JAMJournal of Applied Mathematics1687-00421110-757XHindawi Publishing Corporation16147010.1155/2012/161470161470Research ArticleThe Existence of Fixed Points for Nonlinear Contractive Maps in Metric Spaces with w-DistancesLakzianHossein1LinIng-Jer2LiouYeong-Cheng1Department of Mathematics, Payame Noor University, Tehran 19395-4697Iranpnu.ac.ir2Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824Taiwannknu.edu.tw20121622012201220112011041220112012Copyright © 2012 Hossein Lakzian and Ing-Jer Lin.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.

Some fixed point theorems for (φ,ψ,p)-contractive maps and (φ,k,p)-contractive maps on a complete metric space are proved. Presented fixed point theorems generalize many results existing in the literature.

1. Introduction and Preliminaries

Branciari  established a fixed point result for an integral type inequality, which is a generalization of Banach contraction principle. Kada et al.  introduced and studied the concept of w-distance on a metric space. They give examples of w-distances and improved Caristi’s fixed point theorem, Ekeland’s ϵ-variational’s principle, and the nonconvex minimization theorem according to Takahashi (see many useful examples and results on w-distance in  and in references therein). Kada et al.  defined the concept of w-distance in a metric space as follows.

Definition 1.1 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

Let X be a metric space endowed with a metric d. A function p:X×X[0,) is called a w-distance on X if it satisfies the following properties:

p(x,z)p(x,y)+p(y,z) for any x,y,zX,

p is lower semicontinuous in its second variable, that is, if xX and yny in X then p(x,y)liminfnp(x,yn),

for each ϵ>0, there exists δ>0 such that p(z,x)δ and p(z,y)δ imply d(x,y)ϵ.

We denote by Φ the set of functions φ:[0,+)[0,+) satisfying the following hypotheses:

φ is continuous and nondecreasing,

φ(t)=0 if and only if t=0.

We denote by Ψ the set of functions ψ:[0,+)[0,+) satisfying the following hypotheses:

ψ is right continuous and nondecreasing,

ψ(t)<t for all t>0.

Let p be a w-distance on metric space (X,d), φΦ and ψΨ. A map T from X into itself is a (φ,ψ,p)-contractive map on X if for each x,yX, φp(Tx,Ty)ψφp(x,y).

The following lemmas are used in the next section.

Lemma 1.2 (see [<xref ref-type="bibr" rid="B4">3</xref>]).

If ψΨ, then limnψn(t)=0 for each t>0, and if φΦ,{an}[0,) and limnφ(an)=0, then limnan=0.

Lemma 1.3 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

Let (X,d) be a metric space and let p be a w-distance on X.

If {xn} is a sequence in X such that limnp(xn,x)=limnp(xn,y)=0, then x=y. In particular, if p(z,x)=p(z,y)=0, then x=y.

If p(xn,yn)αn  p(xn,y)βn for any n, where {αn} and {βn} are sequences in [0,) converging to 0, then {yn} converges to y.

Let p be a w-distance on metric space (X,d) and {xn} a sequence in X such that for each ɛ>0 there exist NɛN such that m>n>Nɛ implies p(xn,xm)<ɛ (or limm,np(xn,xm)=0), then {xn} is a Cauchy sequence.

Note that if p(a,b)=p(b,a)=0 and p(a,a)p(a,b)+p(b,a)=0, then p(a,a)=0 and, by Lemma 1.3, a=b.

In , Razani et al. proved a fixed point theorem for (φ,ψ,p)-contractive mappings, which is a new version of the main theorem in , by considering the concept of the w-distance.

The main aim of this paper is to present some generalization fixed point Theorems by Kada et al. , Hicks and Rhoades  and several other results with respect to (φ,ψ,p)-contractive maps on a complete metric space.

2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M89"><mml:mo stretchy="false">(</mml:mo><mml:mi>φ</mml:mi><mml:mo>,</mml:mo><mml:mi>ψ</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>-Contractive Maps

In the next theorem we state one of the main results of this paper generalizing Theorem 4 of . In what follows, we use φp to denote the composition of φ with p.

Theorem 2.1.

Let p be a w-distance on complete metric space (X,d),φΦ and ψΨ. Suppose T:XX is a map that satisfies φp(Tx,T2x)ψ(φp(x,Tx)), for each xX and that inf{p(x,y)+p(x,Tx):xX}>0 for every yX with yTy. Then there exists uX such that u=Tu. Moreover, if v=Tv, then p(v,v)=0.

Proof.

Fix xX. Set xn+1=Txn with x0=x. Then by (2.1) φp(xn,xn+1)ψφp(xn-1,xn)ψ2φp(xn-2,xn-1)  ψn(φp(x0,x1)), thus limnφp(xn,xn+1)=0 and Lemma 1.2 implies limnp(xn,xn+1)=0, and similarly limnp(xn+1,xn)=0.

Now we proof that {xn} is a Cauchy sequence. By triangle inequality, continuity of φ and (2.4), we have φp(xn,xn+2)ψφ[p(xn,xn+1)+p(xn+1,xn+2)]0, as n and so limnφp(xn,xn+2)=0 which concludes limnp(xn,xn+2)=0.

By induction, for any k>0 we have limnp(xn,xn+k)=0.

So, by Lemma 1.3, {xn} is a Cauchy sequence, and since X is complete, there exists uX such that xnu in  X.

Now we prove that u is a fixed point of T.

From (2.8), for each ɛ>0, there exists Nɛ such that n>Nɛ implies p(xNɛ,xn)<ɛ but xnu and p(x,·) is lower semicontinuous, thus p(xNɛ,u)limninf  p(xNɛ,xn)ɛ. Therefore, p(xNɛ,u)ɛ. Set ɛ=1/k,  Nɛ=nk and we have limkp(xnk,u)=0.

Now, assume that uTu. Then by hypothesis, we have0<inf{p(x,u)+p(x,Tx):xX}inf{p(xn,u)+p(xn,xn+1):nN}  0as n by (2.4) and (2.10). This is a contradiction. Hence u=Tu.

If v=Tv, we haveφp(v,v)=φp(Tv,T2v)ψφp(v,Tv)=ψφp(v,v)<φp(v,v).

This is a contradiction. So φp(v,v)=0, and by hypothesis p(v,v)=0.

Here we give a simple example illustrating Theorem 2.1. In this example, we will show that Theorem  4 in  cannot be applied.

Example 2.2.

Let X={(1/n)n}{0}, which is a complete metric space with usual metric d of reals. Moreover, by defining p(x,y)=y, p is a w-distance on (X,d). Let T:XX be a map as T(1/n)=1/(n+1), T0=0. Suppose φ(t)=t1/t is a continuous and strictly nondecreasing map and ψ(t)=(1/3)t, for any t>0. We have supxXp(Tx,T2x)p(x,Tx)=1, and so there is not any r[0,1) such that p(Tx,T2x)rp(x,Tx), and hence Theorem 4 in  dose not work. But φp(Tx,T2x)=p(Tx,T2x)1/p(Tx,T2x)=(1n+2)n+213(1n+1)n+1=13p(x,Tx)1/p(x,Tx)=ψφp(x,Tx), because for any n we have ((n+1)/(n+2))n+11/(n+2)1/3. Also for any n we have 1/nT(1/n). So for arbitrary n, inf{p(1/m,1/n)+p(1/m,1/(m+1)):m}=1/n>0, hence T is satisfied in Theorem 2.1. We note that 0 is a fixed point for T.

The next examples show the role of the conditions (2.1) and (2.2).

Example 2.3.

Let X=[-1,1], d(x,y)=|x-y|, and define p:XX by p(x,y)=|3x-3y|, where x,yX. Set ψ(t)=rt and φ(t)=t for all t[0,). Let us define T:XX by T0=1 and Tx=x/10 if x0. We have φp(T0,T20)=p(T0,T20)=p(1,110)=3-3103=13p(0,T0)=ψφp(0,T0).

If x0, then φp(Tx,T2x)=p(Tx,T2x)=p(x10,x100)=110|3x-3x10|13p(x,Tx)=ψφp(x,Tx) and hence (2.1) holds.

Now, we remark that 0T(0), and inf  nN  p(Tn(x),0)+p(Tn(x),TTn(x))=0for  every  xX. Thus, the condition (2.2) is not satisfied, and there is no zX with Tz=z. In this case we observe that Theorem 2.1 is invalid without condition (2.2).

Example 2.4.

Let X=[2,){0,1}, d(x,y)=|x-y|, x,yX, and set p=d. Let ψ,φ be as Example 2.3. Let us define T:XX by T0=1 and Tx=0 if x0. Clearly, T has no fixed point in X. Now, for each xX and that inf{d(x,y)+d(x,Tx):xX}>0 for every yX with yTy, so condition (2.2) is satisfied. But, for x=0, d(Tx,T2x)>rd(x,Tx) for any r[0,1). Hence, condition (2.1) dose not hold. We note that Theorem 2.1 dose not work without condition (2.1).

Suppose θ:++ is Lebesgue-integrable mapping which is summable and 0ɛθ(η)dη>0, for each ɛ>0. Now, in the next corollary, set φ(t)=0tθ(η)dη and ψ(t)=ct, where c[0,1[. Then, φΦ and ψΨ. Hence we can conclude the following corollary as a special case.

Corollary 2.5.

Let T be a selfmap of a complete metric space (X,d) satisfying 0d(Tx,T2x)θ(t)dtc0d(x,Tx)θ(t)dt for all xX. Suppose that inf{d(x,y)+d(x,Tx):xX}>0for  every  yX with yTy. Then there exists a uX such that Tu=u.

Note that Corollary 2.5 is invalid without condition (2.20). For example, take X={0}{1/2n:n1}, which is a complete metric space with usual metric d of reals. Define T:XX by T(0)=1/2 and T(1/2^n  )=1/2n-1   for n1. Set φ(t)1. It is easy to check that 0d(Tx,T2x)φ(t)dt(1/2)0d(x,Tx)φ(t)dt, for any xX; however, yTy for any yX and inf{d(x,y)+d(x,Tx):xX}=0. Clearly, T has got no fixed point in X.

Remark 2.6.

From Theorem 2.1, we can obtain Theorem 4 in  as a special case. For this, in the hypotheses of Theorem 2.1, set ψ(t)=rt and φ(t)=t for all t[0,).

Corollary 2.7.

Let p be a w-distance on complete metric space (X,d), φΦ and ψΨ. Suppose T is a continuous mapping for X into itself such that (2.1), is satisfied. Then there exists uX such that u=Tu. Moreover, if v=Tv, then p(v,v)=0.

Proof.

Assume that there exists yX with yTy and inf{p(x,y)+p(x,Tx):xX}=0. Then there exists a sequence {xn} such that p(xn,y)+p(xn,Txn)0 as n. Hence p(xn,y)0 and p(xn,Txn)0 as n. Lemma 1.3 implies that Txny as n. Now by assumption φp(Txn,T2xn)ψ(φp(xn,Txn)) and so φp(Txn,T2xn)0 as n. By Lemma 1.2, p(Txn,T2xn)0 as n. We also have p(xn,T2xn)p(xn,Txn)+p(Txn,T2xn), hence p(xn,T2xn)0 as n. By Lemma 1.3, we conclude that {T2xn} converges to y. Since T is continuous, we have Ty=T(limnTxn)=limnT2xn=y. This is a contradiction. Therefore, if yTy, then inf{p(x,y)+p(x,Tx):xX}>0. So, Theorem 2.1 gives desired result.

In Example 2.3,  T is satisfied in condition (2.1), but it is not continuous. So, the hypotheses in Corollary 2.7are not satisfied. We note that T has no fixed point.

It is an obvious fact that, if f:XX is a map which has a fixed point xX, then x is also a fixed point of fn for every natural number n. However, the converse is false. If a map satisfies F(f)=F(fn) for each n, where F(f) denotes a set of all fixed points of f, then it is said to have property P [7, 8]. The following theorem extends and improves Theorem 2 of .

Theorem 2.8.

Let (X,d) be a complete metric space with w-distance p on X. Suppose T:XX satisfies

φp(Tx,T2x)ψφp(x,Tx),xX, or

with strict inequality, ψ1 and for all xX, xTx. If F(T), then T has property P.

Proof.

We shall always assume that n>1, since the statement for n=1 is trivial. Let uF(Tn). Suppose that T satisfies (i). Then, φp(u,Tu)=φp(Tnu,TTnu)ψφp(Tn-1u,TTn-1u)ψnφp(u,Tu), and so p(u,Tu)=0. Now from φp(u,u)=ψφp(u,Tnu)i=0n-1ψφp(Tiu,Ti+1u)=0, we have p(u,u)=0. Hence, by Lemma 1.3, we have u=Tu, and uF(T). Suppose that T satisfies (ii). If Tu=u, then there is nothing to prove. Suppose, if possible, that Tuu. Then a repetition of the argument for case (i) leads to φp(u,Tu)<ψφp(u,Tu), that is a contradiction. Therefore, in all cases, u=Tu and F(Tn)=F(T).

The following theorem extends Theorem 2.1 of . A function G mapping X into the real is T-orbitally lower semicontinuous at z if {xn} is a sequence in O(x,) and xnz implies that G(p)liminfnG(xn).

Theorem 2.9.

Let (X,d) be a complete metric space with w-distance p on X. Suppose T:XX and there exists an x such that φp(Ty,T2y)ψφp(y,Ty),yO(x,). Then,

limTnx=z exists,

φp(Tnx,z)ψn1-ψφp(x,Tx)  for  n1,

p(z,Tz)=0 if and only if G(x)=p(x,Tx) is T-orbitally lower semicontinuous at z.

Proof.

Observe that (i) and (ii) are immediate from the proof of Theorem 2.1. We prove (iii). It is clear that p(z,Tz)=0 impling G(x) is T-orbitally lower semicontinuous at z.

xn=Tnxz and G is T-orbitally lower semicontinuous at x implies 0φp(z,Tz)=φG(z)liminfnφG(xn)=liminfnψφp(xn,Txn)liminfnψnφp(x,Tx)=0. So, p(z,Tz)=0.

The mapping T is orbitally lower semicontinuous at uX if limkTnkx=u implies that limkTnk+1x=Tu. In the following, we improve Theorem 2 of  that it is correct form Theorem 1 of .

Theorem 2.10.

Let p be a w-distance on complete metric space (X,d),φΦ and ψΨ. Suppose T:XX is orbitally lower semicontinuous map on X that satisfies φp(Tx,T2x)ψ(φp(x,Tx)) for each xX. Then there exists uX such that uF(T). Moreover, if v=Tv, then p(v,v)=0.

Proof.

Observe that the sequence {xn} is a Cauchy sequence immediate from the proof of Theorem 2.1 and so there exists a point u in X such that xnu as n. Since T is orbitally lower semicontinuous at u, we have p(u,Tu)liminfnp(xn,xn+1)=0. Now, we have φp(u,Tu)φ  liminfnp(xn,xn+1)=φ(0)=0, and so p(u,Tu)=0. Similarly, p(Tu,u)=0. Hence, uF(T). By Theorem 2.1 we can conclude that if v=Tv, then p(v,v)=0.

The following example shows that Theorem  2 in  cannot be applicable. So our generalization is useful.

Example 2.11.

Let =[0,) be a metric space with metric d defined by d(x,y)=(40/3)|x-y|,x,yX, which is complete. We define p:XX by p(x,y)=(1/3)|y|. Let φ be as defined before in Corollary 2.5 and ψ(t)=(1/10)t,t>0. Assume that T:XX by Tx=x/10 for any xX. We have, d(Tx,T2x)=(4/3)d(x,Tx),  xX, and so Theorem  2 in  dose not work. But φp(Tx,T2x)ψ(φp(x,Tx)) for each xX. Hence by Theorem 2.10 there exists a fixed point for T. We note that 0 is fixed point for T.

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M401"><mml:mo stretchy="false">(</mml:mo><mml:mi>φ</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>-Contractive Maps

In this section we obtain fixed points for (φ,k,p)-contractive maps (i.e., (φ,ψ,p)-contractive maps that ψ(t)=k for all t[0,), where k[0,1)).

In 1969, Kannan  proved the following fixed point theorem. Contractions are always continuous and Kannan maps are not necessarily continuous.

Theorem 3.1 (see [<xref ref-type="bibr" rid="B9">10</xref>]).

Let (X,d) be a complete metric space. Let T be a Kannan mapping on X, that is, there exists k[0,1/2) such that d(Tx,Ty)k(d(x,Tx)+d(y,Ty)) for all x,yX. Then, T has a unique fixed point in X. For each xX, the iterative sequence {Tnx}n1 converges to the fixed point.

In the next theorem, we generalize this theorem as follows.

Theorem 3.2.

Let (X,d) be a complete metric space. Let T be a (φ,k)-Kannan mapping on X, that is, there exists k[0,1/2) such that φd(Tx,Ty)k(φd(x,Tx)+φd(y,Ty)) for all x,yX. Then, T has a unique fixed point in X. For each xX, the iterative sequence {Tnx}n1 converges to the fixed point.

Proof.

Let xX and define xn+1=Tnx for any nN, and set r=k/(1-k). Then, r[0,1), φd(Tx,T2x)k(φd(x,Tx)+φd(Tx,T2x)) and so φd(Tx,T2x)rφd(x,Tx).

Then, from the proof of Theorem 2.1, limTnx=z exists. From (3.4), we have φd(Tnx,Tz)rφd(Tn-1x,z)rn1-rφd(x,Tx)for  n1. Thus, limTnx=Tz, and so z=Tz. Clearly, z is unique. This completes the proof.

The set of all subadditive functions φ in Φ is denoted by Φ. In the following theorems, we generalize Theorems 3.4 and 3.5 due to Suzuki and Takahashi .

Theorem 3.3.

Let p be a w-distance on complete metric space (X,d),φΦ and T be a selfmap. Suppose there exists k[0,1/2) such that

φp(Tx,T2x)kφp(x,T2x) for each xX,

inf{p(x,z)+p(x,Tx):xX}>0 for every zX with zTz.

Then T has a fixed point in X. Moreover, if v is a fixed point of T, then p(v,v)=0.

Proof.

Fix xX. Define x0=x and xn=Tnx0 for every n. Put r=k/(1-k). Then, 0r<1. By hypothesis, since φΦ, we have φp(xn,xn+1)kφp(xn-1,xn+1)kφp(xn-1,xn)+kφp(xn,xn+1), for all n. It follows that φp(xn,xn+1)rφp(xn-1,xn)rnφp(x0,x1), for all n. Using the similar argument as in the proof of Theorem 2.1, we can prove that the sequence {un} is Cauchy and so there exists uX such that xnu as n. Also, we have uF(T). Since φp(v,v)=φp(Tv,T2v)kφp(v,T2v)=kφp(v,v), we have φp(v,v)=0 and so p(v,v)=0. The proof is completed.

Corollary 3.4.

Let p be a w-distance on complete metric space (X,d),φΦ and let T be a continuous map. Suppose there exists k[0,1/2) such that φp(Tx,T2x)kφp(x,T2x), for each xX.

Then T has a fixed point in X. Moreover, if v is a fixed point of T, then p(v,v)=0.

Proof.

It suffices to show that inf{p(x,z)+p(x,Tx):xX}>0 for every uX with uTu. Assume that there exists uX with uTu and inf{p(x,u)+p(x,Tx):xX}=0. Then there exists a sequence {xn} in X such that limn[p(xn,u)+p(xn,Txn)]=0. It follows that p(xn,u)0 and p(xn,Txn)0 as n. Hence, Txnu. On the other hand, since φΦ and (3.9), we have φp(xn,T2xn)φp(xn,Txn)+φp(Txn,T2xn)φp(xn,Txn)+kφp(xn,T2xn), and hence φp(xn,T2xn)11-kφp(xn,Txn), for all n. Thus, p(xn,T2xn)0 as n. Therefore, T2xnu. Since T:XX is continuous, we have T(u)=T(limnTxn)=limnT2xn=u, which is a contradiction. Therefore, using Theorem 3.3, p(v,v)=0. This completes the proof.

Question 1.

Can we generalize Theorems 3.2, 3.3, and Corollary 3.4 for (φ,ψ,p)-contractive maps?

BranciariA.A fixed point theorem for mappings satisfying a general contractive condition of integral typeInternational Journal of Mathematics and Mathematical Sciences2002299531536190034410.1155/S0161171202007524ZBL0993.54040KadaO.SuzukiT.TakahashiW.Nonconvex minimization theorems and fixed point theorems in complete metric spacesMathematica Japonica19964423813911416281ZBL0897.54029RazaniA.Mazlumi NezhadZ.BoujaryM.A fixed point theorem for w-distanceApplied Sciences2009111141172534062SuzukiT.TakahashiW.Fixed point theorems and characterizations of metric completenessTopological Methods in Nonlinear Analysis1996823713821483635ZBL0902.47050DuW.-S.Fixed point theorems for generalized Hausdorff metricsInternational Mathematical Forum2008321–24101110222415182ZBL1158.54020HicksT. L.RhoadesB. E.A Banach type fixed-point theoremMathematica Japonica1979/80243327330550217RhoadesB. E.AbbasM.Maps satisfying generalized contractive conditions of integral type for which F(T)=F(Tn)International Journal of Pure and Applied Mathematics20084522252312421863ZBL1161.54024JeongG. S.RhoadesB. E.Maps for which F(T)=F(Tn)Fixed Point Theory and Applications2005687131LakzianH.RhoadesB. E.Maps satisfying generalized contractive contractions of integral type for which F(T)=F(Tn)submitted to International Journal of Pure and Applied Mathematical SciencesKannanR.Some results on fixed points. IIThe American Mathematical Monthly196976405408025783810.2307/2316437ZBL0179.28203