Abstract: - Banach contraction principle is the beginning of the Metric fixed point theory. This principle gives
existence and uniqueness of fixed points and methods for obtaining approximate fixed points. It is the basic
tool of finding fixed points of all contraction type maps. It has a constructive proof which makes the theorem
worthy because it yields an algorithm for computing a fixed point. Banach fixed point result has been extended
by various authors in many directions either by weakening the conditions of contraction mapping or by changing
the abstract structure. Several generalizations and extensions of metric spaces have been introduced. Among
these, the prominent extensions are b-metric space, fuzzy metric space, partial metric space and a lot more of their
combinations. In particular, a new structure namely Super metric space is introduced. In the present paper, we
generalize and extend the fixed point results of fixed point theory in literature in the framework of super metric
space.
Key-Words: - Super metric space, fixed point, contraction, weakly compatible maps
Received: March 8, 2024. Revised: July 2, 2024. Accepted: August 19, 2024. Published: September 20, 2024.
1 Introduction and Preliminaries
The well-known Banach’s contraction mapping prin-
ciple states that if f:X→Xis a contraction on
X(i.e. ρ(fγ, fβ)≤qρ(γ, β)for some q < 1and
all γ, β ∈Xand Xis complete, then fhas a unique
fixed in X.
A number of generalizations of this result have ap-
peared. Particularly in 1971, [1], a new generalized
contraction was defined as follows:
A mapping f:X→Xis said to be generalized
contraction iff for every γ, β ∈Xthere exist numbers
q, r, s and twhich may depend on both γand β, such
that
sup[q+r+s+ 2t:γ, β ∈X]<1
and
ρ(fγ, fβ)≤qρ(γ, β) + rρ(γ, f γ)
+sρ(β, f β) + tρ(γ, f β)
+ρ(β, f γ).(1)
The idea of generalized contraction was further ex-
tended, [2], by defining quasi-contraction. A map-
ping f:X→Xof a metric space Xinto itself is
said to be a quasi-contraction iff there exists a num-
ber q,0≤q < 1, such that
ρ(fγ, fβ)≤qmax[ρ(γ, β), ρ(γ, f γ), ρ(β, f β),
ρ(γ, fβ), ρ(β, f γ)] (2)
holds for every γ, β ∈X. The condition (2) implies
condition (1) was supported by an example.
Definition 1.1 ([3]).Let Xbe a non-empty set and
ρ:X×X→[0,+∞)be a mapping which satisfies
(ρ1)ρ(γ, β) = 0 if and only if γ=βfor all γ, β ∈X,
(ρ2)ρ(γ, β) = ρ(β, γ)for all γ, β, ∈X,
(ρ3)ρ(γ, β)≤ρ(γ, α) + ρ(α, β)for all γ, β, α ∈X.
(triangular inequality)
Then, the pair (X, ρ)is called a Euclidean metric
space or a metric space.
For the convenience of the reader, let us recall the
following results:
Proposition 1.2 ([4]).Let (X, ρ)be a complete met-
ric space. Let fbe a continuous self-map on Xand g
be any self-map on Xthat commutes with f. Further
let fand gsatisfy g(X)⊂f(X)and there exists a
constant λin (0,1) such that for every γ, β ∈X,
ρ(gγ, gβ)≤λρ(fγ, fβ).
Then fand ghave a unique common fixed point.
Proposition 1.3 ([5]).Let (X, ρ)be a complete met-
ric space. Let fbe a continuous self-map on X, and
Super Metric Space and Fixed Point Results
MONIKA SIHAG, PARDEEP KUMAR, NAWNEET HOODA
Department of Mathematics
Deenbandhu Chhoturam University of Science and Technology
Murthal 131039, Haryana
INDIA
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gbe any self-map on X, that commutes with f. Fur-
ther, let g(X)⊂f(X)and there exists a constant λ
in (0,1) such that for every γ, β ∈X,
ρ(gγ, gβ)≤λMρ(γ, β),
where
Mρ(γ, β) = max[ρ(fγ, fβ), ρ(fγ, gγ), ρ(fγ, gβ),
ρ(fβ, gβ), ρ(fβ, gγ)].
Then fand ghave a unique fixed point.
The concept of metric space has been generalized
and extended by various authors. Recently in, [6],
a new extension of metric space is introduced and it
is named as Super metric space, and an analogue re-
sult of Banach’s contraction principle in super metric
space is established.
Definition 1.4 ([6]).Let Xbe a nonempty set and
m:X×X→[0,+∞)be a mapping satisfying
(m1) if m(γ, β) = 0, then γ=βfor all γ, β ∈X,
(m2)m(γ, β) = m(β, γ)for all γ, β ∈X,
(m3) there exists s≥1such that for all β∈X, there
exist distinct sequences {γn},{βn} ⊂ X, with
m(γn, βn)→0when ntends to infinity, such
that
lim
n→∞
sup m(βn, y)≤slim
n→∞
sup m(γn, y).
Then, the pair (X, m)is called a super metric space.
Definition 1.5 ([6]).Let (X, m)be a super metric
space and let {γn}be a sequence in X. We say
(i) {γn}converges to γin Xif and only if
m(γn, γ)→0, as n→ ∞.
(ii) {γn}is a Cauchy sequence in Xif and only if
lim
n→∞
sup{m(γn, γm) : m > n}= 0.
(iii) (X, m)is a complete super metric space if and
only if every Cauchy sequence is convergent in
X.
Proposition 1.6 ([6]).Let (X, m)be a complete su-
per metric space and let T:X→Xbe a mapping.
Suppose that 0< k < 1such that
m(T γ, T β)≤km(γ, β),for all γ, β ∈X.
Then Thas a unique fixed point in X.
Proposition 1.7 ([7]).On a super metric space, the
limit of a convergent sequence is unique.
Proposition 1.8 ([7]).Let (X, m)be a complete su-
per metric space and T:X→Xbe an asymptoti-
cally regular mapping. If there exists k∈[0,1), such
that
m(T γ, T β)
≤kmax m(γ, β),(m(γ, T β) + m(β, T γ)
2s,
(m(γ, T γ)m(γ, T β) + m(β, T β)m(β, T γ)
(m(γ, T β) + m(β, T γ) + 1) .
Then Thas a unique fixed point.
Proposition 1.9 ([7]).Let (X, m)be a complete su-
per metric space and let T:X→Xbe a mapping
such that there exists k∈(0,1) and
m(T γ, T β)
≤kmax m(γ, β),m(γ, T γ)m(β, T β)
m(γ, β)+1 .
Then, Thas a unique fixed point.
Definition 1.10 ([8]).A pair (f, g)of self mappings
of metric space (X, d)is said to be weakly compatible
if the mappings commute at all of their coincidence
points, that is, f γ =gγ for some γ∈Ximplies
fgγ =gf γ.
Definition 1.11 ([9]).Let fand gbe self-maps of a
set X. If w=fγ =gγ for some γin X, then γis
called a coincidence point of fand g, and w is called
a point of coincidence of fand g.
Proposition 1.12 ([9]).Let fand gbe weakly com-
patible self-maps of a set X. If fand ghave a unique
point of coincidence w=fγ =gγ, then wis the
unique common fixed point of fand g.
In [10], a new concept of the Φ-map was intro-
duced as the following: Let Φbe the set of all func-
tions ϕsuch that ϕ: [0,+∞)→[0,+∞)is a non
decreasing function satisfying:
lim
n→∞
ϕn(t) = 0,for all t∈(0,+∞).
If ϕ∈Φ, then ϕis called a Φ-map. Furthermore, if
ϕis a Φ-map, then
(i) ϕ(t)< t for all t∈(0,∞),
(ii) ϕ(0) = 0.
From now on, unless otherwise stated, ϕis meant
the Φ-map.
Recently in, [11], using the notion of Φ-map a gen-
eralization of Proposition 1.8 is proved in the setting
of Super Metric Space as the following:
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Proposition 1.13 ([11]).Let (X, m)be a complete
super metric space. Suppose that the mappings f, g :
X→Xsatisfy
m(fγ, fβ)
≤kmax m(gγ, gβ),m(gγ, f β) + m(gβ, f γ)
2s,
m(gγ, fγ)m(gγ, fβ) + m(gβ, fβ)m(gβ, fγ)
m(gγ, fβ) + m(gβ, fγ)+1
for all γ, β ∈X. If f(X)⊂g(X)and g(X)is a
complete subspace of X, then fand ghave a unique
point of coincidence in X. Moreover, if fand gare
weakly compatible, then fand ghave a unique fixed
point.
Our aim is to generalize and extend the fixed point
results of, [2], [4], [5], in the framework of Super met-
ric space. Further, the fixed point results of, [6], [7],
[11] are generalized.
2 Main Results
Theorem 2.1. Let (X, m)be a complete super metric
space and the mappings f, g :X→Xsatisfy
m(fγ, fβ)≤ϕmax m(gγ, gβ),
m(gγ, fγ),m(gγ, fβ)
2s,
m(gβ, fβ), m(gβ, fγ)(3)
for all γ, β ∈X. If f(X)⊂g(X)
and g(X)is a complete subspace of X, then fand g
have a unique point of coincidence in X. Moreover,
if fand gare weakly compatible, then fand ghave
a unique fixed point.
Proof. Let γ0∈Xbe an arbitrary point of X. Since
f(X)⊂g(X), there exists γ1∈Xsuch that gγ1=
fγ0. In this way, we can construct two distinct se-
quences {fγn}and {gγn}such that gγn+1 =f γnfor
all n∈N. If for some n∈N, we have gγn=gγn+1,
then fand ghave a point of coincidence. On the con-
trary, let gγn6=gγn+1 for all n∈N.
Thus, for each n∈N, we have
m(gγn, gγn+1)
=m(fγn−1, fγn)
≤ϕmax m(gγn−1, gγn), m(gγn−1, f γn−1),
m(gγn−1, fγn)
2s, m(gγn, fγn),
m(gγn, fγn−1)
=ϕmax m(gγn−1, gγn), m(gγn−1, gγn),
m(gγn−1, gγn+1)
2s, m(gγn, gγn+1),
m(gγn, gγn)
=ϕmax m(gγn−1, gγn),m(gγn−1, gγn+1)
2s,
m(gγn, gγn+1).
If
max m(gγn−1, gγn),m(gγn−1, gγn+1)
2s,
m(gγn, gγn+1)
=m(gγn, gγn+1),
then
m(gγn, gγn+1)≤ϕm(gγn, gγn+1)
< m(gγn, gγn+1),
which is not possible.
Further, if
m(gγn, gγn+1)≤m(gγn−1, gγn+1)
2s,
then using (m3)
lim
n→∞
sup m(gγn, gγn+1)
≤1
2slim
n→∞
sup m(gγn−1, gγn+1)
≤s
2slim
n→∞
sup m(fγn−1, gγn+1)
=1
2lim
n→∞
sup m(gγn, gγn+1),
which is again a contradiction. Therefore,
m(gγn, gγn+1) = m(fγn−1, fγn)
≤ϕm(gγn−1, gγn)
≤ϕ2m(gγn−2, gγn−1)
.
.
.
≤ϕnm(gγ0, gγ1).(4)
Our aim is to prove that {gγn}is Cauchy sequence.
Let > 0.
Since lim
n→∞
ϕnm(gγ0, gγ1) = 0, there exists N∈
Nsuch that
ϕn[m(gγ0, gγ1)] < for all n≥N.
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Therefore, using (4) for all n≥N
m(gγn, gγn+1)< . (5)
Let m, n ∈Nwith m > n. We will prove that
m(gγn, gγm)< for all m≥n≥N. (6)
Now from (5), we get that the result is true for m=
n+ 1. If γn=γm, (6) is trivially true.
Without loss of generality, we can take γn6=γm.
Suppose (6) is true for m=ki.e.
lim
n→∞
sup m(gγn, gγk) = 0.
Therefore, by using (3) for m=k+ 1 we have
m(gγn, gγk+1)
=m(fγn−1, fγk)
≤ϕmax m(gγn−1, gγk), m(gγn−1, f γn−1),
m(gγn−1, fγk)
2s, m(gγk, fγk),
m(gγk, fγn−1)
=ϕmax m(gγn−1, gγk), m(gγn−1, gγn),
m(gγn−1, gγk+1)
2s, m(gγk, gγk+1),
m(gγk, gγn).
Let
Ω = m(gγn−1, gγk), m(gγn−1, gxn),
m(gγn−1, gγk+1)
2s, m(gγk, gγk+1),
m(gγk, gγn).
If max Ω = m(gγn−1, gγk), then
m(gγn, gγk+1) = m(fγn−1, fγk)
≤ϕm(gγn−1, gγk).
Taking n→ ∞, we have
lim
n→∞
sup m(gγn, gγk+1)
≤ϕlim
n→∞
sup m(gγn−1, gγk).
Using (m3), we get
lim
n→∞
sup m(gγn, gγk+1)
≤sϕ lim
n→∞
sup m(fγn−1, gγk)
=sϕ lim
n→∞
sup m(gγn, gγk)
= 0.
Hence, by induction lim
n→∞
sup m(gγn, gγk+1)=0,
since ϕ(t)< t and s≥1is finite.
If max Ω = m(gγn−1, gγn), then
lim
n→∞
sup m(gγn, gγk+1)
≤ϕlim
n→∞
sup m(gγn−1, gγn)
<lim
n→∞
sup m(gγn−1, gγn)
≤slim
n→∞
sup m(fγn−1, gγn)(by m3)
=slim
n→∞
sup m(gγn, gγn)
= 0.
If max Ω = m(gγn−1,gγk+1 )
2s, then
lim
n→∞
sup m(gγn−1, gγk+1)
≤1
2sϕlim
n→∞
sup m(gγn−1, gγk+1)
<1
2slim
n→∞
sup m(gγn−1, gγk+1)
≤s
2slim
n→∞
sup m(fγn−1, gγk+1)(by m3)
=1
2lim
n→∞
sup m(gγn, gγk+1),
which is a contradiction.
If max Ω = m(gγk, gγk+1), then
lim
n→∞
sup m(gγn, gγk+1)
≤ϕlim
n→∞
sup m(gγk, gγk+1)
<lim
n→∞
sup m(gγk, gγk+1)
≤slim
n→∞
sup m(fγk, gγk+1)(by m3)
≤slim
n→∞
sup m(gγk+1, gγk+1)
= 0.
If max Ω = m(gγk, gγn) = m(gγn, gγk), then the
result is clear.
Hence, by induction lim
n→∞
sup m(gγn, gγk+1) = 0.
It follows {gγn}is a Cauchy sequence. Since we have
assumed g(X)to be complete, {gγn}converges to a
point, say q∈g(X). So gp =q=lim
n→∞
gγn, for a
point pof X. Now we will prove gp =fp.
We have,
m(gp, fp) = lim
n→∞
m(gγn, fp)
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=lim
n→∞
m(fγn−1, fp).
Consider m(fγn−1, fp)and applying (3), we obtain
m(fγn−1, fp)
≤ϕmax m(gγn−1, gp), m(gγn−1, f γn−1),
m(gγn−1, fp)
2s, m(gp, fp),
m(gp, fγn−1)
<max m(gγn−1, gp), m(gγn−1, f γn−1),
m(gγn−1, fp)
2s, m(gp, fp),
m(gp, fγn−1)
=max m(gγn−1, gp), m(gγn−1, gγn),
m(gγn−1, fp)
2s, m(gp, fp),
m(gp, gγn).
Taking n→ ∞, gives
m(gp, fp)<max m(gp, f p), m(gp, gp),
m(gp, fp)
2s), m(gp, fp),
m(gp, gp)
=m(gp, fp),
which is a contradiction.
Therefore gp =fp. We will now show that fand
ghave a unique point of coincidence. Suppose that
fq =gq for some q∈X. By applying (3), it follows
that
m(gp, gq) = m(f p, f q)
≤ϕmax m(gp, gq), m(gp, f p),
m(gp, fq)
2s, m(gq, f q),
m(gq, f p)
≤ϕm(gp, gq)< m(gp, gq),
which is a contradiction. Hence we have gp =gq.
This implies that fand ghave a unique point of
coincidence. By Proposition 1.12, we conclude that
fand ghave a unique common fixed point.
This complete the proof of theorem.
Remark 2.2. Let g=IX,be Identity map on Xin
Theorem 2.1, we get a generalization and extension
of Proposition 1.6.
Proof. Define ϕ: [0,∞)→[0,∞)by ϕ(t) =
kt. Therefore, ϕis a non decreasing function and
lim
n→∞
ϕn(t) = 0 for all t∈(0,+∞). It follows that
the contractive conditions of Theorem 2.1 are now
satisfied. This completes the proof.
Remark 2.3. Taking g=IX, the Identity map on X
in Theorem 2.1, one can deduce an extended analogue
of Proposition 1.2 in super metric space.
Example 2.4. Let X= [1,3] and define
m(γ, β) = γβ, γ 6=β,
0, γ =β.
It has been shown in [8] that (X, m)is a super metric
space. Further, let ϕ=1
2. Now consider f, g :X→
Xas follows
fγ =2, γ 6= 3,
3
2, γ = 3 and gγ = 4 −γ.
Here g(X) = [1,3],f(X)⊂g(X)and g(X)is com-
plete space.
We obtain that fand gsatisfy the contractive condi-
tions of Theorem 2.1. Indeed for γ6= 3,β= 3 and
s= 6, we obtain
m(fγ, fβ) = m2,3
2= 2 ×3
2= 3.
We calculate the right hand side of Theorem 2.1.
(i) ϕ[m(gγ, gβ)] = 1
2m(gγ, 1) = 1
2gγ,
where gγ ∈(1,3].
(ii) ϕ[m(gγ, fγ)] = 1
2m(gγ, 2) = 1
22gγ,
where gγ ∈(1,3].
(iii) ϕ[m(gγ,f β)]
2s=1
2
m(gγ, 3
2)
s=1
2(3gγ
2s)≤1
2(3
2gγ),
using (m3)and where gγ ∈(1,3].
(iv) ϕ[m(gγ, fβ)] = ϕm(1,3
2) = 1
2(3).
(v) ϕ[m(gβ, fγ)] = ϕm(1,2) = 1
2(4).
The other cases are straightforward. Now for γ= 2,
fγ =gγ and fgγ =gf γ. So, 2is the unique point
of coincidence of fand g. Thus all the conditions of
Theorem 2.1 are satisfied. Therefore, 2is the unique
common fixed point by Theorem 2.1.
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Theorem 2.5. Let (X, m)be a complete super metric
space. Suppose that the mappings f, g :X→X
satisfy
m(fγ, fβ)≤ϕmax m(gγ, gβ),
m(gγ, fβ) + m(gβ, fγ)
2s,
m(gγ, fγ)m(gγ, fβ)
+m(gβ, fβ)m(gβ, fγ)
m(gγ, fβ) + m(gβ, fγ)+1
(7)
for all γ, β ∈X. If f(X)⊂g(X)and g(X)is a
complete subspace of X, then fand ghave a unique
point of coincidence in X. Moreover, if fand gare
weakly compatible, then fand ghave a unique fixed
point.
Proof. Let γ0∈Xbe an arbitrary point. Since
f(X)⊂g(X), there exists γ1∈Xsuch that gγ1=
fγ0. Inductively, we can construct two distinct se-
quences {fγn}and {gγn}such that gγn+1 =f γnfor
all n∈N. If there is n∈Nsuch that gγn=gγn+1,
then fand ghave a point of coincidence. Thus, we
can suppose that gγn6=gγn+1, for all n∈N. There-
fore, for each n∈N, we obtain that
m(gγn, gγn+1)
=m(fγn−1, fγn)
≤ϕmax m(gγn−1, gγn),
m(gγn−1, fγn) + m(gγn, f γn−1)
2s,
m(gγn−1, fγn−1)m(gγn−1, f γn)
+m(gγn, fγn)m(gγn, f γn−1)
m(gγn−1, fγn) + m(gγn, f γn−1)+1
=ϕmax m(gγn−1, gγn),
m(gγn−1, gγn+1) + m(gγn, gγn)
2s,
m(gγn−1, gγn)m(gγn−1, gγn+1)
+m(gγn, gγn+1)m(gγn, gγn)
m(gγn−1, gγn+1) + m(gγn, fγn−1)+1
≤ϕmax m(gγn−1, gγn),m(gγn−1, gγn+1)
2s.
If
max m(gγn−1, gγn),m(gγn−1, gγn+1)
2s
=m(gγn−1, gγn+1)
2s,
then
m(gγn, gγn+1)≤ϕm(gγn−1, gγn+1)
2s
<m(gγn−1, gγn+1)
2s.
Taking limit as n→ ∞ on both sides implies that
lim
n→∞
sup m(gγn, gγn+1)
≤1
2slim
n→∞
sup m(gγn−1, gγn+1)
≤s
2slim
n→∞
sup m(fγn−1, gγn+1)(by m3)
=1
2lim
n→∞
sup m(gγn, gγn+1),
giving a contradiction.
Therefore,
m(gγn, gγn+1)≤ϕ m(gγn−1, gγn)
That is, for each n∈N, we have
m(gγn, gγn+1) = m(fγn−1, fγn)
≤ϕ m(gγn−1, gγn)
≤ϕ2m(gγn−2, gγn−1)
.
.
.
≤ϕnm(gγ0, gγ1).
We will show that {gγn}is a Cauchy sequence.
Since lim
n→∞
ϕnm(gγ0, gγ1) = 0, there exists N∈
N, such that
ϕnm(gγ0, gγ1)< for all n≥N.
This implies that
m(gγn, gγn+1)< for all n≥N. (8)
Let m, n ∈Nwith m > n. We will prove that
m(gγn, gγm)< for all m≥n≥N(9)
by induction on m. From (8), the result is true for
m=n+1. Suppose that (9) holds for m=k. There-
fore, for m=k+ 1, we have
m(gγn, gγk+1)
=m(fγn−1, fγk)
≤ϕmax m(gγn−1, gγk),
m(gγn−1, fγk) + m(gγk, f γn−1)
2s,
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m(gγn−1, fγn−1)m(gγn−1, f γk)
+m(gγk, fγk)m(gγk, f γn−1)
m(gγn−1, fγk) + m(gγk, f γn−1)+1.
Denote
A=m(gγn−1, gγk),
m(gγn−1, fγk) + m(gγk, f γn−1)
2s,
m(gγn−1, fγn−1)m(gγn−1, f γk)
+m(gγk, fγk)m(gγk, f γn−1)
m(gγn−1, fγk) + m(gγk, f γn−1)+1.
If max A= [m(gγn−1, gγk)], then
m(gγn, gγk+1)≤ϕm(gγn−1, gγk)
< m(gγn−1, gγk).
Using (m3),
lim
n→∞
sup m(gγn, gγk+1)
≤slim
n→∞
sup m(fxn−1, gγk)
=slim
n→∞
sup m(gγn, gγk)
= 0.
Hence
m(gγn, gγk+1)< . (10)
If max A=m(gγn−1,fγk)+m(gγk,f γn−1)
2s, then
m(gγn, gγk+1)
≤ϕm(gγn−1, fγk) + m(gγk, f γn−1)
2s
<m(gγn−1, fγk) + m(gγk, f γn−1)
2s
=m(gγn−1, gγk+1) + m(gγk, gγn)
2s.
Taking n→ ∞ and using (m3),
lim
n→∞
sup m(gγn, gγk+1)
<1
2slim
n→∞
sup m(gγn−1, gγk+1)
+1
2slim
n→∞
sup m(gγn, gγk)
≤1
2lim
n→∞
sup m(fγn−1, gγk+1)
=1
2lim
n→∞
sup m(gγn, gγk+1)
which gives a contradiction.
If
max A=m(gγn−1, fγn−1)m(gγn−1, f γk)
m(gγn−1, fγk) + m(gγk, f γn−1)+1
+m(gγk, fγk)m(gγk, f γn−1)
m(gγn−1, fγk) + m(gγk, f γn−1)+1,
then
m(gγn, gγk+1)
≤ϕm(gγn−1, fγn−1)m(gγn−1, f γk)
+m(gγk, fγk)m(gγk, f γn−1)
m(gγn−1, fγk) + m(gγk, f γn−1)+1
=ϕm(gγn−1, gγn)m(gγn−1, gγk+1)
+m(gγk, gγk+1)m(gγk, gγn)
m(gγn−1, gγk+1) + m(gγk, gγn)+1
=ϕm(gγn−1, gγn)m(gγn−1, gγk+1)
+m(gγk, gγk+1)m(gγk, gγn)
m(gγn−1, gγk+1) + m(gγk, gγn)+1
=ϕm(gγn−1, gγn)m(gγn−1, gγk+1)
m(gγn−1, gγk+1)m(gγk, gγn)+1
+m(gγk, gγk+1)m(gγk, gγn)
m(gγn−1, gγk+1)m(gγk, gγn)+1
≤ϕm(gγn−1, gγn) + m(gγk, gγk+1).
Taking n→ ∞ and using (m3), we have
lim
n→∞
sup m(gγn, gγk+1)
≤ϕlim
n→∞
sup[m(gγn−1, gγn) + m(gγk, gγk+1)]
≤slim
n→∞
sup[m(fγn−1, gγn) + m(fγk, gγk+1)]
=slim
n→∞
sup[m(gγn, gγn) + m(gγk+1, gγk+1)]
= 0,since s≥1is finite.
Therefore,
m(gγn, gγk+1)< . (11)
Thus (10) holds for all m≥n≥N. It follows that
{gγn}is a Cauchy sequence. By the completeness
of g(X), we obtain that {gγn}is convergent to some
q∈g(X). So there exists p∈Xsuch that gp =q=
lim
n→∞
gγn. We will show that gp =f p. Suppose that
gp 6=fp.
Now,
m(gp, fp) = lim
n→∞
m(gγn, fp)
=lim
n→∞
m(fγn−1, fp).
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Consider m(fγn−1, fp)and applying (7), we obtain
m(fγn−1, fp)
≤ϕmax m(gγn−1, gp),
m(gγn−1, fp) + m(gp, f γn−1)
2s,
m(gγn−1, fγn−1)m(gγn−1, f p)
+m(gp, fp)m(gp, f γn−1)
m(gγn−1, fp) + m(gp, f γn−1)+1
=ϕmax m(gγn−1, gp),
m(gγn−1, fp) + m(gp, gγn)
2s,
m(gγn−1, gγn)m(gγn−1, f p)
+m(gp, fp)m(gp, gγn)
m(gγn−1, fp) + m(gp, gγn)+1 .
Taking limit as n→ ∞
m(gp, fp)
≤ϕmax m(gp, gp),m(gp, f p) + m(gp, gp)
2s,
m(gp, gp)m(gp, f p)
+m(gp, fp)m(gp, gp)
m(gp, fp) + m(gp, gp)+1
=ϕm(gp, fp)
2s<m(gp, fp)
2s,
giving a contradiction, since s≥1.So, gp =fp.
We now show that fand ghave a unique point of
coincidence. Let fq =gq for some q∈X.
Assume that gp 6=gq. By applying (7), it follows that
m(gp, gq)
=m(fp, fq)
≤ϕmax m(gp, gq),m(gq, fq) + m(gq, f p)
2s,
m(gp, fp)m(gp, f q)
+m(gq, f q)m(gq, fp)
m(gp, fq)m(gp, f p)+1
=ϕmax m(gp, gq),m(gq, gq) + m(gq, gp)
2s,
m(gp, gp)m(gp, gq)
+m(gq, gq)m(gq, gp)
m(gp, gq)m(gp, gp)+1
=ϕmax m(gp, gq),m(gp, gq)
s
=ϕ m(gp, gq).
Therefore,
m(gp, gq)≤ϕ m(gp, gq)< m(gp, gq)
which leads to a contradiction. Hence gp =gq.
This implies that fand ghave a unique point of
coincidence. By Proposition 1.12, we can conclude
that fand ghave a unique common fixed point.
Remark 2.6. If we take g=IX, the Identity map on
Xin Theorem 2.5, we get a generalization and exten-
sion of Proposition 1.13.
Proof. Define ϕ: [0,∞)→[0,∞)by ϕ(t) =
kt. Therefore, ϕis a non decreasing function and
lim
n→∞
ϕn(t) = 0 for all t∈(0,+∞). It follows that
the contractive conditions of Theorem 2.5 are now
satisfied. This completes the proof.
3 Conclusion
We have studied the results of, [1, 2], [4], and [5], in
Generalized Metric space. Further, the results of, [6],
and [7], have also been studied. We have generalized
and extended the above results in the framework of
Super Metric Space.
References:
[1] Lj. B. Ciric, Generalized Contractions and Fixed
Point Theorem, Publs. Ins. Math., Vol. 12, No.
26, 1971, pp. 19-26.
[2] Lj. B. Ciric, A Generalization of Banach Con-
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[3] M. R. Fréchet, Surquelques Points Du Calcul
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[4] G. Jungck, Commuting Maps and Fixed Points,
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[5] K. M. Das and K. V. Naik, Common Fixed Point
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[6] E. Karapinar and F. Khojasteh, Super Met-
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[7] E. Karapinar and A. Fulga, Contraction in Ra-
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pages.
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Volume 23, 2024
[8] G. Jungck and B. E. Rhoades, Fixed Points for
Set Valued Functions Without Continuity, In-
dian J. Pure Appl. Math., Vol. 29, 1998, pp. 227-
238.
[9] M. Abbas and B. E. Rhoades, Common
Fixed Point Results for Non Commuting Map-
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[10] J. Mathkowski, Fixed Point Theorems for Map-
pings With a Contractive Iterate at a Point, Proc.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present re-
search, at all stages from the formulation of the prob-
lem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This work is supported by Council of Scientific
and Industrial Research, India for providing Ju-
nior Research Fellowship to Mrs. Monika (CSIR-
HRDG Ref No:Sept/06/22(i)EUV) and University
Grant Commission, India for providing Junior Re-
search Fellowship to Mr. Pardeep Kumar (NTA Ref
No. 201610143126).
Conflicts of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
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Creative Commons Attribution License 4.0
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