# An Extension of Generalized(ψ,φ)-Weak Contractions

Authors: Tran Van An; Kieu Phuong Chi; Erdal Karapınar; Tran Duc Thanh
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We prove a fixed-point theorem for a class of maps that satisfy generalized(ψ,φ)-weak contractions depending on a given function. An example is given to illustrate our extensions.


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## Body

## 1. Introduction

Because fixed-point theory has a wide array of applications in many areas such as economics, computer science, and engineering, it plays evidently a crucial role in nonlinear analysis. One of the cornerstones of this theory is the Banach fixed-point theorem, also known as the Banach contraction mapping theorem [1], which can be stated as follows.LetT:X→X be a contraction on a compete metric space (X;d); that is, there is a nonnegative real number k<1 such that d(T(x),T(y))≤kd(x,y) for all x,y∈X. Then the map T admits one and only one point x*∈X such that Tx*=x*. Moreover, this fixed point is the limit of the iterative sequence xn+1=T(xn) for n=0,1,2,…, where x0 is an arbitrary starting point in X. This theorem attracted a lot of attention because of its importance in the field. Many authors have started studying on fixed-point theory to explore some new contraction mappings to generalize the Banach contraction mapping theorem. In particular, Boyd and Wong [2] introduced the notion of Φ-contractions. In 1997 Alber and Guerre-Delabriere [3] defined the φ-weak contraction which is a generalization of Φ-contractions (see also [4–8]).On the other hand, the notion ofT-contractions introduced and studied by the authors of the interesting papers in [9–11]. Following this trend, we explore in this paper another extension of (ψ,φ)-weak contractions in the context of T-contractions.


## 2. Preliminaries

Let(X,d) be a metric space. Boyd and Wong [2] introduced the notion of Φ-contraction as follows. A map T:X→X is called a Φ-contraction if there exists an upper semicontinuous function Φ:[0,+∞)→[0,+∞) such that
(2.1)d(Tx,Ty)≤Φ(d(x,y))
for all x,y∈X. The concept of the φ-weak contraction was defined by Alber and Guerre-Delabriere [3] as a generalization of Φ-contraction under the setting of Hilbert spaces and obtained fixed-point results. A map T:X→X is a φ-weak contraction, if there exists a function φ:[0,+∞)→[0,+∞) such that
(2.2)d(Tx,Ty)≤d(x,y)-φ(d(x,y))
for all x,y∈X provided that the function φ satisfies the following condition:
(2.3)φ(t)=0ifft=0.
Later Rhoades [7] proved analogs of the result in [3] in the context of metric spaces.Theorem 2.1.
Let(X,d) be a complete metric space. Let φ:[0,+∞)→[0,+∞) be a continuous and nondecreasing function such that φ(t)=0 if and only if t=0. If T:X→X is a φweak contraction, then T has a unique fixed point.In [12], Dutta and Choudhury proved an extension of  Rhoades.Theorem 2.2.
Let(X,d) be a complete metric space, and let T:X→X be a self-mapping satisfying
(2.4)ψ(d(Tx,Ty))≤ψ(d(x,y))-φ(d(x,y)),∀x,y∈X,
where ψ,φ:[0,+∞)→[0,+∞) are continuous and nondecreasing functions with φ(t)=ψ(t)=0 if and only if t=0. Then T has a unique fixed point.Zhang and Song [8] improved Theorem 2.1 and gave the following result which states the existence of common fixed points of certain maps in metric spaces.Theorem 2.3.
Let(X,d) be a complete metric space, and let f,g:X→X be self-mappings satisfying
(2.5)d(fx,gy)≤M(x,y)-φ(M(x,y)),∀x,y∈X,
where
(2.6)M(x,y)=max⁡{d(x,y),d(x,fx),d(y,gy),d(x,fy)+d(y,gx)2}
and φ:[0,+∞)→[0,+∞) are lower semicontinuous functions with φ(t)=0 if and only if t=0. Then f,g have a unique common fixed point.Combining the theorems above with the results of Dutta and Choudhury [12], Đoricorić [13] obtained the following theorem.Theorem 2.4.
Let(X,d) be a complete metric space, and let T,S:X→X be self-mappings satisfying
(2.7)ψ(d(fx,gy))≤ψ(M(x,y))-φ(M(x,y)),∀x,y∈X,
where
(2.8)M(x,y)=max⁡{d(x,y),d(x,fx),d(y,gy),d(x,gy)+d(y,fx)2},ψ:[0,+∞)→[0,+∞) is a continuous and nondecreasing function with ψ(t)=0 if and only if t=0, and φ:[0,+∞)→[0,+∞) is a lower semicontinuous function with φ(t)=0 if and only if t=0. Then f,g have a unique common fixed point.The notion of theT-contraction is defined in ([10, 11]) as follows.Definition 2.5.
LetT and S be two self-mappings on a metric space (X,d). The mapping S is said to be a T-contraction if there exists k∈(0,1) such that
(2.9)d(TSx,TSy)≤kd(Tx,Ty),∀x,y∈X.It can be easily seen that ifT is the identity map, then the T-contraction coincides with the usual contraction.Example 2.6.
LetX=(0,∞) with the usual metric d(x,y)=|x-y| induced by (ℝ,d). Consider the following self-mappings T(x)=1/x and Sx=3x on X. It is clear that S is not a contraction. On the contrary,
(2.10)d(TSx,TSy)=|13x-13y|=13|1y-1x|≤13d(Tx,Ty),∀x,y∈X.Definition 2.7 (see, e.g., [9, 11]).
Let(X,d) be a metric space. If {yn} is a convergent sequence whenever {Tyn} is convergent, then T:X→X is called sequentially convergent.The aim of this work is to give a proper extension of Đoricorić’s result of using the concept ofT-contraction, that is, the contraction depending on a given function. We will show the existence of a common fixed point for a class of certain maps.


## 3. Main Results

We start this section by recalling the following two classes of functions.LetΨ denote the set of all functions ψ:[0,∞)→[0,∞) which satisfy  (i)
ψis continuous and nondecreasing,(ii)
ψ(t)=0 if and only if t=0.SimilarlyΦ denotes the set of all functions φ:[0,∞)→[0,∞) which satisfy  (i)
φ is lower semi continuous,(ii)
φ(t)=0 if and only if t=0.It is easy to see thatψ1(t)=t,ψ2(t)=t/(t+1),ψ3(t)=t2 belong to Ψ and φ1(t)=min{t,1},  φ2(t)=ln(1+t) belong to Φ.We are ready to state our main theorem that is a proper extension of Theorem2.4.Theorem 3.1.
Let(X,d) be a complete metric space and T:X→X an injective, continuous, and sequentially convergent mapping. Let f,g:X→X be self-mappings. If there exist ψ∈Ψ and φ∈Φ such that
(3.1)ψ(d(Tfx,Tgy))≤ψ(M(Tx,Ty))-φ(M(Tx,Ty)),
for all x,y∈X, where
(3.2)M(Tx,Ty)=max⁡{d(Tx,Ty),d(Tx,Tfx),d(Ty,Tgy),d(Tx,Tgy)+d(Ty,Tfx)2},
then f,g have a unique common fixed point.Proof.
We will follow the lines in the proof of the main result in [13]. By injection of T, we easily check that M(Tx,Ty)=0 if and only if x=y is a common fixed point of f and g. Let x0∈X. We define two iterative sequences {xn} and {yn} in the following way:
(3.3)x2n+2=fx2n+1,x2n+1=gx2n,yn=Txn,∀n=0,1,2,….
We prove {yn} is a Cauchy sequence. For this purpose, we first claim that limn→∞d(yn+1,yn)=0. It follows from property of φ that if n is odd
(3.4)ψ(d(yn+1,yn))=ψ(d(Txn+1,Txn))=ψ(d(Tfxn,Tgxn-1))≤ψ(M(Txn,Txn-1))-φ(M(Txn,Txn-1))≤ψ(M(Txn,Txn-1)),
where
(3.5)M(Txn,Txn-1)=max⁡{d(Tgxn-1,Txn)+d(Tfxn,Txn-1)2d(Txn,Txn-1),d(Tfxn,Txn),d(Tgxn-1,Txn-1),d(Tgxn-1,Txn)+d(Tfxn,Txn-1)2}=max⁡{d(yn,yn-1),d(yn+1,yn),d(yn,yn-1),d(yn-1,yn+1)2}≤max⁡{d(yn,yn-1),d(yn+1,yn),d(yn-1,yn)+d(yn,yn+1)2}.
Hence, we have (3.6)ψ(d(yn+1,yn))≤ψ(max⁡{d(yn,yn-1),d(yn+1,yn),d(yn-1,yn)+d(yn,yn+1)2}).
If d(yn,yn+1)>d(yn-1,yn)≥0 then M(Txn,Txn-1)=d(yn,yn+1), hence
(3.7)ψ(d(yn,yn+1))≤ψ(d(yn,yn+1))-φ(d(yn,yn+1))
and which contradicts with d(yn,yn+1)>0 and the property of φ. Thus, it follows from (3.5) that
(3.8)d(yn+1,yn)≤M(Txn,Txn-1)=d(yn,yn-1).
If n is even then by the same argument above, we obtain
(3.9)d(yn+1,yn)≤M(Txn-1,Txn)=d(yn,yn-1).
Therefore,
(3.10)d(yn+1,yn)≤M(Txn,Txn-1)=d(yn,yn-1)
for all n and {d(yn,yn+1)} is a nonincreasing sequence of nonnegative real numbers. Hence, there exists r≥0 such that
(3.11)lim⁡n→∞d(yn,yn+1)=lim⁡n→∞M(Txn,Txn-1)=r.
By the lower semicontinuity of φ, we have
(3.12)φ(r)≤lim⁡n→∞inf⁡φ(M(Txn,Txn-1)).
Taking the upper limits as n→∞ on either side of
(3.13)ψ(d(yn,yn+1))≤ψ(M(Txn,Txn-1))-φ(M(Txn,Txn-1)),
we get
(3.14)ψ(r)≤ψ(r)-lim⁡n→∞inf⁡φ(M(Txn,Txn-1))≤ψ(r)-φ(r),
that is, φ(r)≤0. By the property of φ, this implies that φ(r)=0. It follows that r=0 and
(3.15)lim⁡n→∞d(yn,yn+1)=0.
It is implied from (3.10) that
(3.16)lim⁡n→∞M(Txn,Txn-1)=0.
Now, we claim that {yn} is a Cauchy sequence. Since limn→∞d(yn,yn+1)=0, it is sufficient to prove that {y2n} is a Cauchy sequence. Suppose on the contrary that {y2n} is not a Cauchy sequence. Then, there exist ɛ>0 and subsequences {y2n(k)} and {y2m(k)} of {y2n} such that n(k) is the smallest index for which
(3.17)n(k)>m(k)>k,d(y2m(k),y2n(k))>ɛ.
This means that
(3.18)d(y2m(k),y2n(k)-2)<ɛ.
From (3.18) and the triangle inequality, we get
(3.19)ɛ≤d(y2m(k),y2n(k))≤d(y2m(k),y2n(k)-2)+d(y2n(k)-2,y2n(k)-1)+d(y2n(k)-1,y2n(k))<ɛ+d(y2n(k)-2,y2n(k)-1)+d(y2n(k)-1,y2n(k)).
Letting k→∞ and using (3.15), we get
(3.20)lim⁡k→∞d(y2m(k),y2n(k))=ɛ.
By the fact
(3.21)|d(y2m(k),y2n(k)+1)-d(y2m(k),y2n(k))|≤d(y2n(k),y2n(k)+1)|d(y2m(k)-1,y2n(k))-d(y2m(k),y2n(k))|≤d(y2m(k)-1,y2m(k))
and using (3.15) and (3.20), we obtain
(3.22)lim⁡k→∞d(y2m(k)-1,y2n(k))=lim⁡k→∞d(y2m(k),y2n(k)+1)=ɛ.
Moreover, from
(3.23)|d(y2m(k)-1,y2n(k)+1)-d(y2m(k)-1,y2n(k))|≤d(y2n(k),y2n(k)+1)
and combining with (3.15) and (3.22), we conclude that
(3.24)lim⁡k→∞d(y2m(k)-1,y2n(k)+1)=ɛ.
Now, by the definition of M(Tx,Ty) and from (3.10), (3.15), and (3.20)–(3.24), we can deduce that
(3.25)lim⁡k→∞M(Tx2m(k)-1,Tx2n(k))=ɛ.
Due to (3.1), we have (3.26)ψ(d(y2m(k),y2n(k)+1))=ψ(d(Tx2m(k),Tx2n(k)+1))=ψ(d(Tfx2m(k)-1,Tgx2n(k)))≤ψ(M(Tx2m(k)-1,Tx2n(k)))-φ(M(Tx2m(k)-1,Tx2n(k))).
Letting k→∞ and using (3.22) and (3.25), we have
(3.27)ψ(ɛ)≤ψ(ɛ)-φ(ɛ).
It is a contradiction to φ(t)>0 for every t>0. This proves that {yn} is a Cauchy sequence.
SinceX is a complete metric space, there exists u∈X such that limn→∞yn=u. Since T is sequentially convergent, we can deduce that {xn} converges to v∈X. By the continuity of T, we infer that
(3.28)u=lim⁡n→∞yn=lim⁡n→∞Txn=Tv.
We will show thatv=fv=gv. Indeed, suppose that v≠fv, since T is injective, we have u=Tv≠Tfv. Hence, d(Tv,Tfv)>0. Since
(3.29)lim⁡n→∞y2n+1=lim⁡n→∞y2n=u,lim⁡n→∞d(y2n,y2n+1)=0,
we can seek N0∈ℕ such that for any n≥N0(3.30)d(y2n+1,u)<d(Tv,Tfv)4,d(y2n,u)<d(Tv,Tfv)4,d(y2n,y2n+1)<d(Tv,Tfv)4.
Then, we have
(3.31)d(Tv,Tfv)≤M(Tv,Tx2n)=max⁡{d(Tv,Tgx2n)+d(Tx2n,Tfv)2d(Tv,Tx2n),d(Tv,Tfv),d(Tx2n,Tgx2n),d(Tv,Tgx2n)+d(Tx2n,Tfv)2}=max⁡{d(u,y2n+1)+d(y2n,Tfv)2d(u,y2n),d(Tv,Tfv),d(y2n,y2n+1),d(u,y2n+1)+d(y2n,Tfv)2}≤max⁡{d(u,y2n+1)+d(y2n,Tv)+d(Tv,Tfv)2d(u,y2n),d(Tv,Tfv),d(y2n,y2n+1),d(u,y2n+1)+d(y2n,Tv)+d(Tv,Tfv)2}≤max⁡{d(Tv,Tfv)4,d(Tv,Tfv),d(Tv,Tfv)4,d(Tv,Tfv)/4+d(Tv,Tfv)/4+d(Tv,Tfv)2}≤max⁡{d(Tv,Tfv),34d(Tv,Tfv)}=d(Tv,Tfv).
Therefore, M(Tv,Tx2n)=d(Tv,Tfv) for every n≥N0. Since
(3.32)ψ(d(Tfv,y2n+1))=ψ(d(Tfv,Tx2n+1))=ψ(d(Tfv,Tgx2n))≤ψ(M(Tv,Tx2n))-φ(M(Tv,Tx2n))=ψ(d(Tv,Tfv))-φ(d(Tv,Tfv))
and letting n→∞, we arrive at
(3.33)ψ(d(Tfv,Tv))≤ψ(d(Tv,Tfv))-φ(d(Tv,Tfv)).
We get a contradiction. Hence, v=fv. By the same argument, we get v=gv.
Letw∈X such that w=fw=gw. Then, we have
(3.34)M(Tv,Tw)=max⁡{d(Tv,Tw),d(Tv,Tfv),d(Tw,Tgw),d(Tv,Tgw)+d(Tfv,Tw)2}=max⁡{d(Tv,Tw),d(Tv,Tw)+d(Tv,Tw)2}=d(Tw,Tv).
Thus
(3.35)ψ(d(Tv,Tw))=ψ(d(Tfv,Tgw))≤ψ(M(Tv,Tw))-φ(M(Tv,Tw))=ψ(d(Tv,Tw))-φ(d(Tv,Tw)).
This implies that d(Tv,Tw)=0, or Tv=Tw. Since T is injective, we have w=v. The theorem is proved.Remark 3.2.
(1) In Theorem3.1, if we choose Tx=x for all x∈X, then we get Theorem 2.4.
(2) In Theorem3.1, if we fix ψ(t)=t for all t, then we obtain another extension of Theorem 2.3.
(3) In Theorem3.1, if we choose f=g, then we get the uniqueness and existence of fixed point of generalized φ-weak T-contractions.
The following example shows that Theorem3.1 is a proper extension of Theorem 2.4.Example 3.3.
LetX=[1,+∞) and d be the usual metric in X. Consider the maps f(x)=g(x)=4x. It is easy to see that 16 is the unique fixed point of f and g. We claim that f and g are not generalized φ-weak contraction. Indeed, if there exist lower semicontinuous functions ψ,φ:[0,∞)→[0,∞) with ψ(t)>0,φ(t)>0 for t∈(0,∞) and φ(0)=ψ(0)=0, such that
(3.36)ψ(d(fx,gy))≤ψ(M(x,y))-φ(M(x,y)),∀x,y∈X,
then
(3.37)ψ(4|x-y|)≤ψ(M(x,y))-φ(M(x,y)),∀x,y∈X,
where M(x,y)=max{d(x,y),d(fx,x),d(gy,y),(1/2)[d(gx,y)+d(fy,x)]}. For x=4 and y=1, we obtain
(3.38)M(x,y)=max⁡{3,4,3,72}=4.
It follows from (3.37) that
(3.39)ψ(4)≤ψ(4)-φ(4).
Hence, φ(4)≤0. We arrive at a contradiction with φ(t)>0 for t∈(0,∞).
Consider the mapTx=lnx+1, for all  x∈X. It is easy to see that T is injective, continuous, and sequentially convergent. Let ψ(t)=t and φ(t)=t/3, for all  t∈[0,+∞). Now, we show that f and g are generalized φ-weak T-contractions. It reduces to check the following inequality:
(3.40)|ln⁡4x-ln⁡4y|≤23M(Tx,Ty),∀x,y∈[1,+∞).
We have
(3.41)|ln⁡4x-ln⁡4y|=12|ln⁡xy|(3.42)M(Tx,Ty)=max⁡{|ln⁡4y-ln⁡x|+|ln⁡4x-ln⁡y|2|ln⁡x-ln⁡y|,|ln⁡4x-ln⁡x|,|ln⁡4y-ln⁡y||ln⁡4y-ln⁡x|+|ln⁡4x-ln⁡y|2}≥|ln⁡x-ln⁡y|=|ln⁡xy|.
It follows from (3.41) and (3.42) that
(3.43)|ln⁡4x-ln⁡4y|≤12M(Tx,Ty)
for every x,y∈X. This proves that (3.40) is true.By the same method used in the proof of Theorem3.1, we get the following theorem.Theorem 3.4.
Let(X,d) be a complete metric space and T:X→X an injective, continuous, and sequentially convergent mapping. Let f,g:X→X be self-mappings. If there exist ψ∈Ψ and φ∈Φ such that
(3.44)ψ(d(fTx,gTy))≤ψ(d(Tx,Ty))-φ(d(Tx,Ty))
for all x,y∈X, then f,g have a unique common fixed point.Proof.
It follows from the proof of Theorem3.1 with necessary modifications.



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*Source: 431872-2012-08-08.xml*