We prove some fixed point theorems for a T-Hardy-Rogers contraction
in the setting of partially ordered partial metric spaces. We apply our results to study
periodic point problems for such mappings. We also provide examples to illustrate the
results presented herein.

1. Introduction and Preliminaries

The notion of a partial metric space was introduced by Matthews in [1]. In partial metric spaces, the distance of a point in the self may not be zero. After the definition of a partial metric space, Matthews proved the partial metric version of Banach fixed point theorem. A motivation behind introducing the concept of a partial metric was to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle, more suitable in this context [1]. Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions (e.g., [2–21], [22]). Existence of fixed points in partially ordered metric spaces has been initiated in 2004 by Ran and Reurings [23]. Subsequently, several interesting and valuable results have appeared in this direction [14]. The aim of this paper is to study the necessary conditions for existence of fixed point of mapping satisfying T-Hardy-Rogers conditions in the framework of partially ordered partial metric spaces. Our results extend and strengthen various known results [8, 24]. In the sequel, the letters ℝ, ℝ+, and ℕ will denote the set of real numbers, the set of nonnegative real numbers, and the set of nonnegative integer numbers, respectively. The usual order on ℝ (resp., on ℝ+) will be indistinctly denoted by ≤ or by ≥.

Consistent with [1, 8] (see [25–29]) the following definitions and results will be needed in the sequel.

Definition 1.1 (see [<xref ref-type="bibr" rid="B21">1</xref>]).

A partial metric on a nonempty set X is a mapping p:X×X→ℝ+ such that for all x,y,z∈X,

x=y⇔p(x,x)=p(x,y)=p(y,y),

p(x,x)≤p(x,y),

p(x,y)=p(y,x),

p(x,y)≤p(x,z)+p(z,y)-p(z,z).

A partial metric space is a pair (X,p) such that X is a nonempty set and p is a partial metric on X. If p(x,y)=0, then (p1) and (p2) imply that x=y. But converse does not hold always. A trivial example of a partial metric space is the pair (ℝ+,p), where p:ℝ+×ℝ+→ℝ+ is defined as p(x,y)=max{x,y}. Each partial metric p on X generates a T0 topology τp on X which has as a base the family open p-balls {Bp(x,ε):x∈X,ε>0}, where Bp(x,ε)={y∈X:p(x,y)<p(x,x)+ε}.

On a partial metric space the concepts of convergence, Cauchy sequence, completeness, and continuity are defined as follows.

Definition 1.2 (see [<xref ref-type="bibr" rid="B21">1</xref>]).

Let (X,p) be a partial metric space and let {xn} be a sequence in X. Then (i) {xn} converges to a point x∈X if and only if p(x,x)=limn→∞p(x,xn) (we may still write this as limn→∞xn=v or xn→v); (ii) {xn} is called a Cauchy sequence if there exists (and is finite) limn,m→∞p(xn,xm).

Definition 1.3 (see [<xref ref-type="bibr" rid="B21">1</xref>]).

A partial metric space (X,p) is said to be complete if every Cauchy sequence {xn} in X converges to a point x∈X, such that p(x,x)=limn,m→∞p(xn,xm). If p is a partial metric on X, then the function ps:X×X→R+ given by ps(x,y)=2p(x,y)-p(x,x)-p(y,y) is a metric on X.

Lemma 1.4 (see [<xref ref-type="bibr" rid="B21">1</xref>, <xref ref-type="bibr" rid="B25">20</xref>]).

Let (X,p) be a partial metric space. Then

{xn} is a Cauchy sequence in (X,p) if and only if it is a Cauchy sequence in the metric space (X,ps);

(X,p) is complete if and only if the metric space (X,ps) is complete. Furthermore, limn→∞ps(xn,x)=0 if and only if
(1.1)p(x,x)=limn→∞p(xn,x)=limn,m→∞p(xn,xm).

Remark 1.5.

(1) (see [19]) Clearly, a limit of a sequence in a partial metric space does not need to be unique. Moreover, the function p(·,·) does not need to be continuous in the sense that xn→x and yn→y implies p(xn,yn)→p(x,y). For example, if X=[0,+∞) and p(x,y)=max{x,y} for x,y∈X, then for {xn}={1},p(xn,x)=x=p(x,x) for each x≥1 and so, for example, xn→2 and xn→3 when n→∞.

(2) (see [7]) However, if p(xn,x)→p(x,x)=0 then p(xn,y)→p(x,y) for all y∈X.

Definition 1.6 (see [<xref ref-type="bibr" rid="B27">30</xref>]).

Suppose that (X,p) is a partial metric space. Denote τ(p) its topology. We say T:(X,p)→(X,p) is continuous if both T:(X,τ(p))→(X,τ(p)) and T:(X,τ(ps))→(X2,τ(ps)) are continuous.

Remark 1.7.

It is worth to notice that the notions p-continuous and ps-continuous of any function in the context of partial metric spaces are incomparable, in general. Indeed, if X=[0,+∞), p(x,y)=max{x,y}, ps(x,y)=|x-y|, f0=1, and fx=x2 for all x>0 and gx=|sinx|, then f is a p-continuous and ps-discontinuous at point x=0; while g is a p-discontinuous and ps-continuous at x=π.

According to [31], we state the following definition.

Definition 1.8.

Let (X,p) be a partial metric space. A mapping T:X→X is said to be

sequentially convergent if for any sequence {yn} in X such that {Tyn} is convergent in (X,ps) implies that {yn} is convergent in (X,ps),

subsequentially convergent if for any sequence {yn} in X such that {Tyn} is convergent in (X,ps) implies that {yn} has a convergent subsequence in (X,ps).

Consistent with [24, 31] we define a T-Hardy-Rogers contraction in the framework of partial metric spaces.

Definition 1.9.

Let (X,p) be a partial metric space and T,f:X→X be two mappings. A mapping f is said to be a T-Hardy-Rogers contraction if there exist ai≥0, i=1,…,5 with a1+a2+a3+a4+a5<1 such that for all x,y∈X(1.2)p(Tfx,Tfy)≤a1p(Tx,Ty)+a2p(Tx,Tfx)+a3p(Ty,Tfy)+a4p(Tx,Tfy)+a5p(Ty,Tfx).

Putting a1=a4=a5=0 and a2=a3≠0, (resp., a1=a2=a3=0 and a4=a5≠0) in the previous definition, then the inequality (1.2) is said a T-Kannan (resp., T-Chatterjea) type contraction. Also, if a4=a5=0 and a1,a2,a3≠0, (1.2) is said the T-Reich type contraction.

Definition 1.10.

Let X be a nonempty set. Then (X,p,⪯) is called a partially ordered partial metric space if and only if (i) p is a partial metric on X and (ii) ⪯ is a partial order on X.

Let (X,p) be a partial metric space endowed with a partial order ⪯ and let f:X→X be a given mapping. We define sets Δ,Δ1⊂X×X by
(1.3)Δ={(x,y)∈X×X:x⪯yory⪯x},Δ1={(x,x)∈X×X:x⪯fxorfx⪯x}.

A point x∈X is called a fixed point of mapping f:X→X if x=fx. The set of all fixed points of the mapping f is denoted by Ff.

2. Fixed Point Results

In this section, we obtain fixed point results for a mapping satisfying a T-Hardy-Rogers contractive condition defined on a partially ordered partial metric space which is complete.

We start with the following result.

Theorem 2.1.

Let (X,⪯,p) be an partially ordered partial metric space which is complete. Let T:X→X be a continuous, injective mapping and f:X→X a nondecreasing T-Hardy-Rogers contraction for all (x,y)∈Δ. If there exists x0∈X with x0⪯fx0, and one of the following two conditions is satisfied

f is a continuous self-map on X;

for any nondecreasing sequence {xn} in (X,⪯) with limn→∞ps(z,xn)=0 it follows xn⪯z for all n∈ℕ;

then Ff≠ϕ provided that T is subsequentially or sequentially convergent. Moreover, f has a unique fixed point if Ff×Ff⊂Δ.

Proof.

As f is nondecreasing, therefore by given assumption, we have
(2.1)x1=fx0⪯f2x0⪯⋯⪯fnx0⪯fn+1x0⪯⋯
Define a sequence {xn} in X with xn=fnx0 and so xn+1=fxn for n∈ℕ. Since (xn-1,xn)∈Δ therefore by replacing x by xn-1 and y by xn in (1.2), we have
(2.2)p(Txn,Txn+1)=p(Tfxn-1,Tfxn)≤a1p(Txn-1,Txn)+a2p(Txn-1,Tfxn-1)+a3p(Txn,Tfxn)+a4p(Txn-1,Tfxn)+a5p(Txn,Tfxn-1)=a1p(Txn-1,Txn)+a2p(Txn-1,Txn)+a3p(Txn,Txn+1)+a4p(Txn-1,Txn+1)+a5p(Txn,Txn)≤(a1+a2)p(Txn-1,Txn)+a3p(Txn,Txn+1)+a4[p(Txn-1,Txn)+p(Txn,Txn+1)-p(Txn,Txn)]+a5p(Txn,Txn)=(a1+a2+a4)p(Txn-1,Txn)+(a3+a4)p(Txn,Txn+1)+(a5-a4)p(Txn,Txn),
that is,
(2.3)(1-a3-a4)p(Txn,Txn+1)≤(a1+a2+a4)p(Txn-1,Txn)+(a5-a4)p(Txn,Txn).
Similarly, replacing x by xn and y by xn-1 in (1.2), we obtain
(2.4)(1-a2-a5)p(Txn,Txn+1)≤(a1+a3+a5)p(Txn-1,Txn)+(a4-a5)p(Txn,Txn).
Summing (2.3) and (2.4), we obtain p(Txn,Txn+1)≤δp(Txn-1,Txn), where δ=(2a1+a2+a3+a4+a5)/(2-a2-a3-a4-a5). Obviously 0≤δ<1. Therefore, for all n≥1,
(2.5)p(Txn,Txn+1)≤δp(Txn-1,Txn)≤⋯≤δnp(Tx0,Tx1).
Now, for any m∈N with m>n, we have
(2.6)p(Txn,Txm)≤p(Txn,Txn+1)+p(Txn+1,Txn+2)+⋯+p(Txm-1,Txm)≤(δn+δn+1+⋯+δm-1)p(Tx0,Tx1)≤δn1-δp(Tx0,Tx1),
which implies that p(Txn,Txm)→0 as n,m→∞. Hence {Txn} is a Cauchy sequence in (X,p) and in (X,ps). Since (X,p) is complete, therefore from Lemma 1.4, (X,ps) is a complete metric space. Hence {Txn} converges to some v∈X with respect to the metric ps, that is,
(2.7)limn→∞ps(Txn,v)=0,
or equivalently,
(2.8)p(v,v)=limn→∞p(Txn,v)=limn,m→∞p(Txn,Txm)=0.
Suppose that T is subsequentially convergent, therefore convergence of {Txn} in (X,ps) implies that {xn} has a convergent subsequence {xni} in (X,ps). So
(2.9)limi→∞ps(xni,u)=0,
for some u∈X. As T is continuous, so (2.9) and Definition 1.6. imply that limi→∞ps(Txni,Tu)=0. From (2.7) and by the uniqueness of the limit in metric space (X,ps), we obtain Tu=v. Consequently,
(2.10)0=p(Tu,Tu)=limi→∞p(Txni,Tu)=limi,j→∞p(Txni,Txnj).

If f is a continuous self-map on X, then fxni→fu and Tfxni→Tfu as i→∞. Since Txni→Tu as i→∞, we obtain that Tfu=Tu. As T is injective, so we have fu=u.

If f is not continuous then by given assumption we have xn⪯u for all n∈ℕ. Thus for a subsequence {xni} of {xn} we have xni⪯u and (xni,u)∈Δ. Now,
(2.11)p(Tfu,Tu)≤p(Tfu,Tfxni)+p(Tfxni,Tu)-p(Tfxni,Tfxni)≤a1p(Txni,Tu)+a2p(Txni,Tfxni)+a3p(Tu,Tfu)+a4p(Txni,Tfu)+a5p(Tu,Tfxni)+p(Tfxni,Tu)-p(Tfxni,Tfxni)=a1p(Txni,Tu)+a2p(Txni,Txni+1)+a3p(Tu,Tfu)+a4p(Txni,Tfu)+a5p(Tu,Txni+1)+p(Txni+1,Tu)-p(Txni+1,Txni+1)≤a1p(Txni,Tu)+a2p(Txni,Txni+1)+a3p(Tu,Tfu)+a4[p(Txni,Tu)+p(Tu,Tfu)-p(Tu,Tu)]+a5p(Tu,Txni+1)+p(Txni+1,Tu)-p(Txni+1,Txni+1).

On taking limit as i→∞ and applying Remark 1.5. (2) we get
(2.12)p(Tu,Tfu)≤a1p(Tu,Tu)+a2p(Tu,Tu)+a3p(Tu,Tfu)+a4[p(Tu,Tu)+p(Tu,Tfu)-p(Tu,Tu)]+a5p(Tu,Tu)+p(Tu,Tu)-p(Tu,Tu)=(a1+a2+a5)p(Tu,Tu)+(a3+a4)p(Tu,Tfu)≤(a1+a2+a3+a4+a5)p(Tu,Tfu)<p(Tu,Tfu),
which implies that p(Tu,Tfu)=0, and so Tu=Tfu. Now injectivity of T gives u=fu. Following similar arguments to those given above, the result holds when T is sequentially convergent.

Suppose that Ff×Ff⊂Δ. Let w be a fixed point of f. As Ff×Ff⊂Δ, therefore (u,w)∈Δ. From (1.2), we have
(2.13)p(Tu,Tw)=p(Tfu,Tfw)≤a1p(Tu,Tw)+a2p(Tu,Tfu)+a3p(Tw,Tfw)+a4p(Tu,Tfw)+a5p(Tw,Tfu)=a1p(Tu,Tw)+a2p(Tu,Tu)+a3p(Tw,Tw)+a4p(Tu,Tw)+a5p(Tw,Tu)≤(a1+a2+a3+a3+a4+a5)p(Tu,Tw)<p(Tu,Tw),
and hence p(Tu,Tw)=0, which further implies that u=w as T is injective.

Example 2.2.

Let X=[0,1] be endowed with usual order and let p be the complete partial metric on X defined by p(x,y)=max{x,y} for all x,y∈X. Let T,f:X→X be defined by Tx=4x/5 and fx=x/4. Note that Δ=X×X. For any (x,y)∈Δ, we have
(2.14)p(Tfx,Tfy)=max{x5,y5}=15max{x,y}≤1235max{x,y}≤27max{4x5,4y5}+17max{4x5,x5}+17max{4y5,y5}+17max{4x5,y5}+17max{4y5,x5}=a1p(Tx,Ty)+a2p(Tx,Tfx)+a3p(Ty,Tfy)+a4p(Tx,Tfy)+a5p(Ty,Tfx).
Therefore, f is a T-Hardly-Rogers contraction with a1=2/7, a2=a3=a4=a5=1/7. Obviously, T is continuous and sequentially convergent. Thus, all the conditions of Theorem 2.1 are satisfied. Moreover, 0 is the unique fixed point of f.

Example 2.3.

Let X=[0,∞) be endowed with usual order and let p be a partial metric on X defined by p(x,y)=max{x,y} for all x,y∈X. Define T,f:X→X by Tx=x2andfx=x/3. Note that Δ=X×X. For any (x,y)∈Δ, we have
(2.15)p(Tfx,Tfy)=max{x29,y29}=19max{x2,y2}≤16max{x2,y2}=a1p(Tx,Ty)≤a1p(Tx,Ty)+a2p(Tx,Tfx)+a3p(Ty,Tfy)+a4p(Tx,Tfy)+a5p(Ty,Tfx).
Therefore, f is a T-Hardly-Rogers contraction with a1=a2=a3=a4=a5=1/6. Also, T is continuous and sequentially convergent. Thus all the conditions of Theorem 2.1 are satisfied. Moreover, 0 is the unique fixed point of f.

Taking Tx=x in (1.2) and Theorem 2.1, we get the Hardy-Rogers type [32] (and so the Kannan, Chatterjea, and Reich) fixed point theorem on partially ordered partial metric spaces.

Corollary 2.4.

Let (X,⪯,p) be a partially ordered partial metric space which is complete. Let f:X→X be a nondecreasing mapping such that for all (x,y)∈Δ, we have
(2.16)p(fx,fy)≤a1p(x,y)+a2p(x,fx)+a3p(y,fy)+a4p(x,fy)+a5p(y,fx),
where ai≥0, i=1,…,5 with a1+a2+a3+a4+a5<1. If there exists x0∈X with x0⪯fx0, and one of the following two conditions is satisfied.

f is a continuous self map on X;

for any nondecreasing sequence {xn} in (X,⪯) with limn→∞ps(z,xn)=0 it follows xn⪯z for all n∈ℕ; then Ff≠ϕ. Moreover, f has a unique fixed point if Ff×Ff⊂Δ.

Remark 2.5.

Corollary 2.4 corresponds to Theorem 2 of Altun et al. [8] in partially ordered partial metric spaces. For particular choices of the coefficients (ai)i=1,…,5 in Theorem 2.1, we obtain the T-Kannan, T-Chatterjea, and T-Reich type fixed point theorems. Also, Theorem 2.1 is an extension of Theorem 2.1 of Filipović et al. [24] from the cone metric spaces to partial metric spaces.

3. Periodic Point Results

Let f:X→X. If the map f satisfies Ff=Ffn for each n∈ℕ, then it is said to have the property P, for more details see [33].

Definition 3.1.

Let (X,⪯) be a partially ordered set. A mapping f is called (1) a dominating map on X if x⪯fx for each x in X and (2) a dominated map on X if fx⪯x for each x in X.

Example 3.2.

Let X=[0,1] be endowed with usual ordering. Let f:X→X defined by fx=x1/3, then x≤x1/3=fx for all x∈X. Thus f is a dominating map.

Example 3.3.

Let X=[0,∞) be endowed with usual ordering. Let f:X→X defined by fx=xn for x∈[0,1) and let fx=xn for x∈[1,∞), for any n∈ℕ, then for all x∈X, x≤fx that is f is the dominating map. Note that Δ1≠ϕ if f is a dominating or a dominated mapping.

We have the following result.

Theorem 3.4.

Let (X,⪯,p) be a partially ordered partial metric space which is complete. Let T:X→X be an injective mapping and f:X→X a nondecreasing such that for all (x,x)∈Δ1, we have
(3.1)p(Tfx,Tf2x)≤λp(Tx,Tfx),
for some λ∈[0,1) and for all x∈X, x≠fx. Then f has the property P provided that Ff is nonempty and f is a dominating map on Ffn.

Proof.

Let u∈Ffn for some n>1. Now we show that u=fu. Since f is dominating on Ffn, therefore u⪯fu which further implies that fn-1u⪯fnu as f is nondecreasing. Hence (fn-1u,fn-1u)∈Δ1. Now by using (3.1), we have
(3.2)p(Tu,Tfu)=p(Tffn-1u,Tf2fn-1u)≤λp(Tfn-1u,Tfnu)=λp(Tffn-2u,Tf2fn-2u).
Repeating the above process, we get
(3.3)p(Tu,Tfu)≤λnp(Tu,Tfu).
Taking limit as n→∞, we obtain p(Tu,Tfu)=0 and Tu=Tfu. As T is injective, so u=fu, that is, u∈Ff.

Theorem 3.5.

Let (X,⪯,p) be a partially ordered partial metric space which is complete. let T,f:X→X be mappings satisfy the condition of Theorem 2.1. If f is dominating on X, then f has the property P.

Proof.

From Theorem 2.1, Ff≠∅. We will prove that (3.1) is satisfied for all (x,x)∈Δ1. Indeed, f is a dominating map so that x⪯fx and also f is nondecreasing so that fx⪯f2x and hence (x,fx)∈Δ. Now from (1.2),
(3.4)p(Tfx,Tf2x)=p(Tfx,Tffx)≤a1p(Tx,Tfx)+a2p(Tx,Tfx)+a3p(Tfx,Tf2x)+a4p(Tx,Tf2x)+a5p(Tfx,Tfx)≤(a1+a2)p(Tx,Tfx)+a3p(Tfx,Tf2x)+a4(p(Tx,Tfx)+p(Tfx,Tf2x)-p(Tfx,Tfx))+a5p(Tfx,Tfx),
that is,
(3.5)(1-a3-a4)p(Tfx,Tf2x)≤(a1+a2+a4)p(Tx,Tfx)+(a5-a4)p(Tfx,Tfx).
Again by using (1.2), we have
(3.6)p(Tf2x,Tfx)=p(Tffx,Tfx)≤a1p(Tfx,Tx)+a2p(Tfx,Tf2x)+a3p(Tx,Tfx)+a4p(Tfx,Tfx)+a5p(Tx,Tf2x)≤(a1+a3)p(Tx,Tfx)+a2p(Tfx,Tf2x)+a4p(Tfx,Tfx)+a5(p(Tx,Tfx)+p(Tfx,Tf2x)-p(Tfx,Tfx)),
which implies that
(3.7)(1-a2-a5)p(Tfx,Tf2x)≤(a1+a3+a5)p(Tx,Tfx)+(a4-a5)p(Tfx,Tfx).
Summing (3.5) and (3.7) implies p(Tfx,Tf2x)≤λp(Tx,Tfx),λ=(2a1+a2+a3+a4+a5)/(2-a2-a3-a4-a5). Obviously, λ∈[0,1). By Theorem 3.4, f has the property P.

Acknowledgment

S. Radenović is thankful to The Ministry of Science and Technology Development of Serbia.

MatthewsS. G.Partial metric topologyAbbasM.NazirT.RomagueraS.Fixed point results for generalized cyclic contraction mappings in partial metric spacesAbdeljawadT.Fixed points for generalized weakly contractive mappings in partial metric spacesAbdeljawadT.KarapinarE.TaşK.A generalized contraction principle with control functions on partial metric spacesAgarwalP. R.AlghamdiA. M.ShahzadN.Fixed point theory
for cyclic generalized contractions in partial metric spacesAltunI.ErduranA.Fixed point theorems for monotone mappings on partial metric spacesAbdeljawadT.KarapnarE.TaşK.Existence and uniqueness of a common fixed point on partial metric spacesAltunI.SolaF.SimsekH.Generalized contractions on partial metric spacesAydiH.Some coupled fixed point results on partial metric spacesAydiH.Fixed point results for weakly contractive mappings in ordered partial metric
spacesAydiH.Fixed point theorems for generalized weakly contractive condition in ordered
partial metric spacesĆirićL.SametB.AydiH.VetroC.Common fixed points of generalized contractions on partial metric spaces and an applicationDi BariC.VetroP.Fixed points for weak ϕ-contractions on partial
metric spacesDukićD.KadelburgZ.RadenovićS.Fixed point of Geraghty-type mappings in various generalized metric spacesIlićD.PavlovićV.RakočevićV.Some new extensions of Banach's contractions principle in partial metric spacesIlićD.PavlovićV.RakočevićV.Extensions of Zamfirescu theorem to partial metric spacesKarapnarE.ErhanI. M.Fixed point theorems for operators on partial metric spacesHussainN.KadelburgZ.RadenovićS.Comparison functions and fixed point results in partial metric spacesNashineH. K.KadelburgZ.RadenovićS.Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spacesMathematical and Computer Modelling. In press10.1016/j.mcm.2011.12.019OltraS.ValeroO.Banach's fixed point theorem for partial metric spacesValeroO.On Banach fixed point theorems for partial metric spacesSametB.RajovićM.LazovićR.StoiljkovićR.Common fixed point results for nonlinear contractions in ordered partial metric spacesRanA. C. M.ReuringsM. C. B.A fixed point theorem in partially ordered sets and some applications to matrix equationsFilipovićM.PaunovićL.RadenovićS.RajovićM.Remarks on ‘Cone metric spaces and fixed point theorems of T-Kannan and T-Chatterjea contractive mappings’MatthewsS. G.Partial metric topologyRomagueraS.A Kirk type characterization of completeness for partial metric spacesRomagueraS.ValeroO.A quantitative computational model for complete partial metric spaces via formal ballsOltraS.RomagueraS.Saánchez-PérezE. A.Bicompleting weightable quasi-metric spaces and partial metric spacesPaesanoD.VetroP.Suzuki's type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spacesO'NeillS. J.Two topologies are better than one1995Coventry, UKUniversity of WarwickMoralesJ. R.RojasE.Cone metric spaces and fixed point theorems of T-Kannan contractive mappingsHardyG. E.RogersT. D.A generalization of a fixed point theorem of ReichJeongG. S.RhoadesB. E.Maps for which F(T)=F(Tn)