IJMMS International Journal of Mathematics and Mathematical Sciences 1687-0425 0161-1712 Hindawi Publishing Corporation 431872 10.1155/2012/431872 431872 Research Article An Extension of Generalized (ψ,φ)-Weak Contractions An Tran Van 1 Chi Kieu Phuong 1 Karapınar Erdal 2 Thanh Tran Duc 1 Zayed A. 1 Department of Mathematics Vinh University 182 Le Duan Vinh City Vietnam vinhuni.edu.vn 2 Department of Mathematics Atilim University İncek 06836 Ankara Turkey atilim.edu.tr 2012 8 08 2012 2012 06 02 2012 23 05 2012 2012 Copyright © 2012 Tran Van An et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 , which can be stated as follows.

Let T:XX 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,yX. 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  introduced the notion of Φ-contractions. In 1997 Alber and Guerre-Delabriere  defined the φ-weak contraction which is a generalization of Φ-contractions (see also ).

On the other hand, the notion of T-contractions introduced and studied by the authors of the interesting papers in . 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  introduced the notion of Φ-contraction as follows. A map T:XX 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,yX. The concept of the φ-weak contraction was defined by Alber and Guerre-Delabriere  as a generalization of Φ-contraction under the setting of Hilbert spaces and obtained fixed-point results. A map T:XX 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,yX provided that the function φ satisfies the following condition: (2.3)φ(t)=0ifft=0. Later Rhoades  proved analogs of the result in  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:XX is a φ  weak contraction, then T has a unique fixed point.

In , Dutta and Choudhury proved an extension of Rhoades.

Theorem 2.2.

Let (X,d) be a complete metric space, and let T:XX be a self-mapping satisfying (2.4)ψ(d(Tx,Ty))ψ(d(x,y))-φ(d(x,y)),x,yX, 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  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:XX be self-mappings satisfying (2.5)d(fx,gy)M(x,y)-φ(M(x,y)),x,yX, 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 , Đoricorić  obtained the following theorem.

Theorem 2.4.

Let (X,d) be a complete metric space, and let T,S:XX be self-mappings satisfying (2.7)ψ(d(fx,gy))ψ(M(x,y))-φ(M(x,y)),x,yX, 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 the T-contraction is defined in ([10, 11]) as follows.

Definition 2.5.

Let T 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,yX.

It can be easily seen that if T is the identity map, then the T-contraction coincides with the usual contraction.

Example 2.6.

Let X=(0,) with the usual metric d(x,y)=|x-y| induced by (,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,yX.

Definition 2.7 (see, e.g., [<xref ref-type="bibr" rid="B6">9</xref>, <xref ref-type="bibr" rid="B11">11</xref>]).

Let (X,d) be a metric space. If {yn} is a convergent sequence whenever {Tyn} is convergent, then T:XX is called sequentially convergent.

The aim of this work is to give a proper extension of Đoricorić’s result of using the concept of T-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.

Let Ψ denote the set of all functions ψ:[0,)[0,) which satisfy

ψ  is continuous and nondecreasing,

ψ(t)=0 if and only if t=0.

Similarly Φ denotes the set of all functions φ:[0,)[0,) which satisfy

φ is lower semi continuous,

φ(t)=0 if and only if t=0.

It is easy to see that ψ1(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 Theorem 2.4.

Theorem 3.1.

Let (X,d) be a complete metric space and T:XX an injective, continuous, and sequentially convergent mapping. Let f,g:XX be self-mappings. If there exist ψΨ and φΦ such that (3.1)ψ(d(Tfx,Tgy))ψ(M(Tx,Ty))-φ(M(Tx,Ty)), for all x,yX, 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 . 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 x0X. 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 limnd(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 r0 such that (3.11)limnd(yn,yn+1)=limnM(Txn,Txn-1)=r. By the lower semicontinuity of φ, we have (3.12)φ(r)limninfφ(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)-limninfφ(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)limnd(yn,yn+1)=0. It is implied from (3.10) that (3.16)limnM(Txn,Txn-1)=0. Now, we claim that {yn} is a Cauchy sequence. Since limnd(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)ɛd(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)limkd(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)limkd(y2m(k)-1,y2n(k))=limkd(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)limkd(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)limkM(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.

Since X is a complete metric space, there exists uX such that limnyn=u. Since T is sequentially convergent, we can deduce that {xn} converges to vX. By the continuity of T, we infer that (3.28)u=limnyn=limnTxn=Tv.

We will show that v=fv=gv. Indeed, suppose that vfv, since T is injective, we have u=TvTfv. Hence, d(Tv,Tfv)>0. Since (3.29)limny2n+1=limny2n=u,limnd(y2n,y2n+1)=0, we can seek N0 such that for any nN0(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 nN0. 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.

Let wX 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 Theorem 3.1, if we choose Tx=x for all xX, then we get Theorem 2.4.

(2) In Theorem 3.1, if we fix ψ(t)=t for all t, then we obtain another extension of Theorem 2.3.

(3) In Theorem 3.1, 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 Theorem 3.1 is a proper extension of Theorem 2.4.

Example 3.3.

Let X=[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,yX, then (3.37)ψ(4|x-y|)ψ(M(x,y))-φ(M(x,y)),x,yX, 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 map Tx=lnx+1, for all  xX. 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)|ln4x-ln4y|23M(Tx,Ty),x,y[1,+). We have (3.41)|ln4x-ln4y|=12|lnxy|(3.42)M(Tx,Ty)=max{|ln4y-lnx|+|ln4x-lny|2|lnx-lny|,|ln4x-lnx|,|ln4y-lny||ln4y-lnx|+|ln4x-lny|2}|lnx-lny|=|lnxy|. It follows from (3.41) and (3.42) that (3.43)|ln4x-ln4y|12M(Tx,Ty) for every x,yX. This proves that (3.40) is true.

By the same method used in the proof of Theorem 3.1, we get the following theorem.

Theorem 3.4.

Let (X,d) be a complete metric space and T:XX an injective, continuous, and sequentially convergent mapping. Let f,g:XX be self-mappings. If there exist ψΨ and φΦ such that (3.44)ψ(d(fTx,gTy))ψ(d(Tx,Ty))-φ(d(Tx,Ty)) for all x,yX, then f,g have a unique common fixed point.

Proof.

It follows from the proof of Theorem 3.1 with necessary modifications.

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