# Some Generalizations of Jungck's Fixed Point Theorem

Authors: J. R. Morales; E. M. Rojas
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We are going to generalize the Jungck's fixed point theorem for commuting mappings by mean of the concepts of altering distance functions and compatible pair of mappings, as well as, by using contractive inequalities of integral type and contractive inequalities depending on another function.


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## Body

## 1. Introduction and Preliminary Facts

In 1922, Banach introduced his famous result in the metric fixed point theory, the Banach contraction Principle (BCP), as follows.Theorem 1.1 (see [1]).
Let(M,d) be a complete metric space and let S:M→M be a self-map that satisfies the following condition: there is a∈[0,1) such that
(BC)d(Sx,Sy)≤ad(x,y)
for all x,y∈M. Then, S has a unique fixed point z0∈M such that for each x∈M,limn→∞Snx=z0. One says that a mapping S belongs to the class BC if it satisfies the condition (BC).Since then, several generalizations of the BCP have been appeared, some of them can be found, for instance, in [2–7] and into the references therein. In this paper we will focus our attention on an extension of the BCP introduced in 1976 by  Jungck. More precisely, we are going to improve and generalize the following extension of the BCP.Theorem 1.2 (see [8]).
LetS1,S2 be two self-maps on a complete metric space, (M,d) such that (J1)(S1,S2) is a commuting pair,(J2)S2 is continuous,(J3)S1(M)⊂S2(M),(J4) there is a∈[0,1) such that
(JC)d(S1x,S1y)≤ad(S2x,S2y)
for allx,y∈M.
Then,S1 and S2 have a unique common fixed point z0∈M.The class of all pair of mappings satisfying condition(JC) will be denoted by JC.Notice that by takingS2x=x, for all x∈M, in (JC) we obtain the condition (BC). That is, the Theorem 1.1 is obtained from Theorem 1.2.In order to attain our goals, we are going to use the notions of altering distance functions [9] and compatible pair of mappings [10], also we will use some contractive inequalities of integral type [11] and contractive inequalities depending on another function [12].We would like to start recalling that in 1984 Khan et al. [9] introduced the concept of altering distance function as follows. A function ψ:ℝ+→ℝ+ is called an altering distance function if the following conditions are satisfied:(ψ1)
ψ(0)=0,(ψ2)
ψ is monotonically nondecreasing,(ψ3)
ψ is continuous.We will denote by Ψ the set of all altering distance functions. Using this notion the same authors introduce the following class of mappings and studied the conditions for the existence of fixed points.Definition 1.3 (see [9]).
Let(M,d) be a metric space, ψ∈Ψ and S:M→M be a self-map. The map S is called a ψ-Banach contraction, (ψ-BC), if for each x,y∈M there exists a∈[0,1) such that
(ψ-BC)ψ[d(Sx,Sy)]≤aψ[d(x,y)].
The class of all mappings satisfying condition (ψ-BC) will be called the ψ-BC class. Notice that letting ψ(x)=iMx=x in the inequality (ψ-BC) we obtain(BC), so the ψ-BC class generalizes the BC ones.In 2002, Branciari [12] introduced a new generalization of Banach contraction mappings, which satisfies a general contractive condition of integral type. For that generalization first let us consider the set Φ={φ/φ:ℝ+→ℝ+} where φ satisfies the following conditions:(φ1)
φ is nonnegative,(φ2)
φ is Lebesgue integrable mapping which is summable on each compact subset of ℝ+,(φ3)
for eachɛ>0,∫0ɛφ(t)dt>0.Definition 1.4 (see [12]).
Let(M,d) be a metric space. A mapping S1:M→M is said to be a ∫φ-Banach contraction, (∫φ-BC), if there is a∈[0,1) such that for all x,y∈M, one has
(∫φ-BC)∫0d(S1x,S1y)φ(t)dt≤a∫0d(x,y)φ(t)dt,
where φ∈Φ.The class of all the mappings satisfying the definition above will be denoted by∫φ-BC.Notice that the Definition1.4 is a generalization of the BCP. In fact, letting φ(t)=1 for each t>0 in (∫φ-BC), we have
(1.1)∫0d(S1x,yS1)φ(t)dt=d(S1x,S1y)≤ad(x,y)=a∫0d(x,y)φ(t)dt.
Thus, a Banach contraction mapping also satisfies the condition (∫φ-BC). The converse is not true as we will see in [12, Example 3.6].Theorem 1.5 (see [12]).
Let(M,d) be a complete metric space and let S1:M→M be a ∫φ-BC. Then S1 has a unique fixed point z0∈M such that for each x∈M,limn→∞S1nx=z0.General contractive inequalities of integral type are becoming popular for extend classes of contractive mappings with fixed points. See for example, [13–20] and references therein.In 2009, Beiranvand et al. [11] introduced a new class of contraction mappings, by generalizing the contraction condition (BC) in terms of another function.Definition 1.6 (see [11]).
Let(M,d) be a metric space and T,S1:M→M be two mappings. The mapping S1 is said to be a T-Banach contraction, (or belongs to the T-BC class). If there is a∈[0,1) such that
(T-BC)d(TS1x,TS1y)≤ad(Tx,Ty)
for all x,y∈M.By takingTx=x, for all x∈M, we get that the (T-BC) and (BC) are equivalent conditions.Contraction conditions depending on another function have been used in order to generalize other well-known contraction type maps as the Contractive, Chatterjea, Ćirić, Hardy-Rogers, Kannan, Reich and Rhoades mappings [11, 21–25].Definition 1.7 (see [11]).
Let(M,d) be a metric space. A mapping T:M→M is said sequentially convergent if one has, for every sequence (yn), if T(yn) is convergent, then (yn) also is convergent. T is said subsequentially convergent if one has, for every sequence (yn), if (T(yn)) is convergent, then (yn) has a convergent subsequence.The conditions for the existence of a unique fixed point for mappings in the classT-BC are given in the following result.Theorem 1.8 (see [11]).
Let(M,d) be a complete metric space and T:M→M be a one to one, continuous, and subsequentially convergent mapping. Then, every T-BC continuous mapping S1:M→M, has a unique fixed point z0∈M. Moreover, if T is sequentially convergent, then for each x∈M,limn→∞Snx=z0.


## 2. A Version of the Jungck's Fixed Point Theorem Using Altering Distance Functions

In this section we are going to generalize the Jungck’s fixed point Theorem1.2 by using the altering distance function and the JC class. More precisely, we will introduce the class of ψ-J-contraction mappings which generalize the JC class, and therefore the BC class of mappings.Definition 2.1.
Let(M,d) be a metric space, S,T:M→M be self-mappings and ψ∈Ψ. The pair (S,T) is called a ψ-Jungck contraction, (ψ-J-Contraction) if for each x,y∈M there exists a∈[0,1) such that
(ψ-JC)ψ[d(Sx,Sy)]≤aψ[d(Tx,Ty)].It is clear that lettingTx=x in (ψ-JC) we obtain (ψ-BC).Example 2.2.
LetM=[0,1]⊂ℝ endowed with the Euclidean metric. We define Sx=x2, Tx=(2/a)x, with 0<a<1 and ψ(x)=x2. Then,
(2.1)ψ(d(Sx,Sy))=(x2-y2)2,ψ(d(Tx,Ty))=4a(x-y)2.
From here we get that the pair (S,T) belongs to the class of mappings ψ-JC. On the other hand,
(2.2)ψ(d(Sx,Sy))=(x2-y2)2,ψ(d(x,y))=(x-y)2.
Hence, the pair (S,T) does not belong to the class ψ-BC.In order to prove the fixed point theorem for pair of mappings belonging to the classψ-JC we will need the following notions.Definition 2.3 (see [26]).
Let(M,d) be a metric space and let S1,S2:M→M be two mappings. Suppose that S1(M)⊂S2(M) and for every x0∈M one considers the sequence (xn)⊂M defined by, S1(xn-1)=S2(xn), for all n∈ℕ, one says that (S2(xn)) is a (S1,S2)-sequence of initial point x0.On the other hand, recall that a pair of mappings(T,S) is said to be compatible [10] if and only if d(TSxn,STxn)=0, whenever (xn)⊂M is such that
(2.3)limn→∞S(xn)=limn→∞T(xn)=t
for some t∈M.Theorem 2.4.
Let(M,d) be a complete metric space and let S,T:M→M be self-mappings such that(a)
T is a continuous function,(b)
S(M)⊂T(M),(c)
(S,T) is compatible,(d)
the pair(S,T) belongs to the class of mappings ψ-JC.Then, S and T have a unique common fixed point z0∈M.Proof.
Letx0∈M be an arbitrary point, we will prove that the (S,T)-sequence (T(xn)) of initial point x0 is a Cauchy sequence in M. For each n∈ℕ from inequality contraction (ψ-JC) we have
(2.4)ψ[d(Txn+1,Txn)]=ψ[d(Sxn,Sxn-1)]≤aψ[d(Txn,Txn-1)]≤⋯≤anψ[d(Tx1,Tx0)]
since 0≤a<1, it follows that
(2.5)limn→∞ψ[d(Txn+1,Txn)]=0.
From conditions (ψ1) and (ψ3) of the altering distance function ψ we obtain
(2.6)limn→∞d(Txn+1,Txn)=0,limn→∞d(Sxn,Sxn-1)=0.
Now, we want to prove that (Txn)⊂M is a Cauchy sequence in M. Suppose that (Txn) is not a Cauchy sequence in M. So, there exists ɛ0>0 for which we can find subsequences (Txm(k)) and (Txn(k)) of (Txn) with k<m(k)<n(k) such that
(2.7)d(Txm(k),Txn(k))≥ɛ0
and for this m(k) we can choose n(k) in such a way that it is the smallest integer with n(k)>m(k) satisfying (2.7). Then
(2.8)d(Txm(k),Txn(k)-1)<ɛ0.
Thus, we have
(2.9)ɛ0≤d(Txm(k),Txn(k))≤d(Txm(k),Txn(k)-1)+d(Txn(k)-1,Txn(k))<ɛ0+d(Txn(k)-1,Txn(k)).
From (2.6), (2.7), and (2.9), it follows that
(2.10)limk→∞d(Txm(k),TSxn(k))=ɛ0.
In a similar way we obtain
(2.11)limk→∞d(Txn(k)-1,Txm(k)-1)=ɛ0.
Now, using the definition of ψ-JC mappings for x=Txm(k) and y=Txn(k) we have
(2.12)ψ(ɛ0)≤ψ[d(Txm(k),Txn(k))]=ψ[d(Sxm(k)-1,Sxn(k)-1)]≤aψ[d(Txm(k)-1,Txn(k)-1)].
Letting k→∞ and using (2.11) we have
(2.13)ψ(ɛ0)≤aψ(ɛ0),
which is a contraction if ɛ0>0. This shows that (Txn) is a Cauchy sequence in M and hence it is convergent in the complete metric space M. So there is z0∈M such that
(2.14)limn→∞T(xn+1)=limn→∞S(xn)=z0.
Since T is a continuous function
(2.15)limn→∞T2xn+1=limn→∞TSxn=Tz0,
which proves that the pair (S,T) is compatible. Thus
(2.16)limn→∞d(STxn,TSxn)=0.
Now
(2.17)ψ[d(TSxn,Sxn)]≤aψ[d(T2xn,Txn)]
letting n→∞ we have
(2.18)ψ[d(Tz0,z0)]≤aψ[d(Tz0,z0)]
then, it follows that
(2.19)ψ[d(Tz0,z0)]=0,
which implies that d(Tz0,z0)=0, so Tz0=z0. Again,
(2.20)ψ[d(STxn,Sz0)]≤aψ[d(T2xn,Tz0)]
letting n→∞(2.21)ψ[d(Tz0,Sz0)]≤aψ[d(Tz0,Tz0)]=0
we have that ψ[d(Tz0,Sz0)]=0, so it follows that d(Tz0,Sz0)=0. Then z0=Tz0=Sz0.
Therefore,z0 is a common fixed point of S and T. Now, we are going to suppose that z1 is another common fixed point of S and T. Then
(2.22)ψ[d(z0,z1)]=ψ[d(Sz0,Sz1)]≤aψ[d(z0,z1)]
thus we conclude that ψ[d(z0,z1)]=0, therefore d(z0,z1)=0, or equivalently, z0=z1.Notice that lettingψ=iM, in the definition of the ψ-JC class, we obtain a natural generalization of Jungck’s fixed point Theorem 1.2 to compatible pair of mappings, by replacing condition (J1) for the following:(
J
1
′
)
(
S
1
,
S
2
) is compatible.Also, notice that the fixed point criterion for the class of mappings introduced by Khan et al. in [9] is obtained now as a consequence of Theorem 2.4 by letting Tx=x.


## 3. The Jungck's Fixed Point Theorem with Contractive Inequality Depending on Another Function

In this section our main goal is to generalize the Theorem1.2 by considering its contractive condition depending on another function. Now, using the ideas of Beiranvand et al. given in [11], we introduce the following class of pair of mappings.Definition 3.1.
Let(M,d) be a metric space and let T,S1,S2:M→M be mappings. The pair (S1,S2) is said to be a T-J-contraction, (T-JC) if there is a∈[0,1) such that
(T-JC)d(TS1x,TS1y)≤ad(TS2x,TS2y)
for all x,y∈M.Notice that takingTx=x in the inequality above, we get that (T-JC) and (JC) are equivalent conditions. The next example shows that the class T-JC is more general than the JC ones.Example 3.2.
LetM=[1,+∞)⊂ℝ endowed with the Euclidean metric. We define S1x=4x, S2x=x, and Tx=1+lnx for all x,y∈M. Then,
(3.1)d(S1x,S1y)=|S1x-S1y|=|4x-4y|≤2|x-y|=2d(S2x,S2y).
Thus, the pair (S1,S2) does not belong to JC. However,
(3.2)d(TS1x,TS1y)=|TS1x-TS1y|=|T(4x)-T(4y)|=12|lnx-lny|=12d(TS2x,TS2y).
Hence, the pair (S1,S2) belongs to the class T-JC.Theorem 3.3.
Let(M,d) be a complete metric space and let T,S1,S2:M→M be mappings such that(a)
T is one to one, continuous, and sequentially convergent,(b)
S2 is continuous,(c)
S1(M)⊂S2(M),(d)
S1 and S2 are commuting mappings,(e)
the pair(S1,S2) is a member of the class T-JC.Then, S1 and S2 have a unique common fixed point z0∈M.Proof.
Letx0∈M be an arbitrary point. We will prove that the (S1,S2)-sequence (S2(xn)) of initial point x0 is a Cauchy sequence in M. For each n∈ℕ from the (T-JC) condition we get
(3.3)d(TS2xn+1,TS2xn)=d(TS1xn,TS1xn-1)≤ad(TS2xn,TS2xn-1)
consequently, by repeating the argument above we can conclude that
(3.4)d(TS2xn+1,TS2xn)≤and(TS2x1,TS2x0).
Taking limits in the last inequality we have that
(3.5)limn→∞d(TS2xn+1,TS2xn)=limn→∞d(TS1xn,TS1xn-1)=0.
From (3.3), we have that (TS2xn) is a Cauchy sequence in M. Since M is a complete metric space, then (TS2xn) is convergent in M, and due to the fact that T is sequentially convergent, then (S2xn) is convergent in M. So, there exists z0∈M such that
(3.6)limn→∞S2xn=z0,limn→∞S1xn=z0.
Since T and S2 are continuous maps, then TS2 is a continuous mapping, and using the contractive inequality (T-JC) we conclude then that TS1 is continuous. Thus
(3.7)limn→∞TS1S2xn=TS1z0,limn→∞TS1S1xn=TS1z0.
Using that S1 and S2 are commuting mappings we obtain
(3.8)limn→∞TS1S2xn=TS1z0,limn→∞TS2S1xn=TS2z0.
Therefore
(3.9)TS1z0=TS2z0,TS12z0=TS2S1z0.
Now(3.10)d(TS12z0,TS1z0)≤ad(TS2(S1z0),TS2z0)=ad(TS12z0,TS1z0).
Then, it follows that
(3.11)d(TS12z0,TS1z0)=0
so, TS12z0=TS1z0. Since T is one to one, we obtain that S1(S1z0)=S1z0, which implies that S1z0 is a fixed point of S1 and also that S1z0 is a fixed point of S2. Hence, S1z0 is the common fixed point of S1 and S2.
Now, we will prove thatz0∈M is a common fixed point of S1 and S2. Using the (T-JC) condition, we have
(3.12)d(Tz0,TS1z0)≤d(Tz0,TS1xn)+d(TS1xn,TS1z0)≤d(Tz0,TS1xn)+ad(TS2xn,TS2z0)
taking the limit as n→∞ we get
(3.13)d(Tz0,TS1z0)≤d(Tz0,Tz0)+ad(Tz0,TS2z0).
That is,
(3.14)d(Tz0,TS1z0)≤ad(Tz0,TS1z0).
Therefore, d(Tz0,TS1z0)=0, then Tz0=TS1z0. Since T is one to one, we conclude that z0=S1z0=S2z0. Finally, we suppose that y0=S1y0=S2y0 and z0≠y0. Then
(3.15)d(Tz0,Ty0)=d(TS1z0,TS1z0)≤ad(TS2z0,TS2y0)=ad(Tz0,Ty0),
which implies that Tz0=Ty0, or equivalently, z0=y0. Therefore, the common fixed point of S1 and S2 is unique.Example 3.4.
As in Example3.2, let M=[1,+∞)⊂ℝ be endowed with the euclidean metric and we define S1(x)=4x, S2x=x, and Tx=1+lnx, for all x∈M. Then,(a)
T is one to one, continuous and sequentially convergent,(b)
S2 is continuous,(c)
S1(M)⊂S2(M),(d)
S1 and S2 are commuting mappings,(e)
In the Example3.2, we proved that the pair (S1,S2) does not belong to JC, thus we cannot apply the Theorem 1.2.(f)
In the Example3.2 we have proved that the pair (S1,S2) is a T-JC, so we apply the Theorem 3.3,(g)
It is clear thatz0=16 is the unique common fixed point of S1 and S2.Notice that if in the Theorem3.3 we take Tx=x, for all x∈M, then we obtain Theorem 1.2.


## 4. Contractive Conditions of Integral Type for the Jungck's  Fixed Point Theorem

This section is devoted to generalize the Theorem1.2 using the ideas of Branciari [12]. More precisely, for a pair of mappings we introduce contractive conditions of integral type depending on another function. Afterwards, we will give (common) fixed point results for this new class of mappings.Definition 4.1.
Let(M,d) be a metric space and let T,S1,S2:M→M be mappings. The pair (S1,S2) is called T-∫‍φ-JC if there is a∈[0,1) such that for all x,y∈M, we have
(T-∫φ-JC)∫0d(TS1x,TS1y)φ(t)dt≤a∫0d(TS2x,TS2y)φ(t)dt,
where φ∈Φ.The class of pair of mappings satisfying the definition above will be denoted byT-∫‍φ-JC.Example 4.2.
Let us considerT,S1,S2 and (M,∥·∥) as in Example 3.2 and consider the function φ∈Φ defined by φ(t)=t. So, the pair (S1,S2) belongs to the class T-∫‍φ-JC if we have that
(4.1)∫0d(TS1x,TS1y)tdt≤a∫0d(TS2x,TS2y)tdt,0≤a<1.
Or, equivalently
(4.2)12(d(TS1x,TS1y))2≤a2(d(TS2x,TS2y))2,
which is hold if and only if
(4.3)d(TS1x,TS1y)≤ad(TS2x,TS2y).
On the other hand, from Example 3.2 we have that the inequality above is valid if we take a=1/4. Therefore, the pair (S1,S2) belongs to T-∫φ-JC with contractive constant a=1/4.Theorem 4.3.
Let(M,d) be a complete metric space and let T,S1,S2:M→M be mappings such that(a)
T is one to one, continuous, and sequentially convergent,(b)
S1 and S2 are continuous mappings,(c)
S1(M)⊂S2(M),(d)
S1 and S2 are commuting mappings,(e)
The pair(S1,S2) is a T-∫‍φ-JC. Then, S1 and S2 have a unique common fixed point.Proof.
Letx0∈M be an arbitrary point. We must prove that the (S1,S2)-sequence (S1xn-1)=(S2xn) of initial point x0 is a Cauchy sequence in M. For each n∈ℕ from the condition (T-∫φ-JC) we have
(4.4)∫0d(TS1xn-1,TS1xn)φ(t)dt≤a∫0d(TS2xn-1,TS2xn)φ(t)dt=a∫0d(TS1xn-2,TS1xn-1)φ(t)dt.
Consequently
(4.5)∫0d(TS1xn-1,TS1xn)φ(t)dt≤an∫0d(TS2x0,TS2x1)φ(t)dt
since 0≤a<1, it follows from (4.5) that
(4.6)limn→∞∫0d(TS1xn-1,TS1xn)φ(t)dt≤0
using that ∫0ɛφ(t)dt>0 for each ɛ>0, we conclude that
(4.7)limn→∞d(TS1xn-1,TS1xn)=0.
Now, we can prove that (TS1xn-1) is a Cauchy sequence in M. We are going to follow the proof of the Theorem 2.1 of [27].
In first place, we prove that for allɛ>0, d(TS1xm(p),TS1xn(p)-1)≤ɛ. Suppose that it is not true. Then, there exist ɛ>0 and subsequences (mp(k)) and (np(k)) such that for each positive integer k, np(k) is minimal in the sense that
(4.8)d(TS1xmp(k),TS1xnp(k))≥ɛ,d(TS1xmp(k),TS1xnp(k)-1)<ɛ.
Now,
(4.9)ɛ<d(TS1xmp(k)-1,TS1xnp(k)-1)≤d(TS1xmp(k)-1,TS1xmp(k))+d(TS1xmp(k),TS1xnp(k)-1)<ɛ+d(TS1xmp(k)-1,TS1xmp(k)).
From (4.7) and (4.8), letting k→∞, we have
(4.10)limk→∞d(TS1xmp(k)-1,TS1xnp(k)-1)=ɛ,∫0ɛφ(t)dt≤∫0d(TS1xmp(k),TS1xnp(k))φ(t)dt≤a∫0d(TS2xmp(k),TS2xnp(k))φ(t)dt=a∫0d(TS1xmp(k)-1,TS1xnp(k)-1)φ(t)dt
letting k→∞, we have the contradiction
(4.11)∫0ɛφ(t)dt≤a∫0ɛφ(t)dt.
Thus
(4.12)d(TS1xmp(k)-1,TS1xnp(k)-1)≤ɛ.
Now, we prove that (TS1xn) is a Cauchy sequence in M. From (4.7)–(4.12),
(4.13)∫0ɛφ(t)dt≤∫0d(TS1xmp(k),TS1xnp(k))φ(t)dt≤a∫ad(TS2xmp(k),TS2xnp(k))‍=a∫0d(TS1xmp(k)-1,TS1xnp(k)-1)φ(t)dt≤a∫0ɛφ(t)dt,
which is a contradiction. Therefore (TS1xn) is a Cauchy sequence in M, and since M is a complete metric space, then (TS1xn) is a convergent sequence in M. Now, from the fact that T is sequentially convergent, we get that (S1xn) is a convergent sequence in M. Thus, there exists z0∈M satisfying
(4.14)limn→∞S1xn=z0,limn→∞S2xn=z0
using that T, S1 and S2 are continuous mappings, we conclude that TS1 and TS2 are continuous, hence from (4.14) we get
(4.15)limn→∞TS2S1xn=TS2z0,limn→∞TS1S2xn=TS1z0.
On the other hand, by the commutative property of S1 and S2, it follows that
(4.16)limn→∞TS1S2xn=TS2z0,limn→∞TS1S2xn=TS1z0
therefore, TS1z0=TS2z0 and because T is injective, we have
(4.17)S1z0=S2z0.
It follows for the commuting of the pair (S1,S2) that
(4.18)TS1S2z0=TS2S1z0=TS2S2z0.
Now, suppose that TS1z0≠TS1(S1z0). Using the condition (T-∫φ-JC) we have
(4.19)∫0d(TS1z0,TS1S1z0)φ(t)dt≤a∫0d(TS2z0,TS2S1z0)φ(t)dt=a∫0d(TS1z0,TS1S1z0)φ(t)dt.
That is,
(4.20)(1-a)∫0d(TS1z0,TS1S1z0)φ(t)dt≤0,
or equivalently
(4.21)∫0d(TS1z0,TS1S1z0)φ(t)dt≤0,
which is a contradiction. Therefore,
(4.22)∫0d(TS1z0,TS1S1z0)φ(t)dt=0.
Since ∫0ɛφ(t)dt>0, for each ɛ>0, then d(TS1z0,TS1S1z0)=0 which implies that
(4.23)TS1z0=TS1S1z0
again, since T is one to one, we have that S1z0=S1(S1z0). Thus, S1z0 is a fixed point of S1. Now, due to the equalities TS2S1z0=TS1S1z0=TS1z0 and because T is one to one, we get that
(4.24)S2(S1z0)=S1z0
therefore, S1z0 is a fixed point of S2, hence S1z0 is a common fixed point of S1 and S2.
Now we are going to prove thatz0∈M is a common fixed point of S1 and S2. Using the inequality condition (T-∫φ-JC), we have
(4.25)∫0d(Tz0,TS1z0)φ(t)dt≤∫0d(Tz0,TS1xn)φ(t)dt+∫0d(TS1xn,TS1z0)φ(t)dt≤∫0d(Tz0,TS1xn)φ(t)dt+a∫0d(TS2xn,TS2z0)φ(t)dt.
Taking the limit as n→∞ we obtain
(4.26)∫0d(Tz0,TS1z0)φ(t)dt≤∫0d(Tz0,Tz0)φ(t)dt+a∫0d(Tz0,TS2z0)φ(t)dt=a∫0d(Tz0,TS1z0)φ(t)dt
thus we have
(4.27)(1-a)∫0d(Tz0,TS1z0)φ(t)dt≤0
since 0≤a<1, then we get that
(4.28)∫0d(Tz0,TS1z0)φ(t)dt≤0,
which is a contradiction, therefore
(4.29)∫0d(Tz0,TS1z0)φ(t)dt=0.
From here we conclude that
(4.30)d(Tz0,TS1z0)=0
therefore, Tz0=TS1z0, or equivalently, z0=S1z0=S2z0. Now, we will prove the uniqueness of the common fixed point. Let us suppose that y0=S1y0=S2y0 with y0≠z0. Then
(4.31)∫0d(Tz0,Ty0)φ(t)dt=∫0d(TS1z0,TS1y0)φ(t)dt≤a∫0d(TS2z0,TS2y0)φ(t)dt=a∫0d(Tz0,Ty0)φ(t)dt
then
(4.32)(1-a)∫0d(Tz0,Ty0)φ(t)dt≤0,
which implies that
(4.33)∫0d(Tz0,Ty0)‍φ(t)dt=0
thus, d(Tz0,Ty0)=0. That is, Tz0=Ty0 and since T is one to one, then z0=y0. Therefore, we have proved that z0 is the unique common fixed point of S1 and S2.


## 5. Further Generalizations

Using contractive conditions of integral type and altering distance functions we can introduce new classes of mapping that generalize theJC class.Definition 5.1.
Let(M,d) be a metric space, ψ∈Ψ and S,T:M→M be self-mappings. The pair (S,T) is called a ∫‍φ-ψ-Jungck contraction, (∫φ-ψ-JC) if for each x,y∈M there exists a∈[0,1) such that
(∫φ-ψ-JC)∫0ψ[d(Sx,Sy)]φ(t)dt≤a∫0ψ[d(Tx,Ty)]‍φ(t)dt,
where φ∈Φ.The class of pairs of mappings fulfilling inequality above will be denoted by∫‍φ-ψ-JC.Similarly, we introduce the following class of mappings.Definition 5.2.
Let(M,d) be a metric space, ψ∈Ψ and S,T:M→M be self-mappings. The pair (S,T) is called a ψ-∫‍φ-Jungck contraction, (ψ-∫φ-JC) if for each x,y∈M there exists a∈[0,1) such that
(ψ-∫φ-JC)ψ[∫0d(Sx,Sy)φ(t)dt]≤aψ[∫0d(Tx,Ty)φ(t)dt],
where φ∈Φ.Byψ-∫‍φ-JC we mean the class of mappings given by definition above.The existence and uniqueness of the common fixed point of pair of mappings in these new classes of self-maps is a consequence of the Theorem2.4.Proposition 5.3.
Let(M,d) be a complete metric space and S,T:M→M be self-mappings satisfying the following conditions: (a)
T is a continuous function,(b)
S(M)⊂T(M),(c)
The pair(S,T) is compatible,(d)
The pair(S,T) belongs to ∫φ-ψ-JC. Then, S and T have a unique common fixed point z0∈M.Proof.
Letψ0:ℝ+→ℝ+ be a function defined by
(5.1)ψ0(S)=∫0Sφ(t)dt,
where φ∈Φ. It is clear that ψ0∈Ψ. Moreover,
(5.2)ψ0[ψ(d(Sx,Sy))]=∫0ψ(d(Sx,Sy))‍φ(t)dt≤a∫0ψ(d(Tx,Ty))‍φ(t)dt=aψ0(ψ(d(Tx,Ty)))
it follows that
(5.3)ψ0∘ψ(d(Sx,Sy))≤aψ0∘ψ(d(Tx,Ty))
for all x,y∈M and ψ0∘ψ∈Ψ. Hence by the Theorem 2.4, the mappings S and T have a unique common fixed point z0∈M.In similar form we can prove the following result.Proposition 5.4.
Let(M,d) be a complete metric space and S,T:M→M be self-mappings satisfying the conditions:(a)
T is a continuous mapping,(b)
S(M)⊂T(M),(c)
The pair(S,T) is compatible,(d)
The pair(T,S) belongs to ψ-∫φ-JC.Then, S and T have a unique common fixed point z0∈M.Notice that takingψ=iM in (∫φ-ψ-JC) or (ψ-∫φ-JC) we obtain the results given by Kumar et al. in [28].



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*Source: 213876-2012-11-01.xml*