Common Fixed Point Theorems for Four Mappings on Cone Metric Type Space

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Volume 2011, Article ID 589725,15pages doi:10.1155/2011/589725

Research Article

Common Fixed Point Theorems for Four Mappings on Cone Metric Type Space

Aleksandar S. Cvetkovi´c,

¹

Marija P. Stani´c,

²

Sladjana Dimitrijevi´c,

²

and Suzana Simi´c

²

1Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia

2Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, Radoja Domanovi´ca 12, 34000 Kragujevac, Serbia

Correspondence should be addressed to Aleksandar S. Cvetkovi´c,[email protected] Received 9 December 2010; Revised 26 January 2011; Accepted 3 February 2011

Academic Editor: Fabio Zanolin

Copyrightq2011 Aleksandar S. Cvetkovi´c 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.

In this paper we consider the so called a cone metric type space, which is a generalization of a cone metric space. We prove some common fixed point theorems for four mappings in those spaces.

Obtained results extend and generalize well-known comparable results in the literature. All results are proved in the settings of a solid cone, without the assumption of continuity of mappings.

1. Introduction

Replacing the real numbers, as the codomain of a metric, by an ordered Banach space we obtain a generalization of metric space. Such a generalized space, called a cone metric space, was introduced by Huang and Zhang in1. They described the convergence in cone metric space, introduced their completeness, and proved some fixed point theorems for contractive mappings on cone metric space. Cones and ordered normed spaces have some applications in optimization theorysee2. The initial work of Huang and Zhang 1inspired many authors to prove fixed point theorems, as well as common fixed point theorems for two or more mappings on cone metric space, for example,3–14.

In this paper we consider the so-called a cone metric type space, which is a generalization of a cone metric space and prove some common fixed point theorems for four mappings in those spaces. Obtained results are generalization of theorems proved in13. For some special choices of mappings we obtain theorems which generalize results from1,8,15.

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All results are proved in the settings of a solid cone, without the assumption of continuity of mappings.

The paper is organized as follows. InSection 2we repeat some definitions and well- known results which will be needed in the sequel. InSection 3we prove common fixed point theorems. Also, we presented some corollaries which show that our results are generalization of some existing results in the literature.

2. Definitions and Notation

LetEbe a real Banach space andP a subset ofE. Byθwe denote zero element ofEand by intP the interior ofP. The subsetP is called a cone if and only if

iPis closed, nonempty andP /{θ};

iia, b∈^Ê,a, b≥0, andx, y∈P implyaxby∈P;

iiiP∩−P {θ}.

For a given coneP, a partial orderingwith respect toPis introduced in the following way:x yif and only if y−x ∈ P. One writes x ≺ yto indicate thatx y, butx /y. If y−x∈intP, one writesxy.

If intP /∅, the conePis called solid.

In the sequel we always suppose thatEis a real Banach space,Pis a solid cone inE, andis partial ordering with respect toP.

Analogously with definition of metric type space, given in 16, we consider cone metric type space.

Definition 2.1. LetX be a nonempty set and Ea real Banach space with coneP. A vector- valued functiond:X×X → Eis said to be a cone metric type function onX with constant K≥1 if the following conditions are satisfied:

d1θdx, yfor allx, y∈Xanddx, y θif and only ifxy;

d2dx, y dy, xfor allx, y∈X;

d3dx, yKdx, z dz, yfor allx, y, z∈X.

The pairX, dis called a cone metric type spacein brief CMTS.

Remark 2.2. ForK1 inDefinition 2.1we obtain a cone metric space introduced in1.

Definition 2.3. LetX, dbe a CMTS and{xn}a sequence inX.

c1{xn}converges tox∈Xif for everyc∈Ewithθcthere existsn₀ ∈^Æ such that dxn, xcfor alln > n₀. We write lim_n_{→ ∞}x_nx, orx_n → x,n → ∞.

c2If for everyc ∈ Ewithθ cthere existsn0 ∈ ^Æ such thatdxn, xm cfor all n, m > n0, then{xn}is called a Cauchy sequence inX.

If every Cauchy sequence is convergent inX, thenXis called a complete CMTS.

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Example 2.4. LetB{ei|i1, . . . , n}be orthonormal basis of^Êⁿ with inner product·,·. Let p >0, and define

Xp

x|x:0,1−→^Êⁿ, ₁

0

|xt, ek|^pdt∈^Ê, k1, . . . , n

, 2.1

wherexrepresents class of elementxwith respect to equivalence relation of functions equal almost everywhere. We chooseE^Êⁿ and

PB

y∈^Êⁿ | y, ei

≥0, i1, . . . , n

. 2.2

We show thatP_Bis a solid cone. Lety_k∈P_B,k∈^Æ, with property lim_k_→_∞y_ky. Since scalar product is continuous, we get lim_k_→_∞yk, e_i limk→∞y_k, e_i y, ei,i1, . . . , n. Clearly, it must bey, ei ≥ 0,i 1, . . . , n, and, hence,y ∈PB, that is,PB is closed. It is obvious that θ /e1∈PB/{θ}, and fora, b≥0, and allz, y∈PB, we haveazby, ei az, eiby, ei≥0, i1, . . . , n. Finally, ifz ∈PB∩−PBwe havez, ei ≥0 and−z, ei ≥0,i1, . . . , n, and it follows thatz, ei 0,i 1, . . . , n, and, since Bis complete, we getz 0. Let us choose z ⁿ_i1ei. It is obvious thatz∈intPB, since if not, for everyε >0 there existsy /∈ PBsuch that|1−y, ei| ≤ ⁿ_i1|1−y, ei|²^1/2z−y< ε. If we chooseε1/4, we conclude that it must bey, ei>1−1/4>0, hencey∈PB, which is contradiction.

Finally, defined:Xp×Xp → PBby

d f, g

ⁿ

i1

ei

₁

0

f−g

t, ei^pdt, f, g∈Xp. 2.3 Then it is obvious that Xp, d is CMTS withK 2^p−1. Letf, g,hbe functions such that f, e1 1,g, e1 −2,h, e1 0, andf, ei g, ei h, ei 0,i2, . . . , n, withp2 give df, g 9e1,df, h e1, anddh, g 4e1, which proves 5e1 df, h dh, gdf, g 9e1, but 9e1df, g2df, h dh, g 10e1.

The following properties are well known in the case of a cone metric space, and it is easy to see that they hold also in the case of a CMTS.

Lemma 2.5. LetX, dbe a CMTS over-ordered real Banach spaceEwith a coneP. The following properties holda, b, c∈E.

p1Ifabandbc, thenac.

p2Ifθacfor allc∈intP, thenaθ.

p3Ifaλa, wherea∈Pand 0≤λ <1, thenaθ.

p4Letxn → θinEand letθc. Then there exists positive integern0such thatxncfor eachn > n₀.

Definition 2.6see17. LetF, G:X → Xbe mappings of a setX. IfyFxGxfor some x∈X, thenxis called a coincidence point ofFandG, andyis called a point of coincidence ofFandG.

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Definition 2.7see17. LetFandGbe self-mappings of setXandCF, G {x∈X :Fx Gx}. The pair{F, G}is called weakly compatible if mappingsFandGcommute at all their coincidence points, that is, ifFGxGFxfor allx∈ CF, G.

Lemma 2.8see5. LetFandGbe weakly compatible self-mappings of a setX. IfFandGhave a unique point of coincidenceyFxGx, thenyis the unique common fixed point ofFandG.

3. Main Results

Theorem 3.1. LetX, dbe a CMTS with constant 1 ≤ K ≤ 2 andP a solid cone. Suppose that self-mappingsF, G, S, T :X → X are such thatSX ⊂ GX,TX ⊂ FXand that for some constant λ∈0,1/Kfor allx, y∈Xthere exists

u x, y

∈

Kd

Fx, Gy

, KdFx, Sx, Kd

Gy, Ty , Kd

Fx, Ty d

Gy, Sx 2

, 3.1

such that the following inequality

d Sx, Ty

λ Ku

x, y

, 3.2

holds. If one ofSX,TX,FX, orGXis complete subspace ofX, then{S, F}and{T, G}have a unique point of coincidence inX. Moreover, if{S, F}and{T, G}are weakly compatible pairs, thenF,G,S, andThave a unique common fixed point.

Proof. Let us choosex0 ∈ Xarbitrary. SinceSX ⊂ GX, there existsx1 ∈X such that Gx1 Sx0 z0. SinceTX ⊂FX, there existsx2 ∈X such thatFx2 Tx1 z1. We continue in this manner. In general,x_2n1 ∈Xis chosen such thatGx_2n1 Sx2nz2n, andx_2n2∈Xis chosen such thatFx_2n2 Tx_2n1z_2n1.

First we prove that

dzn, z_n1αdz_n−1, z_n, n≥1, 3.3 whereαmax{λ, λK/2−λK}, which will lead us to the conclusion that{zn}is a Cauchy sequence, sinceα ∈ 0,1 it is easy to see that 0 < λK/2−λK < 1. To prove this, it is necessary to consider the cases of an odd integernand of an evenn.

Forn21,∈^Æ0, we havedz₂₁, z₂₂ dSx₂₂, Tx₂₁, and from3.2there exists

ux22, x₂₁∈

KdFx22, Gx₂₁, KdFx22, Sx₂₂,

KdGx21, Tx₂₁, KdFx22, Tx₂₁ dGx21, Sx₂₂ 2

Kdz₂₁, z₂, Kdz₂₁, z₂₂,Kdz2, z₂₂ 2

,

3.4

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such thatdz21, z₂₂λ/Kux22, x₂₁. Thus we have the following three cases:

idz₂₁, z₂₂λdz₂₁, z2;

iidz₂₁, z₂₂ λdz₂₁, z₂₂, which, because of property p3, implies dz₂₁, z₂₂ θ;

iiidz21, z₂₂λ/2dz2, z₂₂, that is, by usingd3,

dz₂₁, z₂₂ λK

2 dz2, z₂₁ λK

2 dz₂₁, z₂₂, 3.5

which impliesdz21, z₂₂λK/2−λKdz2, z₂₁. Thus, inequality3.3holds in this case.

Forn2,∈^Æ0, we have

dz2, z₂₁ dSx2, Tx₂₁ λ

Kux2, x₂₁, 3.6

where

ux2, x₂₁∈

KdFx2, Gx₂₁, KdFx2, Sx₂,

KdGx21, Tx₂₁, KdFx2, T₂₁ dGx21, Sx2 2

Kdz₂₋₁, z2, Kdz2, z₂₁,Kdz₂₋₁, z₂₁ 2

.

3.7

Thus we have the following three cases:

idz2, z₂₁λdz2−1, z₂;

iidz2, z₂₁λdz2, z₂₁, which impliesdz2, z₂₁ θ;

iiidz2, z₂₁ λ/2dz2−1, z₂₁ λK/2dz2−1, z2 λK/2dz2, z₂₁, which impliesdz2, z₂₁λK/2−λKdz2, z₂₋₁.

So, inequality3.3is satisfied in this case, too.

Therefore,3.3is satisfied for alln∈^Æ0, and by iterating we get

dzn, z_n1αⁿdz0, z1. 3.8

SinceK≥1, form > nwe have

dzn, z_mKdzn, z_n1 K²dzn1, z_n2 · · ·K^m−n−1dzm−1, z_m

KαⁿK²αⁿ¹· · ·K^m−nα^m−1

dz0, z₁

Kαⁿ

1−Kαdz0, z1−→θ, asn−→ ∞.

3.9

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Now, byp4andp1, it follows that for everyc∈intPthere exists positive integern0such thatdzn, zmcfor everym > n > n0, so{zn}is a Cauchy sequence.

Let us suppose that SX is complete subspace of X. Completeness of SX implies existence ofz∈SXsuch that lim_n_{→ ∞}z_2n lim_n_{→ ∞}Sx_2nz. Then, we have

nlim→ ∞Gx_2n1 lim

n→ ∞Sx_2n lim

n→ ∞Fx_2n lim

n→ ∞Tx_2n1 z, 3.10 that is, for anyθc, for suﬃciently largenwe havedzn, zc. Sincez∈SX⊂GX, there existsy∈Xsuch thatzGy. Let us prove thatzTy. Fromd3and3.2, we have

d Ty, z

Kd

Ty, Sx2n

KdSx2n, zλu x2n, y

Kdz2n, z, 3.11

where u

x2n, y

∈

Kd

Fx2n, Gy

, KdFx2n, Sx2n, Kd

Gy, Ty , Kd

Fx_2n, Ty d

Gy, Sx_2n 2

Kdz2n−1, z, Kdz2n−1, z2n, Kd z, Ty

, Kd

z_2n−1, Ty

dz, z2n 2

.

3.12

Therefore we have the following four cases:

idTy, zKλdz2n−1, z Kdz2n, zKλ·c/2Kλ K·c/2K c, asn → ∞;

iidTy, zKλdz2n−1, z_2nKdz2n, zKλ·c/2KλK·c/2K c, asn → ∞;

iiidTy, zKλdz, Ty Kdz2n, z, that is,

d Ty, z

K

1−Kλdz2n, z K

1−Kλ·1−Kλ

K ·cc, asn−→ ∞; 3.13 ivdTy, zKλ/2dz2n−1, Ty dz, z2n Kdz2n, z, that is, because ofd3,

d Ty, z

Kλ 2

Kdz2n−1, z Kd z, Ty

dz, z2n

Kdz2n, z, 3.14

which implies

d Ty, z

1

1−K²λ/2 K²λ

2 dz2n−1, z Kλ

2 K

dz2n, z

K²λ 2−K²λ

2−K²λ K²λ

c

2 Kλ2 2−K²λ

2−K²λ Kλ2

c

2 c, asn−→ ∞,

3.15

since from 1≤K≤2 andλ∈0,1/Kwe haveλ <1/K≤2/K², and therefore 1−K²λ/2>0.

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Therefore,dTy, z cfor eachc ∈ intP. So, byp2we havedTy, z θ, that is, TyGyz,yis a coincidence point, andzis a point of coincidence ofTandG.

SinceTX ⊂ FX, there existsv ∈Xsuch thatz Fv. Let us prove thatSv z. From d3and3.2, we have

dSv, zKdSv, Tx2n1 KdTx2n1, zλuv, x2n1 Kdz2n1, z, 3.16

where uv, x2n1

∈

KdFv, Gx2n1, KdFv, Sv, KdGx2n1, Tx_2n1, KdFv, Tx2n1 dGx2n1, Sv 2

Kdz, z2n, Kdz, Sv, Kdz2n, z_2n1, Kdz, z2n1 dz2n, Sv 2

.

3.17

Therefore we have the following four cases:

idSv, zKλdz, z2n Kdz2n1, z;

iidSv, zKλdz, Sv Kdz_2n1, z;

iiidSv, zKλdz2n, z_2n1 Kdz2n1, z;

ivdSv, zKλ/2dz, z_2n1 dz2n, Sv Kdz_2n1, z.

By the same arguments as above, we conclude thatdSv, z θ, that is,SvFvz.

So,zis a point of coincidence ofSandF, too.

Now we prove thatzis unique point of coincidence of pairs{S, F}and{T, G}. Suppose that there existsz^∗which is also a point of coincidence of these four mappings, that is,Fv^∗ Gy^∗ Sv^∗Ty^∗z^∗. From3.2,

dz, z^∗ d

Sv, Ty^∗ λ

Ku v, y^∗

, 3.18

where

u v, y^∗

∈

Kd

Fv, Gy^∗

, KdFv, Sv, d

Gy^∗, Ty^∗ , Kd

Fv, Ty^∗ d

Gy^∗, Sv 2

{Kdz, z^∗, θ}.

3.19

Usingp3we getdz, z^∗ θ, that is,zz^∗. Therefore,zis the unique point of coincidence of pairs{S, F}and{T, G}. If these pairs are weakly compatible, thenzis the unique common fixed point ofS,F,T, andG, byLemma 2.8.

Similarly, we can prove the statement in the cases whenFX,GX, orTXis complete.

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We give one simple, but illustrative, example.

Example 3.2. LetX ^Ê,E^Ê, andP 0,∞. Let us definedx, y |x−y|²for allx, y∈X.

ThenX, dis a CMTS, but it is not a cone metric space since the triangle inequality is not satisfied. Starting with Minkowski inequalitysee18forp2, by using the inequality of arithmetic and geometric means, we get

|x−z|²≤x−y²y−z²2x−y|x−z| ≤2x−y²y−z²

. 3.20

Here,K2.

Let us define four mappingsS, F, T, G:X → Xas follows:

SxMaxb, Fxaxb, TxMcxd, Gxcxd, 3.21

where x ∈ X,a /0, c /0, and M < 1/√

2. Since SX FX TX GX X we have trivially SX ⊂ GX and TX ⊂ FX. Also, X is a complete space. Further, dSx, Ty

|Maxb−Mcyd|² M²dFx, Gy, that is, there existsλ M² < 1/2 1/K such that3.2is satisfied.

According toTheorem 3.1,{S, F}and{T, G}have a unique point of coincidence inX, that is, there exists uniquez∈Xand there existx, y∈Xsuch thatzSxFxTy Gy. It is easy to see thatx−b/a,y−d/c, andz0.

If{S, F}is weakly compatible pair, we haveSFx FSx, which impliesMb b, that is,b 0. Similarly, if{T, G}is weakly compatible pair, we haveTGy GTy, which implies Mdd, that is,d0. Thenxy 0, andz0 is the unique common fixed point of these four mappings.

The following two theorems can be proved in the same way asTheorem 3.1, so we omit the proofs.

Theorem 3.3. LetX, dbe a CMTS with constantK ≥ 2 andP a solid cone. Suppose that self- mappings F, G, S, T : X → X are such thatSX ⊂ GX,TX ⊂ FX and that for some constant λ∈0,2/K²for allx, y∈Xthere exists

u x, y

∈

Kd

Fx, Gy

, KdFx, Sx, Kd

Gy, Ty , Kd

Fx, Ty d

Gy, Sx 2

, 3.22

d Sx, Ty

λ Ku

x, y

, 3.23

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Theorem 3.4. LetX, dbe a CMTS with constantK ≥ 1 andP a solid cone. Suppose that self- mappings F, G, S, T : X → X are such thatSX ⊂ GX,TX ⊂ FX and that for some constant λ∈0,1/Kfor allx, y∈Xthere exists

u x, y

∈

Kd

Fx, Gy

, KdFx, Sx, Kd

Gy, Ty ,d

Fx, Ty d

Gy, Sx 2

, 3.24

d Sx, Ty

λ Ku

x, y

, 3.25

Theorems3.1and3.4are generalizations of13, Theorem 2.2. As a matter of fact, for K1, from Theorems3.1and3.4, we get13, Theorem 2.2.

If we chooseT SandG F, from Theorems3.1,3.3, and3.4we get the following results for two mappings on CMTS.

Corollary 3.5. LetX, dbe a CMTS with constant 1 ≤ K ≤ 2 andP a solid cone. Suppose that self-mappingsF, S:X → Xare such thatSX⊂FXand that for some constantλ∈0,1/Kfor all x, y∈Xthere exists

u x, y

∈

Kd

Fx, Fy

, KdFx, Sx, Kd

Fy, Sy , Kd

Fx, Sy d

Fy, Sx 2

, 3.26

d Sx, Sy

λ Ku

x, y

, 3.27

holds. IfFXorSXis complete subspace ofX, thenFandShave a unique point of coincidence inX.

Moreover, if{F, S}is a weakly compatible pair, thenFandShave a unique common fixed point.

Corollary 3.6. LetX, dbe a CMTS with constantK ≥ 2 andP a solid cone. Suppose that self- mappingsF, S :X → Xare such thatSX⊂ FXand that for some constantλ ∈0,2/K²for all x, y∈Xthere exists

u x, y

∈

Kd

Fx, Fy

, KdFx, Sx, Kd

Fy, Sy , Kd

Fx, Sy d

Fy, Sx 2

, 3.28

d Sx, Sy

λ Ku

x, y

, 3.29

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Corollary 3.7. LetX, dbe a CMTS with constantK ≥ 1 andP a solid cone. Suppose that self- mappingsF, S :X → X are such thatSX ⊂ FXand that for some constantλ ∈ 0,1/Kfor all x, y∈Xthere exists

u x, y

∈

Kd

Fx, Fy

, KdFx, Sx, Kd

Fy, Sy ,d

Fx, Sy d

Fy, Sx 2

, 3.30

d Sx, Sy

λ Ku

x, y

, 3.31

Theorem 3.8. LetX, dbe a CMTS with constantK ≥ 1 andP a solid cone. Suppose that self- mappingsF, G, S, T :X → X are such thatSX ⊂GX,TX ⊂FXand that there exist nonnegative constantsa_i,i1, . . . ,5, satisfying

a1a2a32Kmax{a4, a5}<1, a3Ka4K²<1, a2Ka5K²<1, 3.32 such that for allx, y∈Xinequality

d

Sx, Ty a₁d

Fx, Gy

a₂dFx, Sx a₃d

Gy, Ty a₄d

Fx, Ty a₅d

Gy, Sx , 3.33

Proof. We define sequences{xn}and{zn}as in the proof ofTheorem 3.1. First we prove that dzn, z_n1αdzn−1, zn, n≥1, 3.34

where

αmax

a₁a₃a₅K

1−a₂−a₅K ,a₁a₂a₄K 1−a₃−a₄K

, 3.35

which implies that{zn}is a Cauchy sequence, since, because of3.32, it is easy to check that α∈0,1. To prove this, it is necessary to consider the cases of an odd and of an even integer n.

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Forn21,∈^Æ0, we havedz21, z₂₂ dSx22, Tx₂₁, and from3.33we have

dSx22, Tx₂₁a1dFx22, Gx₂₁ a2dFx22, Sx₂₂

a3dGx21, Tx₂₁ a4dFx22, Tx₂₁ a5dGx21, Sx₂₂, 3.36

that is,

dz21, z₂₂a₁dz21, z₂ a₂dz21, z₂₂ a₃dz2, z₂₁ a₄dz21, z₂₁ a₅dz2, z₂₂

a1a3dz2, z₂₁ a2dz21, z₂₂ a5dz2, z₂₂ a1a₃dz2, z₂₁ a₂dz21, z₂₂ a₅Kdz2, z₂₁

a5Kdz21, z₂₂

a1a3a5Kdz2, z₂₁ a2a5Kdz₂₁, z₂₂.

3.37

Therefore,

dz21, z₂₂a₁a₃a₅K

1−a2−a5K dz2, z₂₁, 3.38 that is, inequality3.34holds in this case.

Similarly, forn 2, ∈^Æ0, we havedz2, z₂₁ dSx2, Tx₂₁, and from3.33 we have

dSx2, Tx₂₁a1dFx2, Gx₂₁ a2dFx2, Sx2 a3dGx₂₁, Tx₂₁ a4dFx2, Tx₂₁ a₅dGx21, Sx₂,

3.39

that is,

dz2, z₂₁a₁dz2−1, z₂ a₂dz2−1, z₂ a₃dz2, z₂₁ a4dz2−1, z₂₁ a5dz2, z2

a1a2dz₂₋₁, z2 a3dz2, z₂₁ a4dz₂₋₁, z₂₁

a1a₂dz2−1, z₂ a₃dz2, z₂₁ a₄Kdz2−1, z₂ a₄Kdz2, z₂₁ a1a2a4Kdz2−1, z2 a3a4Kdz2, z₂₁.

3.40

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Thus,

dz2, z₂₁ a1a2a4K

1−a₃−a₄K dz2−1, z2, 3.41 and inequality3.34holds in this case, too.

By the same arguments as inTheorem 3.1we conclude that{zn}is a Cauchy sequence.

Let us suppose that SX is complete subspace of X. Completeness of SX implies existence ofz∈SXsuch that limn→ ∞z2n limn→ ∞Sx2nz. Then, we have

nlim→ ∞Gx_2n1 lim

n→ ∞Sx_2n lim

n→ ∞Fx_2n lim

n→ ∞Tx_2n1 z, 3.42 that is, for anyθc, for suﬃciently largenwe havedzn, zc. Sincez∈SX⊂GX, there existsy∈Xsuch thatzGy. Let us prove thatzTy. Fromd3and3.33, we have

d Ty, z

Kd

Ty, Sx_2n

KdSx2n, z a1Kd

Fx2n, Gy

a2KdFx2n, Sx2n a3Kd

Gy, Ty a₄Kd

Fx_2n, Ty

a₅Kd

Gy, Sx_2n

KdSx2n, z a1Kdz2n−1, z a2Kdz2n−1, z2n a3Kd

z, Ty a₄Kd

z_2n−1, Ty

a₅Kdz, z2n Kdz2n, z a1Kdz_2n−1, z a2Kdz_2n−1, z2n a3Kd

z, Ty a4K²dz2n−1, z a4K²d

z, Ty

a5Kdz, z2n Kdz2n, z.

3.43

The sequence{zn}converges toz, so for eachc∈intPthere existsn0 ∈^Æsuch that for every n > n0

d Ty, z

1

1−a₃K−a₄K²

a₁Kdz2n−1, z a₂Kdz2n−1, z_2n a₄K²dz2n−1, z a₅Kdz, z2n Kdz2n, z

a1K

1−a3K−a4K² ·1−a3K−a4K² a₁K ·c

5 a2K

1−a3K−a4K² ·1−a3K−a4K² a₂K ·c

5 a4K²

1−a₃K−a₄K² ·1−a3K−a4K² a₄K² ·c

5 a5K

1−a₃K−a₄K² ·1−a3K−a4K² a5K ·c

5 K

1−a3K−a4K² ·1−a₃K−a₄K²

K · c

5 c,

3.44

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because of3.32. Now, byp2 it follows thatdTy, z θ, that is, Ty z. So, we have Ty Gyz, that is,yis a coincidence point, andzis a point of coincidence of mappingsT andG.

SinceTX ⊂ FX, there existsv ∈ X such thatz Fv. Let us prove thatSv z, too.

Fromd3and3.33, we have

dSv, zKdSv, Tx2n1 KdTx2n1, z

a₁KdFv, Gx2n1 a₂KdFv, Sv a₃KdGx2n1, Tx_2n1 a4KdFv, Tx_2n1 a5KdGx_2n1, Sv KdTx_2n1, z a1Kdz, z2n a2Kdz, Sv a3Kdz2n, z_2n1

a₄Kdz, z2n1 a₅Kdz2n, Sv KdTx2n1, z a1Kdz, z2n a2Kdz, Sv a3Kdz2n, z_2n1

a4Kdz, z2n1 a5K²dz2n, z a5K²dSv, z KdTx2n1, z,

3.45

and by the same arguments as above, we conclude thatdSv, z θ, that is,Sv Fv z.

Thus,zis a point of coincidence of mappingsSandF, too.

Suppose that there exists z^∗ which is also a point of coincidence of these four mappings, that is,Fv^∗Gy^∗Sv^∗Ty^∗z^∗. From3.33we have

dz, z^∗ d

Sv, Ty^∗ a₁Kd

Fv, Gy^∗

a₂KdFv, Sv a₃Kd

Gy^∗, Ty^∗ a4Kd

Fv, Ty^∗

a4Kd

Gy^∗, Sv

a₁Kdz, z^∗ a₂Kdz, z a₃Kdz^∗, z^∗ a₄Kdz, z^∗ a₅Kdz^∗, z a1a4a5Kdz, z^∗,

3.46

andbecause ofp3it follows thatzz^∗. Therefore,zis the unique point of coincidence of pairs{S, F}and{T, G}, and we havezSvFvGyTy. If{S, F}and{T, G}are weakly compatible pairs, thenzis the unique common fixed point ofS,F,T, andG, byLemma 2.8.

The proofs for the cases in whichFX,GX, orTXis complete are similar.

Theorem 3.8 is a generalization of 13, Theorem 2.8. Choosing K 1 from Theorem 3.8we get the following corollary.

Corollary 3.9. Let X, d be cone metric space and P a solid cone. Suppose that self-mappings F, G, S, T :X → X are such thatSX⊂ GX,TX ⊂ FXand that there exist nonnegative constants a_i,i1, . . . ,5, satisfyinga₁a₂a₃2 max{a4, a₅}<1, such that for allx, y∈Xinequality

d

Sx, Ty a₁d

Fx, Gy

a₂dFx, Sx a₃d

Gy, Ty a₄d

Fx, Ty a₅d

Gy, Sx , 3.47

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If we chooseT SandGF, fromTheorem 3.8, we get the following result for two mappings on CMTS.

Corollary 3.10. LetX, dbe a CMTS with constantK ≥ 1 andP a solid cone. Suppose that self- mappingsF, S : X → X are such thatSX ⊂ FXand that there exist nonnegative constants a_i, i1, . . . ,5, satisfying

a₁a₂a₃2Kmax{a4, a₅}<1, a₃Ka₄K²<1, a₂Ka₅K²<1, 3.48 such that for allx, y∈Xinequality

d Sx, Sy

a1d

Fx, Fy

a2dFx, Sx a3d

Fy, Sy a4d

Fx, Sy a5d

Fy, Sx , 3.49

holds. If one ofSXorFXis complete subspace ofX, thenSandFhave a unique point of coincidence inX. Moreover, if{F, S}is a weakly compatible pair, thenFandShave a unique common fixed point.

Acknowledgments

The authors are indebted to the referees for their valuable suggestions, which have contributed to improve the presentation of the paper. The first two authors were supported in part by the Serbian Ministry of Science and Technological DevelopmentsGrant no. 174015.

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