Property P in G-Metric Spaces

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Volume 2010, Article ID 401684,12pages doi:10.1155/2010/401684

Research Article

Property P in G-Metric Spaces

Renu Chugh,

¹

Tamanna Kadian,

¹

Anju Rani,

¹

and B. E. Rhoades

²

1Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, India

2Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

Correspondence should be addressed to Renu Chugh,[email protected] Received 19 December 2009; Revised 1 May 2010; Accepted 13 May 2010 Academic Editor: Brailey Sims

Copyrightq2010 Renu Chugh 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 two general fixed theorems for maps inG-metric spaces and then show that these maps satisfy propertyP.

1. Introduction

Metric fixed point theory is an important mathematical discipline because of its applications in areas such as variational and linear inequalities, optimization, and approximation theory.

Generalizations of metric spaces were proposed by Gahler 1, 2 called 2-metric spaces and Dhage 3, 4 called D-metric spaces. Hsiao 5 showed that, for every contractive definition, with xn : Tⁿx0, every orbit is linearly dependent, thus rendering fixed point theorems in such spaces trivial. Unfortunately, it was shown that certain theorems involving Dhage’s D-metric spaces are flawed, and most of the results claimed by Dhage and others are invalid. These errors were pointed out by Mustafa and Sims in 6, among others. They also introduced a valid generalized metric space structure, which they call G-metric spaces. Some other papers dealing with G-metric spaces are those in 7–11.

LetT be a self-map of a complete metric spaceX, dwith a nonempty fixed point set FT. ThenT is said to satisfy property P if FT FTⁿfor eachn ∈ N. An interesting fact about maps satisfying propertyP is that they have no nontrivial periodic points. Papers dealing with propertyPare those in12–14.

In this paper, we will prove two general fixed point theorems for maps inG-metric spaces and then show that these maps satisfy propertyP. Throughout this paper, we mean byNthe set of all natural numbers.

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Definition 1.1see8. LetXbe a nonempty set, and letG: X×X×X → Rbe a function satisfying the following axioms:

G1Gx, y, z 0 ifxyz,

G20< Gx, x, yfor allx, y∈Xwithx /y,

G3Gx, x, y≤Gx, y, z, for allx, y, z∈X, withz /y,

G4Gx, y, z Gx, z, y Gy, z, x · · · symmetry in all three variables, G5Gx, y, z≤Gx, a, a Ga, y, z, for allx, y, z, a∈Xrectangle inequality.

Then the functionGis called a generalized metric, or, more specifically, aG-metric onX, and the pairX, Gis called aG-metric space.

Definition 1.2 see 8. Let X, G and X, G be G-metric spaces and let f : X, G → X, G be a function, thenf is said to beG-continuous at a point a ∈ X; if given ε > 0, there existsδ >0 such thatx, y ∈ X;Ga, x, y< δ implies thatGfa, fx, fy < ε. A functionfisG-continuous onXif and only if it isG-continuous at alla∈X.

Proposition 1.3see8. LetX, G,X, GbeG-metric spaces, then a functionf : X → Xis G-continuous at a pointx ∈Xif and only if it isG-sequentially continuous atx; that is, whenever {xn}isG-convergent tox,{fxn}isG-convergent tofx.

Definition 1.4see8. LetX, Gbe aG-metric space, and let{xn}be a sequence of points ofX; therefore, we say that{xn}isG-convergent toxif lim_{n,m→ ∞}Gx, xn, xm 0; that is, for anyε >0, there existsN∈Nsuch thatGx, xn, xm< ε, for alln, m≥N. We callxthe limit of the sequence and writexn → xor limxnx.

Proposition 1.5see8. LetX, Gbe aG-metric space. Then the following are equivalent:

1{xn}isG-convergent tox, 2Gxn, xn, x → 0, asn → ∞, 3Gxn, x, x → 0, asn → ∞, 4Gxm, xn, x → 0, asm, n → ∞.

Definition 1.6see8. LetX, Gbe aG-metric space. A sequence{xn}is calledG-Cauchy if, for eachε > 0, there is N ∈ N such thatGxn, xm, xl < ε, for alln,m, l ≥ N; that is, Gxn, xm, xl → 0 asn, m, l → ∞.

Proposition 1.7see8. In aG-metric spaceX, Gthe following are equivalent 1The sequence{xn}isG-Cauchy.

2For everyε >0, there existsN∈Nsuch thatGxn, xm, xm< ε, for all n, m≥N.

Proposition 1.8see8. LetX, Gbe aG-metric space. Then the functionGx, y, zis jointly continuous in all three of its variables.

Definition 1.9see8. AG-metric spaceX, Gis called a symmetricG-metric space if G

x, y, y G

y, x, x

∀x, y∈X. 1.1

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Proposition 1.10see8. EveryG-metric spaceX, Gdefines a metric spaceX, dGby

dG

x, y G

x, y, y G

y, x, x

∀x, y∈X. 1.2

Note that, ifX, Gis a symmetricG-metric space, then

dG

x, y 2G

x, y, y

, ∀x, y∈X. 1.3

However, ifX, Gis not symmetric, then it holds by theG-metric properties that

3 2G

x, y, y

≤dG

x, y

≤3G x, y, y

, ∀x, y∈X. 1.4

In general, these inequalities cannot be improved.

Proposition 1.11 see 8. A G-metric space X, G is G-complete if and only ifX, dG is a complete metric space.

Proposition 1.12see8. LetX, Gbe aG-metric space. Then, for anyx, y, z, a∈X, it follows that

1ifGx, y, z 0, thenxyz, 2Gx, y, z≤Gx, x, y Gx, x, z, 3Gx, y, y≤2Gy, x, x,

4Gx, y, z≤Gx, a, z Ga, y, z,

5Gx, y, z≤2/3{Gx, a, a Gy, a, a Gz, a, a}.

Theorem 1.13see15. LetTbe a self-map of a metric spaceXsuch thatXisT-orbitally complete.

Suppose thatT satisfies

d

Tx, Ty

≤kmax d

x, y

, dx, Tx, d y, Ty

, d x, Ty

, d y, Tx

, 1.5

wherekis a real number satisfying 0≤k <1. ThenThas a unique fixed pointu∈X. Moreover, for eachx∈X, limTⁿxuand

dTⁿx, u≤ qⁿ

1−qdx, Tx. 1.6

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2. Fixed Point Theorems

Theorem 2.1. LetX, Gbe a completeG-metric space, and letTbe a self-mapofXsatisfying, for all x, y, z∈X,

G

Tx, Ty, Tz

≤kmax

G x, y, z

, Gx, Tx, Tx, G

y, Ty, Ty

, Gz, Tz, Tz, G

x, Ty, Ty

Gz, Tx, Tx

2 ,

G

x, Ty, Ty G

y, Tx, Tx

2 ,

G

y, Tz, Tz G

z, Ty, Ty

2 ,Gx, Tz, Tz Gz, Tx, Tx

2 ,

2.1 where k is a constant satisfying 0 ≤ k < 1. ThenT has a unique fixed point (sayp) andT isG- continuous atp.

Proof. Letx0 ∈Xand define the sequence{xn}byxn Tⁿx0. We may assume thatxn/xn1

for eachn∈N∪ {0}. For, if there exists anNsuch thatxNx_N1, thenxNis a fixed point of T.

From2.1, withxxn−1,yzxn, Gxn, xn1, xn1≤kmax

Gxn−1, xn, xn, Gxn−1, xn, xn, Gxn, xn1, xn1,

Gxn, x_n1, x_n1,Gx_n−1, x_n1, x_n1 0

2 ,Gx_n−1, x_n1, x_n1 0

2 ,

Gxn, xn1, xn1,Gxn−1, xn1, xn1 0 2

,

2.2 Gxn, xn1, xn1≤kMn, say.

Suppose that, for somen∈N, MnGxn, x_n1, x_n1. Then we have

Gxn, x_n1, x_n1≤kGxn, x_n1, x_n1, 2.3

which is a contradiction, sincexn’s are distinct.

Suppose that there is ann∈Nfor whichMn Gx_n−1, x_n1, x_n1/2. Using property G5,

Gxn−1, xn1, xn1≤Gxn−1, xn, xn Gxn, xn1, xn1, 2.4

and one obtains

Gxn, x_n1, x_n1≤ k

2{Gx_n−1, xn, xn Gxn, x_n1, x_n1}, 2.5

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which leads to

Gxn, xn1, xn1≤ k

2−kGxn−1, xn, xn< kGxn−1, xn, xn, sincek <1. 2.6 Thus, we get

Gxn, xn1, xn1≤kGxn−1, xn, xn≤ · · · ≤kⁿGx0, x1, x1. 2.7

For everym, n∈N, m > n, usingG5,

Gxn, xm, xm≤Gxn, xn1, xn1 · · ·Gxm−1, xm, xm

≤

kⁿ· · ·k^m−1

Gx0, x1, x1≤ kⁿ

1−kGx0, x1, x1. 2.8 Therefore{xn}isG-Cauchy, henceG-convergent, sinceXisG-complete. Call the limitp.

From2.1withxxn,yzp, G

xn1, Tp, Tp

≤kmax

G

xn, p, p

, Gxn, xn1, xn1, G

p, Tp, Tp , G

p, Tp, Tp , G

xn, Tp, Tp G

p, xn1, xn1

2 ,

G

xn, Tp, Tp G

p, xn1, xn1

2 ,

G

p, Tp, Tp ,

G

xn, Tp, Tp G

p, xn1, xn1

2 .

2.9

Taking the limit of both sides of2.9asn → ∞yields G

p, Tp, Tp

≤kG

p, Tp, Tp

, 2.10

which implies thatGp, Tp, Tp 0 and hencepTp.

Suppose thatqis also a fixed point ofT. Then, from2.1withxp, yzq,

G p, q, q

≤kmax

G p, q, q

,0,0,0, G

p, q, q G

q, p, p

2 ,

G p, q, q

G

q, p, p

2 ,0,

G p, q, q

G

q, p, p

2 ,

2.11

which implies that

G p, q, q

≤ k 2−kG

q, p, p

. 2.12

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Using2.1again, this time withxq,yzp, one obtains

G q, p, p

≤kmax

G q, p, p

,0,0,0, G

q, p, p G

p, q, q

2 ,

G q, p, p

G p, q, q

2 ,0,

G q, p, p

G

p, q, q

2 ,

2.13

which implies that

G q, p, p

≤ k 2−kG

p, q, q

. 2.14

Combining2.12and2.14gives

G p, q, q

≤ k

2−k ₂

G p, q, q

. 2.15

Therefore,pq, sincek/2−k<1.

Let{yn} ⊂Xbe any sequence with limitp. Using2.1withxzyn, yp,

G

Tyn, Tp, Tyn

≤kmax

G

yn, p, yn

, G

yn, Tyn, Tyn

,0, G

yn, Tyn, Tyn

, G

yn, p, p G

yn, Tyn, Tyn

2 ,

G

yn, p, p G

p, Tyn, Tyn

2 ,

2.16 That is,

G

Tyn, p, Tyn

≤kmax

G

yn, p, yn

, G

yn, Tyn, Tyn

, G

yn, p, p G

yn, Tyn, Tyn

2 ,

G

yn, p, p G

p, Tyn, Tyn

2 .

2.17

Using the fact that, fromG5, G

yn, Tyn, Tyn

≤G

yn, p, p G

p, Tyn, Tyn

, G

Tyn, p, Tyn

≤kL, say. 2.18

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If, for somen,Lis equal toGyn, p, yn, then we have G

Tyn, p, Tyn

≤kG

yn, p, yn

. 2.19

If, for somen, Lis equal toGyn, Tyn, Tyn, then, usingG5, G

Tyn, p, Tyn

≤kG

yn, Tyn, Tyn

≤G

yn, p, p G

p, Tyn, Tyn

, 2.20

which implies that

G

Tyn, p, Tyn

≤ k 1−kG

yn, p, p

. 2.21

If, for somen, Lis equal toGyn, p, p Gyn, Tyn, Tyn/2, then, usingG5,

G

Tyn, p, Tyn

≤ k 2

G

yn, p, p G

p, Ty_n, Ty_n

, 2.22

which implies that

G

Tyn, p, Tyn

≤ 2k 2−kG

yn, p, p

. 2.23

If, for somen,Lis equal toGyn, p, p Gp, Tyn, Tyn/2, then, usingG5,

G

Tyn, p, Tyn

≤ k 2

G

yn, p, p G

p, Ty_n, Ty_n

, 2.24

which implies that

G

Tyn, p, Tyn

≤ k 2−kG

yn, p, p

. 2.25

Therefore, for alln, limGp, Tyn, Tyn 0 andT isG-continuous atp.

Special cases of Theorem2.1are Theorem 2.1 of9and Theorems 2.1, 2.4, 2.6, and 2.8 of10.

Theorem 2.2. LetX, Gbe a completeG-metric space, and letTbe a self-map ofXsatisfying, for all x, y, z∈X,

G

Tx, Ty, Tz

≤kmax G

x, y, z

, Gx, Tx, Tx, G

y, Ty, Ty , G

x, Ty, Ty , G

y, Tx, Tx

, Gz, Tz, Tz ,

2.26

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or

G

Tx, Ty, Tz

≤kmax G

x, y, z

, Gx, x, Tx, G

y, y, Ty , G

x, x, Ty , G

y, y, Tx

, Gz, z, Tx ,

2.27

where k is a constant satisfying 0 ≤ k < 1. Then T has a unique fixed point (call itp) and T is G-continuous atp.

Proof. Suppose thatTsatisfies2.26. Using2.26withzy, we have G

Tx, Ty, Ty

≤kmax G

x, y, y

, Gx, Tx, Tx, G

y, Ty, Ty , G

x, Ty, Ty , G

y, Tx, Tx 2.28.

Suppose thatX, Gis symmetric.

From Proposition 1.10,dG, defined by dGx, y 2Gx, y, ymakes, X, dG into a metric space. Substituting into2.28and then multiplying by 2 yield

dG

Tx, Ty

≤kmax dG

x, y

, dGx, Tx, dG

y, Ty , dG

x, Ty , dG

y, Tx

. 2.29

From Theorem1.13,Thas a unique fixed point.

Suppose thatX, Gis not symmetric. Define An

G

Tⁱx, T^jx, T^jx

: 0≤i, j≤n , δnmax

i,j An. 2.30

ThenδnGTⁱx, T^mx, T^mxfor somei, msatisfying 0≤i, m≤n.

Suppose thati >0. Then, from2.26, δnGxi, xm, xm

≤kmax{Gx_i−1, x_m−1, x_m−1, Gx_i−1, xi, xi, Gx_m−1, xm, xm, Gx_i−1, xm, xm, Gx_m−1, xi, xi}

≤kδn,

2.31 a contradiction. Therefore,i0.

Thus, for somemsatisfying 0≤m≤n, using propertyG5and2.26, δnGx0, xm, xm≤Gx0, x1, x1 Gx1, xm, xm

≤Gx0, x1, x1 kmax{Gx0, xm−1, xm−1, Gx0, x1, x1,

Gx_m−1, xm, xm, Gx0, xm, xm, Gx_m−1, x1, x1}

≤Gx0, x1, x1 kδn,

2.32

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which implies that

δn≤ 1

1−kGx0, x1, x1, 2.33

andδnis bounded inn. Call this boundδ.

DefinexnTx_n−1. Without loss of generality, we may assume thatxn/x_n1for eachn.

For, if there exists anNfor whichxNxN1, thenxN1TxNandxNis a fixed point ofT. Again from2.26,

Gxn, xn1, xn1

≤kmax{Gxn−1, xn, xn, Gxn−1, xn, xn, Gxn, xn1, xn1, Gxn−1, xn1, xn1,0}

kmax{Gx_n−1, xn, xn, Gx_n−1, x_n1, x_n1}

≤kmax{Gxn−1, xn, xn, δ}

≤ · · · ≤kⁿmax{Gx0, x1, x1, δ} ≤kⁿδ.

2.34

For anym, n∈N;m > n,

Gxn, xm, xm≤Gxn, x_n1, x_n1 Gx_n1, x_n2, x_n2 · · ·Gxm−1, xm, xm

≤

kⁿkⁿ¹· · ·k^m−1

δ≤ kⁿδ 1−k.

2.35

Therefore, limGxn, xm, xm 0 asm, n → ∞and {xn}isG-Cauchy, henceG-convergent, sinceXisG-complete. Call the limitp.

From2.26, G

xn, Tp, Tp

≤kmax G

xn−1, p, p

, Gxn, xn1, xn1, G

p, Tp, Tp , G

xn−1, Tp, Tp , G

p, xn, xn

.

2.36

Taking the limit of both sides of2.36asn → ∞yields G

p, Tp, Tp

≤kG

p, Tp, Tp

, 2.37

which implies thatpTp.

Suppose thatqis another fixed point ofTwithp /q. Then, from2.26, G

p, q, q

≤kmax G

p, q, q

,0,0, G p, q, q

, G

q, p, p kG

q, p, p

. 2.38

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Again using2.26,

G q, p, p

≤kmax G

q, p, p

,0,0, G q, p, p

, G

p, q, q kG

p, q, q

. 2.39

Combining2.36and2.38givesGp, q, q ≤ k²Gp, q, q,a contradiction. Thereforep q and the fixed point is unique.

Now let{yn} ⊂Xwith limynp. Using2.26,

G

Tyn, p, Tyn

≤kmax G

yn, p, yn

, G

yn, Tyn, Tyn

,0, G

yn, p, p , G

p, Tyn, Tyn

, G

yn, Tyn, Tyn

2.40

But fromG5, we have

G

yn, Tyn, Tyn

≤G

yn, p, p G

p, Tyn, Tyn

. 2.41

Therefore,2.40reduces to

G

Tyn, p, Tyn

≤max

kG

yn, p, yn

, k 1−kG

yn, p, p

. 2.42

Taking the limit of both sides of the above equation asn → ∞gives limGTyn, p, Tyn 0, which implies that limTynp, andTisG-continuous atp.

The proof using2.27is similar. Special cases of Theorem2.2are Theorems 2.5, 2.8, and 2.9 of9.

3. Property P

In this section we shall show that maps satisfying2.1or2.26possess propertyP. Theorem 3.1. Under the conditions of Theorem2.1,Thas propertyP.

Proof. From Theorem2.1,T has a fixed point. ThereforeFTⁿ/∅for eachn∈N. Fixn > 1 and assume thatp∈FTⁿ. We wish to show thatp∈FT.

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Suppose thatp /Tp. Using2.1,

G

p, Tp, Tp G

Tⁿp, Tⁿ¹p, Tⁿ¹p

≤kmax

G

Tⁿ⁻¹p, Tⁿp, Tⁿp , G

Tⁿp, Tⁿ¹p, Tⁿ¹p ,

G

Tⁿp, Tⁿ¹p, Tⁿ¹p ,

G

Tⁿ⁻¹p, Tⁿ¹p, Tⁿ¹p 0

2 ,

G

Tⁿ⁻¹p, Tⁿ¹p, Tⁿ¹p 0

2 ,

G

Tⁿp, Tⁿ¹p, Tⁿ¹p G

2 ,

G

Tⁿ⁻¹p, Tⁿ¹p, Tⁿ¹p 0 2

kG

Tⁿ⁻¹p, Tⁿp, Tⁿp

≤k²G

Tⁿ⁻²p, Tⁿ⁻¹p, Tⁿ⁻¹p

≤ · · · ≤kⁿG

p, Tp, Tp ,

3.1

a contradiction.

Thereforep∈FTandT has propertyP.

Theorem 3.2. Under the conditions of Theorem2.2,Thas propertyP.

Proof. From Theorem2.2,T has a fixed point. ThereforeFTⁿ/∅for eachn ∈N. Fixn > 1 and assume thatp∈FTⁿ. Using2.26and assuming thatp /Tp, we have

G

p, Tp, Tp G

≤kmax G

Tⁿ⁻¹p, Tⁿp, Tⁿp , G

Tⁿ⁻¹p, Tⁿp, Tⁿp , G

Tⁿp, Tⁿ¹p, Tⁿ¹p , G

Tⁿ⁻¹p, Tⁿ¹p, Tⁿ¹p ,0,0

.

3.2

DefineBn{GTⁱp, T^jp, T^jp: 0≤i, j≤n}. Then

δnmax

i,j Bn. 3.3

Then,δnGTⁱp, T^mp, T^mpfor some 0≤i, m≤n.

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Assume thatδn>0. Then from2.26, δnG

Tⁱp, T^mp, T^mp

≤kmax G

Tⁱ⁻¹p, T^m−1p, T^m−1p , G

Tⁱ⁻¹p, Tⁱp, Tⁱp , G

T^m−1p, T^mp, T^mp , G

Tⁱ⁻¹p, T^mp, T^mp , G

T^m−1p, Tⁱp, Tⁱp , G

T^m−1p, T^mp, T^mp

≤kδn,

3.4

a contradiction. Thereforeδn0. In particular,Gp, Tp, Tp 0 andpTp.

Acknowledgment

The authors would like to thank the Editor-in-Chief and referees for the valuable suggestions and corrections for the improvement of this paper.

References

1 S. Gahler, “2-metrische R¨aume und ihre topologische Struktur,” Mathematische Nachrichten, vol. 26, pp. 115–148, 1963.

2 S. Gahler, “Zur geometric 2-metriche raume,” Revue Roumaine de Math´ematiques Pures et Appliqu´ees, vol. 40, pp. 664–669, 1966.

3 B. C. Dhage, “Generalized metric space and mapping with fixed point,” Bulletin of the Calcutta Mathematical Society, vol. 84, pp. 329–336, 1992.

4 B. C. Dhage, “Generalized metric spaces and topological structure. I,” Analele S¸tiint¸ifice ale Universit˘atii

“Al. I. Cuza” din Ias¸i. Serie Nou˘a. Matematic˘a , vol. 46, no. 1, pp. 3–24, 2000.

5 C. R. Hsiao, “A property of contractive type mappings in 2-metric spaces,” J ˜n¯an¯abha, vol. 16, pp. 223–

239, 1986.

6 Z. Mustafa and B. Sims, “Some remarks concerningD-metric spaces,” in Proceedings of the International Conference on Fixed Point Theory and Applications, pp. 189–198, Valencia, Spain, 2003.

7 Z. Mustafa, A new structure for generalized metric spaces—with applications to fixed point theory, Ph.D.

thesis, The University of Newcastle, Callaghan, Australia, 2005.

8 Z. Mustafa and B. Sims, “A new approach to generalized metric spaces,” Journal of Nonlinear and Convex Analysis, vol. 7, no. 2, pp. 289–297, 2006.

9 Z. Mustafa, H. Obiedat, and F. Awawdeh, “Some fixed point theorem for mapping on completeG- metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 189870, 12 pages, 2008.

10 Z. Mustafa and B. Sims, “Fixed point theorems for contractive mappings in complete G-metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 917175, 10 pages, 2009.

11 M. Abbas and B. E. Rhoades, “Common fixed point results for noncommuting mappings without continuity in generalized metric spaces,” Applied Mathematics and Computation, vol. 215, no. 1, pp.

262–269, 2009.

12 G. S. Jeong and B. E. Rhoades, “Maps for whichFT FTⁿ,” in Fixed Point Theory and Applications.

Vol. 6, pp. 71–105, Nova Sci. Publ., New York, NY, USA, 2007.

13 G. S. Jeong and B. E. Rhoades, “More maps for whichFT FTⁿ,” Demonstratio Mathematica, vol.

40, no. 3, pp. 671–680, 2007.

14 B. E. Rhoades and M. Abbas, “Maps satisfying generalized contractive conditions of integral type for whichFT FTⁿ,” International Journal of Pure and Applied Mathematics, vol. 45, no. 2, pp. 225–231, 2008.

15 Lj. B. ´Ciri´c, “A generalization of Banach’s contraction principle,” Proceedings of the American Mathematical Society, vol. 45, pp. 267–273, 1974.