Some Common Fixed Point Theorems in Menger PM Spaces

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Volume 2010, Article ID 819269,14pages doi:10.1155/2010/819269

Research Article

Some Common Fixed Point Theorems in

Menger PM Spaces

M. Imdad,

1

_{M. Tanveer,}

2

_{and M. Hasan}

3

1_{Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India}

2_{School of Computer & Systems Sciences, Jawaharlal Nehru University, New Delhi 110 067, India} 3_{Department of Applied Mathematics, Aligarh Muslim University, Aligarh 202 002, India}

Correspondence should be addressed to M. Tanveer,tanveer [email protected]

Received 11 May 2010; Accepted 11 August 2010

Academic Editor: Tomonari Suzuki

Copyrightq2010 M. Imdad 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.

Employing the common propertyE.A, we prove some common fixed point theorems for weakly

compatible mappings via an implicit relation in Menger PM spaces. Some results on similar lines satisfying quasicontraction condition as well asψ-type contraction condition are also proved in Menger PM spaces. Our results substantially improve the corresponding theorems contained in

Branciari,2002; Rhoades,2003; Vijayaraju et al.,2005and also some others in Menger as well as metric spaces. Some related results are also derived besides furnishing illustrative examples.

1. Introduction and Preliminaries

Sometimes, it is found appropriate to assign the average of several measurements as a measure to ascertain the distance between two points. Inspired from this line of thinking, Menger1,2introduced the notion of Probabilistic Metric spacesin short PM spacesas a generalization of metric spaces. In fact, he replaced the distance functiond :R×R → R with a distribution functionFp,q :R → 0,1wherein for any numberx, the valueFp,qx

describes the probability that the distance betweenpand qis less thanx. In fact the study of such spaces received an impetus with the pioneering work of Schweizer and Sklar 3. The theory of PM spaces is of paramount importance in Probabilistic Functional Analysis especially due to its extensive applications in random diﬀerential as well as random integral equations.

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point and common fixed point theorems in PM spaces. For an idea of this kind of the literature, one can consult the results contained in3–14.

In metric spaces, Jungck 15 introduced the notion of compatible mappings and utilized the same as a tool to improve commutativity conditions in common fixed point theorems. This concept has been frequently employed to prove existence theorems on common fixed points. However, the study of common fixed points of noncompatible mappings is also equally interesting which was initiated by Pant16. Recently, Aamri and Moutawakil17and Liu et al.18respectively, defined the propertyE.Aand the common property E.A and proved some common fixed point theorems in metric spaces. Imdad et al.19 extended the results of Aamri and Moutawakil17to semimetric spaces. Most recently, Kubiaczyk and Sharma20defined the propertyE.Ain PM spaces and used it to prove results on common fixed points wherein authors claim to prove their results for strict contractions which are merely valid up to contractions.

In 2002, Branciari 21 proved a fixed point result for a mapping satisfying an integral-type inequality which is indeed an analogue of contraction mapping condition. In recent past, several authorse.g.,22–26proved various fixed point theorems employing relatively more general integral type contractive conditions. In one of his interesting articles, Suzuki27pointed out that Meir-Keeler contractions of integral type are still Meir-Keeler contractions. In this paper, we prove the fixed point theorems for weakly compatible mappings via an implicit relation in Menger PM spaces satisfying the common property

E.A. Our results substantially improve the corresponding theorems contained in21,24,26,

28along with some other relevant results in Menger as well as metric spaces. Some related results are also derived besides furnishing illustrative examples.

In the following lines, we collect the background material to make our presentation as self-contained as possible.

Definition 1.1 see 3. A mapping F : R → R is called distribution function if it is nondecreasing and left continuous with inf{Ft:t∈R}0 and sup{Ft:t∈R}1.

LetLbe the set of all distribution functions whereasHbe the set of specific distribution functionalso known as Heaviside functiondefined by

Hx ⎧ ⎨ ⎩

0, ifx≤0,

1, ifx >0.

1.1

Definition 1.2see1. LetXbe a nonempty set. An ordered pairX, Fis called a PM space ifFis a mapping fromX×XintoLsatisfying the following conditions:

1Fp,qx Hxif and only ifpq,

2Fp,qx Fq,px,

3Fp,qx 1 andFq,ry 1, thenFp,rxy 1, for allp, q, r∈Xandx, y≥0.

Every metric space X, d can always be realized as a PM space by considering

F : X ×X → L defined by Fp,qx Hx−dp, qfor all p, q ∈ X. So PM spaces oﬀer

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Definition 1.3see3. A mappingΔ:0,1×0,1 → 0,1is called at-norm if

1 Δa,1 a,Δ0,0 0,

2 Δa, b Δb, a,

3 Δc, d≥Δa, bforc≥a, d≥b,

4 ΔΔa, b, c Δa,Δb, cfor alla, b, c∈0,1.

Example 1.4. The following are the four basict-norms.

iThe minimumt-norm:ΔMa, b min{a, b}.

iiThe productt-norm:ΔPa, b a.b.

iiiThe Lukasiewiczt-norm:ΔLa, b max{ab−1,0}.

ivThe weakestt-norm, the drastic product:

ΔDa, b

⎧ ⎨ ⎩

min{a, b} if max{a, b}1,

0, otherwise.

1.2

In respect of above mentionedt-norms, we have the following ordering:

ΔD<ΔL<ΔP <ΔM. 1.3

Throughout this paper,Δstands for an arbitrary continuoust-norm.

Definition 1.5see1. A Menger PM spaceX, F,Δis a triplet whereX, Fis a PM space andΔis at-norm satisfying the following condition:

Fp,r

xy≥ΔFp,qx, Fq,r

y. 1.4

Definition 1.6see6. A sequence{pn}in a Menger PM spaceX, F,Δis said to converge to

a pointpinXif for every >0 andλ >0, there is an integerM, λsuch thatFpn,p>1−λ,

for alln≥M, λ.

Definition 1.7see10. A pairA, Sof self-mappings of a Menger PM spaceX, F,Δis said to be compatible ifFASpn,SApnx → 1 for allx > 0, whenever{pn}is a sequence inX such

thatApn, Spn → t,for sometinXasn → ∞.

Definition 1.8see23. A pairA, Sof self-mappings of a Menger PM spaceX, F,Δis said to be noncompatible if and only if there exists at least one sequence{xn}inXsuch that

lim

n→ ∞Axnnlim→ ∞Sxnt∈X, for somet∈X, 1.5

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Definition 1.9see6. A pairA, Sof self-mappings of a Menger PM spaceX, F,Δis said to satisfy the propertyE.Aif there exist a sequence{xn}inXsuch that

lim

n→ ∞Axnnlim→ ∞Sxnt∈X. 1.6

Clearly, a pair of compatible mappings as well as noncompatible mappings satisfies the propertyE.A.

Inspired by Liu et al.18, we introduce the following.

Definition 1.10. Two pairsA, SandB, Tof self-mappings of a Menger PM spaceX, F,Δ

are said to satisfy the common property E.Aif there exist two sequences{xn},{yn} inX

and sometinXsuch that

lim

n→ ∞Axnnlim→ ∞Sxnnlim→ ∞Tynnlim→ ∞Byn t. 1.7

Example 1.11. LetX, F,Δbe a Menger PM space withX −1,1and,

Fx,yt

⎧ ⎨ ⎩

e−|x−y|/t_, _if _{t >}₀_,

0, if t0,

1.8

for allx, y ∈X. Define self-mappingsA, B, SandT onX asAx x/3,Bx −x/3,Sx x, andTx−xfor allx∈X. Then with sequences{xn}1/nand{yn} −1/ninX, one can

easily verify that

lim

n→ ∞Axnnlim→ ∞Sxnnlim→ ∞Bynnlim→ ∞Tyn0. 1.9

This shows that the pairsA, SandB, Tshare the common propertyE.A.

Definition 1.12 see29. A pair A, S of self-mappings of a nonempty set X is said to be weakly compatible if the pair commutes on the set of coincidence points, that is,ApSpfor somep∈Ximplies thatASpSAp.

Definition 1.13see8. Two finite families of self-mappings{Ai}im1 and{Bk}nk1 of a setX

are said to be pairwise commuting if

1AiAjAjAi, i, j∈ {1,2, . . . , m},

2BkBlBlBk, k, l∈ {1,2, . . . , n},

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2. Implicit Relation

LetF6 be the set of all continuous functionsΦt1, t2, t3, t4, t5, t6 : 0,16 → Rsatisfying the

following conditions:

Φ1 Φu,1, u,1,1, u<0, for allu∈0,1, Φ2 Φu,1,1, u, u,1<0, for allu∈0,1, Φ3 Φu,1, u,1, u, u<0, for allu∈0,1.

Example 2.1. DefineΦt1, t2, t3, t4, t5, t6:0,16 → Ras

Φt1, t2, t3, t4, t5, t6 t1−ψmin{t2, t3, t4, t5, t6}, 2.1

where ψ : 0,1 → 0,1is increasing and continuous function such that ψt > t for all

t∈0,1.Notice that

Φ1 Φu,1, u,1,1, u u−ψu<0, for allu∈0,1, Φ2 Φu,1,1, u, u,1 u−ψu<0, for allu∈0,1, Φ3 Φu,1, u,1, u, u u−ψu<0, for allu∈0,1.

Φt1, t2, t3, t4, t5, t6

_t₁

0

φtdt−ψ

min{t2,t3,t4,t5,t6}

0

φtdt, 2.2

whereψ : 0,1 → 0,1is an increasing and continuous function such thatψt > tfor all

t∈0,1andφ:R → Ris a Lebesgue integrable function which is summable and satisfies

0<

0

φsds <1, ∀0< <1, ₁

0

φsds1. 2.3

Observe that

Φ1 Φu,1, u,1,1, u

_u

0 φtdt−ψ

_u

0 φtdt<0, for allu∈0,1, Φ2 Φu,1,1, u, u,1

u

0 φtdt−ψ

u

0 φtdt<0, for allu∈0,1, Φ3 Φu,1, u,1, u, u

u

0 φtdt−ψ

u

0 φtdt<0, for allu∈0,1.

Φt1, t2, t3, t4, t5, t6

t1

0

φtdt−ψ

min

t2

0

φtdt, t3

0

φtdt, t4

0

φtdt, t5

0

φtdt, t6

0

φtdt ,

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whereψ : 0,1 → 0,1is an increasing and continuous function such thatψt > tfor all

t∈0,1andφ:R → Ris a Lebesgue integrable function which is summable and satisfies

0<

0

φsds <1, ∀0< <1, ₁

0

φsds1. 2.5

Observe that

Φ1 Φu,1, u,1,1, u

_u

0 φtdt−ψ

_u

0 φtdt<0, for allu∈0,1, Φ2 Φu,1,1, u, u,1

_u

0 φtdt−ψ

_u

0 φtdt<0, for allu∈0,1, Φ3 Φu,1, u,1, u, u

_u

0 φtdt−ψ

_u

0 φtdt<0, for allu∈0,1.

3. Main Results

We begin with the following observation.

Lemma 3.1. LetA, B, SandTbe self-mappings of a Menger spaceX, F,Δsatisfying the following:

ithe pairA, S(orB, Tsatisfies the propertyE.A;

iifor anyp, q∈X,Φ∈F6and for allx >0,

ΦFAp,Bqx, FSp,Tqx, FSp,Apx, FTq,Bqx, FSp,Bqx, FTq,Apx

≥0; 3.1

iiiAX⊂TX(orBX⊂SX.

Then the pairsA, SandB, Tshare the common propertyE.A.

Proof. Suppose that the pairA, Sowns the propertyE.A, then there exists a sequence{xn}

inXsuch that

lim

n→ ∞Axnnlim→ ∞Sxnt, for somet∈X. 3.2

SinceAX ⊂ TX, hence for each{xn}there exists{yn} ∈X such thatAxn Tyn.

Therefore,

lim

n→ ∞Tynnlim→ ∞Axnt. 3.3

Thus in all, we haveAxn → t, Sxn → tandTyn → t.Now we assert thatByn → t. Suppose

thatByn→t; then applying inequality3.1, we obtain

ΦFAxn,Bynx, FSxn,Tynx, FSxn,Axnx, FTyn,Bynx, FSxn,Bynx, FTyn,Axnx

≥0 3.4

which on makingn → ∞reduces to

ΦFt,lim

n→ ∞Bynx, Ft,tx, Ft,tx, Ft,nlim→ ∞Bynx, Ft,nlim→ ∞Bynx, Ft,tx

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or

ΦFt,lim

n→ ∞Bynx,1,1, Ft,nlim→ ∞Bynx, Ft,nlim→ ∞Bynx,1

≥0 3.6

which is a contradiction toΦ2, and thereforeByn → t. Hence the pairsA, SandB, T

share the common propertyE.A.

Remark 3.2. The converse ofLemma 3.1is not true in general. For a counter example, one can seeExample 3.17presented in the end.

Theorem 3.3. Let A, B, S and T be self-mappings on a Menger PM space X, F,Δ satisfying inequality3.1. Suppose that

ithe pairA, S(orB, Tenjoys the property (E.A),

iiAX⊂TX(orBX⊂SX,

iiiSX(orTXis a closed subset ofX.

Then the pairsA, SandB, Thave a point of coincidence each. Moreover,A, B, SandThave a unique common fixed point provided that both the pairsA, SandB, Tare weakly compatible.

Proof. In view ofLemma 3.1, the pairsA, SandB, Tshare the common propertyE.A, that is, there exist two sequences{xn}and{yn}inXsuch that

lim

n→ ∞Axnnlim→ ∞Sxnnlim→ ∞Bynnlim→ ∞Tynt, for somet∈X. 3.7

Suppose thatSXis a closed subset ofX, thentSufor someu∈X.If_{t /}Au, then applying inequality3.1, we obtain

ΦFAu,Bynx, FSu,Tynx, FSu,Aux, FTyn,Bynx, FSu,Bynx, FTyn,Aux

≥0 3.8

which on makingn → ∞,reduces to

ΦFAu,tx,1, Ft,Aux,1,1, Ft,Aux≥0 3.9

which is a contradiction toΦ1. HenceAuSut.

SinceAX⊂TX, there existsv∈Xsuch thattAuTv.

If_{t /}Bv, then using inequality3.1, we have

ΦFAu,Bvx, FSu,Tvx, FSu,Aux, FTv,Bvx, FSu,Bvx, FTv,Aux≥0 3.10

or

ΦFt,Bvx,1,1, Ft,Bvx, Ft,Bvx,1≥0 3.11

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Since the pairs A, S and B, T are weakly compatible and Au Su, Bv Tv,

therefore

AtASuSAuSt, BtBTvTBvTt. 3.12

If_{At /}t, then using inequality3.1, we have

ΦFAt,Bvx, FSt,Tvx, FSt,Atx, FTv,Bvx, FSt,Bvx, FTv,Atx≥0 3.13

or

Φ FAt,tx,1, FAt,tx,1, FAt,tx0, Ft,Atx0≥0 3.14

which is a contradiction toΦ3, and thereforeAtStt.

Similarly, one can prove that Bt Tt t.Hence t Bt Tt At St, andt is a common fixed point ofA, B, Sand T. The uniqueness of common fixed point is an easy consequences of inequality3.1.

By choosingA, B, SandT suitably, one can derive corollaries involving two or three mappings. As a sample, we deduce the following natural result for a pair of self-mappings by settingBAandT SinTheorem 3.3.

Corollary 3.4. LetAandSbe self-mappings on a Menger spaceX, F,Δ. Suppose that

ithe pairA, Senjoys the property (E.A),

iifor allp, q∈X,Φ∈F6and for allx >0,

ΦFAp,Aqx, FSp,Sqx, FSp,Apx, FSq,Aqx, FSp,Aqx, FSq,Apx

≥0, 3.15

iiiSXis a closed subset ofX.

Then A and S have a coincidence point. Moreover, if the pair A, S is weakly compatible, thenAandShave a unique common fixed point.

Theorem 3.5. LetA, B, S andT be self-mappings of a Menger PM space X, F,Δsatisfying the inequality3.1. Suppose that

ithe pairsA, SandB, Tshare the common property (E.A),

iiSXandTXare closed subsets ofX.

If the pairsA, SandB, Tare weakly compatible, thenA, B, SandThave a unique common fixed point inX.

Proof. Suppose that the pairsA, SandB, Tsatisfy the common propertyE.A, then there exist two sequences{xn}and{yn}inXsuch that

lim

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SinceSXandTXare closed subsets ofX, we obtaintSuTvfor someu, v∈X. If_{t /}Au,then using inequality3.1, we have

ΦFAu,Bynx, FSu,Tynx, FSu,Aux, FTyn,Bynx, FSu,Bynx, FTyn,Aux

≥0 3.17

ΦFAu,tx,1, Ft,Aux,1,1, Ft,Aux≥0 3.18

which is a contradiction toΦ1, and hencet Au Tv Su.The rest of the proof can be

completed on the lines of the proof ofTheorem 3.3, hence it is omitted.

Remark 3.6. Theorem 3.3extends the main result of Ciric30to Menger PM spaces besides extending the main result of Kubiaczyk and Sharma20to two pairs of mappings without any condition on containment of ranges amongst involved mappings.

Theorem 3.7. The conclusions ofTheorem 3.5remain true if condition (ii) ofTheorem 3.5is replaced by the following:

iii AX⊂TXandBX⊂SX.

Corollary 3.8. The conclusions of Theorems3.3and3.5remain true if conditions (ii) (ofTheorem 3.3) and (iii)’ (ofTheorem 3.7) are replaced by the following:

ivAXandBXare closed subsets ofXwhereasAX⊂TXandBX⊂SX.

As an application ofTheorem 3.3, we prove the following result for four finite families of self-mappings. While proving this result, we utilize Definition 1.13 which is a natural extension of commutativity condition to two finite families of mappings.

Theorem 3.9. Let {A1, A2, . . . , Am}, {B1, B2, . . . , Bp}, {S1, S2, . . . , Sn} and {T1, T2, . . . , Tq} be

four finite families of self-mappings of a Menger PM space X, F,ΔwithA A1A2· · ·Am, B

B1B2· · ·Bp,S S1S2· · ·Snand T T1T2· · ·Tq satisfying condition3.1. If the pairsA, Sand

B, Tshare the common property (E.A) andSXas well asTXare closed subsets ofX, then

ithe pairA, Sas well asB, Thas a coincidence point,

iiAi, Bk, Sr and Tt have a unique common fixed point provided that the pair of families

{Ai},{Sr} and {Bk},{Tt} commute pairwise, where i ∈ {1,2, . . . , m}, k ∈

{1,2, . . . , p},r∈ {1,2, . . . , n}, andt∈ {1,2, . . . , q}.

Proof. The proof follows on the lines of Theorem 4.1 according to M. Imdad and J. Ali31

and Theorem 3.1 according to Imdad et al.19.

Remark 3.10. By restricting four families as{A1, A2},{B1, B2},{S1}and{T1}inTheorem 3.9,

we can derive improved versions of certain results according to Chugh and Rathi 4, Kutukcu and Sharma32, Rashwan and Hedar11, Singh and Jain14, and some others.

Theorem 3.9also generalizes the main result of Razani and Shirdaryazdi12to any finite number of mappings.

By settingA1 A2 · · · Am G, B1 B2 · · · Bp H, S1 S2 · · ·Sn Iand

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Corollary 3.11. LetG, H, I andJ be self-mappings of a Menger spaceX, F,Δsuch that the pairs

Gm_{, I}n_and_Hp_{, J}q_{share the common property (E.A) and also satisfy the condition}

ΦFGm_x,Hp_yz, F_In_x,Jq_yz, F_In_x,Gm_xz, F_In_x,Hp_yz, F_Jq_y,Hp_yz, F_Jq_y,Gm_xz≥0 3.19

for allx, y∈X, for allz >0andm, n, pandqare fixed positive integers.

IfIn_X_and_Jq_X_{are closed subsets of}_X_{, then}_{G, H, I} _and_J_{have a unique common fixed}

point provided,GIIGandHJJH.

Remark 3.12. Corollary 3.11 is a slight but partial generalization of Theorem 3.3 as the commutativity requirementsi.e.,GI IGandHJ JHin this corollary are stronger as compared to weak compatibility inTheorem 3.3.Corollary 3.11also presents the generalized and improved form of a result according to Bryant33in Menger PM spaces.

Our next result involves a lower semicontinuous functionψ :0,1 → 0,1such that

ψt> tfor allt∈0,1along withψ0 0 andψ1 1.

Theorem 3.13. LetA, B, SandTbe self-mappings of a Menger spaceX, F,Δsatisfying conditions (i) and (ii) ofTheorem 3.5and for allp, q∈X, x >0

FAp,Bqx

0

φtdt≥ψ

_mp,q

0

φtdt , 3.20

wheremp, q min{FSp,Tqx,FSp,Apx,FTq,Bqx,FSp,Bqx,FTq,Apx}.

Then the pairsA, SandB, Thave point of coincidence each. Moreover,A, B, Sand

Thave a unique common fixed point provided that both the pairsA, SandB, Tare weakly compatible.

Proof. As both the pairs share the common property E.A, there exist two sequences

{xn},{yn} ⊂Xsuch that

lim

n→ ∞Axnnlim→ ∞Sxnnlim→ ∞Bynnlim→ ∞Tynt, for somet∈X. 3.21

IfSXis a closed subset ofX, then3.21.Therefore, there exists a pointu∈Xsuch thatSut.Now we assert thatAuSu.If it is not so, then settingpu, qynin3.20, we

get

FAu,Bynx

0

φtdt≥ψ

min{FSu,Tynx,FSu,Aux,FTyn,Bynx,FSu,Bynx,FTyn,Aux}

0

φtdt 3.22

which on makingn → ∞, reduces to

FAu,tx

0

φtdt≥ψ

min{Ft,tx,Ft,Aux,Ft,tx,Ft,tx,Ft,Aux}

0

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or

FAu,tx

0

φtdt≥ψ

FAu,tx

0

φtdt >

FAu,tx

0

φtdt 3.24

a contradiction. ThereforeAut, and henceAuSuwhich shows that the pairA, Shas a point of coincidence.

IfTXis a closed subset ofX, then3.21.Hence, there exists a pointw∈Xsuch that

Twt.Now we show thatBwTw.If it is not so, then using3.20withpxn, qw, we

have

_F_Axn,Bw_x

0

φtdt≥ψ

min{FSxn,Twx,FSxn,Axnx,FTw,Bwx,FSxn,Bwx,FTw,Axnx}

0

φtdt 3.25

Ft,Bwx

0

φtdt≥ψ

min{Ft,tx,Ft,tx,Ft,Bwx,Ft,Bwx,Ft,tx}

0

φtdt 3.26

or

_F_t,Bw_x

0

φtdt≥ψ

Ft,Bwx

0

φtdt >

_F_t,Bw_x

0

φtdt 3.27

a contradiction. ThereforeBw tand henceTwBwwhich proves that the pairB, Thas a point of coincidence.

Since the pairsA, SandB, Tare weakly compatible and both the pairs have point of coincidenceuandv, respectively. Following the lines of the proof ofTheorem 3.3, one can easily prove the existence of unique common fixed point of mappingsA, B, S andT. This concludes the proof.

Remark 3.14. Theorem 3.13generalizes the main result of Kohli and Vashistha9to two pairs of self-mappings asTheorem 3.13never requires any condition on the containment of ranges amongst involved mappings besides weakening the completeness requirement of the space to closedness of suitable subspaces along with suitable commutativity requirements of the involved mappings. Here one may also notice that the functionψ is lower semicontinuous whereas all the involved mappings may be discontinuous at the same time.

Remark 3.15. Notice that results similar to Theorems3.5–3.9and Corollaries3.4–3.11can also be outlined in respect ofTheorem 3.13, but we omit the details with a view to avoid any repetition.

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Example 3.16. Let X, F,Δbe a Menger space, whereX 0,2 with at-norm defined by

Δa, b min{a, b}for alla, b∈0,2,ψs √sfor alls∈0,1and

Fp,qt t

tp−q, ∀p, q∈X, t >0. 3.28

DefineA, B, SandTby:AxBx1,

Sx ⎧ ⎪ ⎨ ⎪ ⎩

1 ifxis rational,

2

3 ifxis irrational,

Tx ⎧ ⎪ ⎨ ⎪ ⎩

1 if xis rational,

1

3 if xis irrational.

3.29

Also defineΦt1, t2, t3, t4, t5, t6:0,16 → Ras

Φt1, t2, t3, t4, t5, t6

_t₁

0

φtdt−ψ

min

t2

0

φtdt, _t₃

0

φtdt, _t₄

0

φtdt, _t₅

0

φtdt, _t₆

0

φtdt .

3.30

It is easy to see that for allx, y∈Xandt >0

1

FAp,Bqx

0

φtdt

≥ψ

min

FSp,Tqx

0

φtdt,

FSp,Apx

0

φtdt,

FTq,Bqx

0

φtdt,

FSp,Bqx

0

φtdt,

FTq,Apx

0

φtdt .

3.31

AlsoAX {1} ⊂ {1,2/3}SX. Thus all the conditions ofTheorem 3.3are satisfied, and 1 is the unique common fixed point ofA, B, SandT.

Example 3.17. LetX, F,ΔandFbe the same as inExample 3.16. DefineA, B, SandT by

Ax ⎧ ⎪ ⎨ ⎪ ⎩

1 ifxis rational,

3

4 ifxis irrational,

Bx ⎧ ⎪ ⎨ ⎪ ⎩

1 ifxis rational,

1

2 ifxis irrational,

Sx ⎧ ⎪ ⎨ ⎪ ⎩

1 ifxis rational,

2

3 ifxis irrational,

Tx ⎧ ⎪ ⎨ ⎪ ⎩

1 ifxis rational,

1

3 ifxis irrational.

(13)

By a routine calculation, one can verify that for allx, y∈Xandt >0

_F_Ap,Bq_x

0

φtdt≥ψ

_mp,q

0

φtdt

≥ψ

minFSp,Tqx,FSp,Apx,FTq,Bqx,FSp,Bqx,FTq,Apx

0

φtdt .

3.33

AlsoAX {1,3/4}/⊂{1,2/3} SX,BX {1,1/2}/⊂{1,1/3}TX,ψs > s. Thus all conditions ofTheorem 3.13are satisfied, and 1 is the unique common fixed point ofA, B, S

andT.

References

1 K. Menger, “Statistical metrics,”Proceedings of the National Academy of Sciences of the United States of America, vol. 28, pp. 535–537, 1942.

2 K. Menger, “Probabilistic geometry,”Proceedings of the National Academy of Sciences of the United States of America, vol. 37, pp. 226–229, 1951.

3 B. Schweizer and A. Sklar,Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland, New York, NY, USA, 1983.

4 R. Chugh and S. Rathi, “Weakly compatible maps in probabilistic metric spaces,”The Journal of the Indian Mathematical Society, vol. 72, no. 1–4, pp. 131–140, 2005.

5 J. Fang and Y. Gao, “Common fixed point theorems under strict contractive conditions in Menger spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 184–193, 2009.

6 O. Hadˇzi´c and E. Pap,Fixed Point Theory in Probabilistic Metric Spaces, vol. 536 ofMathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.

7 T. L. Hicks, “Fixed point theory in probabilistic metric spaces,”Univerzitet u Novom Sadu. Zbornik Radova Prirodno-Matematiˇckog Fakulteta. Serija za Matemati, vol. 13, pp. 63–72, 1983.

8 M. Imdad, J. Ali, and M. Tanveer, “Coincidence and common fixed point theorems for nonlinear

contractions in Menger PM spaces,”Chaos, Solitons and Fractals, vol. 42, no. 5, pp. 3121–3129, 2009.

9 J. K. Kohli and S. Vashistha, “Common fixed point theorems in probabilistic metric spaces,”Acta Mathematica Hungarica, vol. 115, no. 1-2, pp. 37–47, 2007.

10 S. N. Mishra, “Common fixed points of compatible mappings in PM-spaces,”Mathematica Japonica, vol. 36, no. 2, pp. 283–289, 1991.

11 R. A. Rashwan and A. Hedar, “On common fixed point theorems of compatible mappings in Menger

spaces,”Demonstratio Mathematica, vol. 31, no. 3, pp. 537–546, 1998.

12 A. Razani and M. Shirdaryazdi, “A common fixed point theorem of compatible maps in Menger

space,”Chaos, Solitons and Fractals, vol. 32, no. 1, pp. 26–34, 2007.

13 V. M. Sehgal and A. T. Bharucha-Reid, “Fixed points of contraction mappings on probabilistic metric spaces,”Mathematical Systems Theory, vol. 6, pp. 97–102, 1972.

14 B. Singh and S. Jain, “A fixed point theorem in Menger space through weak compatibility,”Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 439–448, 2005.

15 G. Jungck, “Compatible mappings and common fixed points,”International Journal of Mathematics and Mathematical Sciences, vol. 9, no. 4, pp. 771–779, 1986.

16 R. P. Pant, “Common fixed points of noncommuting mappings,”Journal of Mathematical Analysis and Applications, vol. 188, no. 2, pp. 436–440, 1994.

17 M. Aamri and D. El Moutawakil, “Some new common fixed point theorems under strict contractive

conditions,”Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 181–188, 2002.

18 Y. Liu, J. Wu, and Z. Li, “Common fixed points of single-valued and multivalued maps,”International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 19, pp. 3045–3055, 2005.

19 M. Imdad, J. Ali, and L. Khan, “Coincidence and fixed points in symmetric spaces under strict

contractions,”Journal of Mathematical Analysis and Applications, vol. 320, no. 1, pp. 352–360, 2006.

20 I. Kubiaczyk and S. Sharma, “Some common fixed point theorems in Menger space under strict

(14)

21 A. Branciari, “A fixed point theorem for mappings satisfying a general contractive condition of integral type,”International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 9, pp. 531– 536, 2002.

22 A. Aliouche, “A common fixed point theorem for weakly compatible mappings in symmetric spaces

satisfying a contractive condition of integral type,”Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 796–802, 2006.

23 A. Djoudi and A. Aliouche, “Common fixed point theorems of Gregus type for weakly compatible

mappings satisfying contractive conditions of integral type,”Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 31–45, 2007.

24 B. E. Rhoades, “Two fixed-point theorems for mappings satisfying a general contractive condition of integral type,”International Journal of Mathematics and Mathematical Sciences, no. 63, pp. 4007–4013, 2003.

25 D. Turkoglu and I. Altun, “A common fixed point theorem for weakly compatible mappings in

symmetric spaces satisfying an implicit relation,”Boletin de la Sociedad Matematica Mexicana. Tercera Serie, vol. 13, no. 1, pp. 195–205, 2007.

26 P. Vijayaraju, B. E. Rhoades, and R. Mohanraj, “A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type,”International Journal of Mathematics and Mathematical Sciences, no. 15, pp. 2359–2364, 2005.

27 T. Suzuki, “Meir-Keeler contractions of integral type are still Meir-Keeler contractions,”International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 39281, 6 pages, 2007.

28 J. Ali and M. Imdad, “An implicit function implies several contraction conditions,”Sarajevo Journal of Mathematics, vol. 417, no. 2, pp. 269–285, 2008.

29 G. Jungck, “Common fixed points for noncontinuous nonself maps on nonmetric spaces,”Far East

Journal of Mathematical Sciences, vol. 4, no. 2, pp. 199–215, 1996.

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

31 M. Imdad and J. Ali, “Jungck’s common fixed point theorem and E.A property,”Acta Mathematica

Sinica, vol. 24, no. 1, pp. 87–94, 2008.

32 S. Kutukcu and S. Sharma, “Compatible maps and common fixed points in Menger probabilistic

metric spaces,”Communications of the Korean Mathematical Society, vol. 24, no. 1, pp. 17–27, 2009.