,Ty) ≤ �;

(ii) d(Tx,Ty) ≤ P(x,y).
Then T has a fixed point, say z, and Tnx → z for each x ∈ X.

Proof. Let x0 be any point in X and let x 6= Tx. Define a sequence {xn} in X
given by the rule xn+1 = T

nx0 = Txn. Then following the proof of Theorem
2.1 we can easily prove that {xn} is a Cauchy sequence. Since X is complete,
there exists a point z ∈ X such that xn → z as n → ∞. Also Txn → z. We
claim that Tz = z. For if Tz 6= z, we get

d(Tz,Txn) ≤
max{d(z,xn),b[d(z,Tz) + d(xn,Txn)]/2,c[d(z,Txn) + d(xn,Tz)]/2}.

On letting n →∞ this yields, d(Tz,z) ≤ max{b[d(Tz,z)]/2,c[d(Tz,z)]/2} <
d(Tz,z), a contradiction since 0 ≤ b,c < 1. Thus z is a fixed point of T . �

Remark 3.2. Theorem 3.1 also remains true if we replace condition (ii) by the
following condition:

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Contractive definitions and discontinuity at fixed point

(i). d(Tx,Ty) ≤ max{d(x,y),d(x,Tx),d(y,Ty),b[d(x,Ty)+d(y,Tx)]/2}, 0 ≤
b < 1.

The following example illustrates Theorem 3.1:

Example 3.3. Let X = [−1, 1] and d be the usual metric on X. Define
T : X → X by

T(x) = −|x|x for each x.

Then T satisfies all the conditions of Theorem 3.1 and has a fixed point x = 0.
The mapping T satisfies condition (i) with δ(�) = 1

4
(
√

(2�) − �)) for � < 2 and
δ(�) = � for � ≥ 2. However, T does not satisfy the contractive condition
d(Tx,Ty) < max{d(x,y), [d(x,Tx) + d(y,Ty)]/2, [d(x,Ty) + d(y,Tx)]/2}.

It may be observed that there exist a large number of Meir-Keeler type
nonexpansive conditions which yield more than one fixed point. The following
example illustrates this fact:

Example 3.4. Let X = [0, 1] and d be the usual metric on X. Define T :
X → X by

Tx = sgn(x) (the signum function), i.e., T0 = 0,Tx = 1 if x > 0.

Then T has two fixed points x = 0 and x = 1.

Acknowledgements. The authors are thankful to the learned referees for
their suggestions which improved the presentation of this paper.

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