Assad-Kirk-Type Fixed Point Theorems for a Pair of Nonself Mappings on Cone Metric Spaces

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Volume 2009, Article ID 761086,16pages doi:10.1155/2009/761086

Research Article

Assad-Kirk-Type Fixed Point Theorems for a Pair of Nonself Mappings on Cone Metric Spaces

S. Jankovi ´c,

¹

Z. Kadelburg,

²

S. Radenovi ´c,

³

and B. E. Rhoades

⁴

1Mathematical Institute SANU, Knez Mihailova 36, 11001 Beograd, Serbia

2Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia

3Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia

4Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

Correspondence should be addressed to S. Radenovi´c,[email protected] Received 7 February 2009; Accepted 27 April 2009

Recommended by William A. Kirk

New fixed point results for a pair of non-self mappings defined on a closed subset of a metrically convex cone metric spacewhich is not necessarily normalare obtained. By adapting Assad-Kirk’s method the existence of a unique common fixed point for a pair of non-self mappings is proved, using only the assumption that the cone interior is nonempty. Examples show that the obtained results are proper extensions of the existing ones.

Copyrightq2009 S. Jankovi´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.

1. Introduction and Preliminaries

Cone metric spaces were introduced by Huang and Zhang in1, where they investigated the convergence in cone metric spaces, introduced the notion of their completeness, and proved some fixed point theorems for contractive mappings on these spaces. Recently, in2–4, some common fixed point theorems have been proved for maps on cone metric spaces. However, in1–3, the authors usually obtain their results for normal cones. In this paper we do not impose the normality condition for the cones.

We need the following definitions and results, consistent with1, in the sequel.

LetEbe a real Banach space. A subsetPofEis a cone if iPis closed, nonempty andP /{0};

iia, b∈R, a, b≥0, andx, y∈Pimplyaxby∈P;

iiiP∩−P {0}.

Given a coneP⊂E, we define the partial ordering≤with respect toPbyx≤yif and only ify−x ∈ P. We writex < y to indicate thatx ≤ ybutx /y, whilex ystands for y−x∈intP the interior ofP.

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There exist two kinds of cones: normal and nonnormal cones. A coneP ⊂Eis a normal cone if

infxy:x, y∈P,xy1

>0 1.1

or, equivalently, if there is a numberK >0 such that for allx, y∈P,

0≤x≤y impliesx ≤Ky. 1.2

The least positive number satisfying1.2is called the normal constant ofP. It is clear that K≥1.

It follows from1.1thatPis nonnormal if and only if there exist sequencesx_n, y_n ∈P such that

0≤x_n≤x_ny_n, x_ny_n−→0 butx_n0. 1.3 So, in this case, the Sandwich theorem does not hold.

Example 1.1see5. LetE C¹_R0,1withx x_∞x_∞ andP {x ∈ E : xt ≥ 0 on0,1}. This cone is not normal. Consider, for example,

x_nt 1−sinnt

n2 , y_nt 1sinnt

n2 . 1.4

Thenxnyn1 andxnyn2/n2 → 0.

Definition 1.2see1. LetXbe a nonempty set. Suppose that the mappingd:X×X → E satisfies

d10≤dx, yfor allx, y∈Xanddx, y 0 if and only ifxy;

d2dx, y dy, xfor allx, y∈X;

d3dx, y≤dx, z dz, yfor allx, y, z∈X.

Thendis called a cone metric onX,andX, dis called a cone metric space.

The concept of a cone metric space is more general than that of a metric space, because each metric space is a cone metric space withERandP 0,∞ see1, Example 1and 4, Examples 1.2 and 2.2.

Let{xn}be a sequence inX, and letx∈X. If, for everycinEwith 0c, there is an n₀ ∈ Nsuch that for alln > n₀,dxn, x c, then it is said thatx_n converges tox, and this is denoted by lim_n_{→ ∞}x_n x, orx_n → x,n → ∞. If for everycinEwith 0c, there is an n0∈Nsuch that for alln, m > n0,dxn, xmc, then{xn}is called a Cauchy sequence inX. If every Cauchy sequence is convergent inX, thenXis called a complete cone metric space.

Huang and Zhang1proved that ifP is a normal cone thenx_n ∈ X converges to x∈Xif and only ifdxn, x → 0,n → ∞, and thatxn ∈Xis a Cauchy sequence if and only ifdxn, x_m → 0,n, m → ∞.

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Let X, d be a cone metric space. Then the following properties are often useful particulary when dealing with cone metric spaces in which the cone needs not to be normal:

p₁ifu≤vandvw, thenuw,

p₂if 0≤ucfor eachc∈intPthenu0, p₃ifa≤bcfor eachc∈intPthena≤b, p₄if 0≤x≤y, anda≥0, then 0≤ax≤ay,

p₅if 0≤x_n≤y_nfor eachn∈N,and lim_n_{→ ∞}x_nx, lim_{n→ ∞}y_n y, then 0≤x≤y, p₆if 0≤dxn, x≤b_nandb_n → 0, thendxn, xcwherex_nandxare, respectively,

a sequence and a given point inX,

p₇ifEis a real Banach space with a coneP and ifa≤λawherea∈P and 0< λ <1, thena0,

p₈ifc∈intP, 0≤anandan → 0, then there existsn0such that for alln > n0we have a_nc.

It follows fromp₈that the sequencex_nconverges tox∈Xifdxn, x → 0 asn → ∞ andx_n is a Cauchy sequence ifdxn, x_m → 0 asn, m → ∞. In the case when the cone is not necessarily normal, we have only one half of the statements of Lemmas 1 and 4 from1.

Also, in this case, the fact thatdxn, y_n → dx, yifx_n → xandy_n → yis not applicable.

There exist a lot of fixed-point theorems for self-mappings defined on closed subsets of Banach spaces. However, for applications numerical analysis, optimization, etc. it is important to consider functions that are not self-mappings, and it is natural to search for suﬃcient conditions which would guarantee the existence of fixed points for such mappings.

In what follows we suppose only thatEis a Banach space, thatPis a cone inEwith intP /∅and that≤is the partial ordering with respect toP.

Rhoades 6 proved the following result, generalizing theorems of Assad 7 and Assad and Kirk8.

Theorem 1.3. LetX be a Banach space,Ca nonempty closed subset ofX,and letT :C → Xbe a mapping fromCintoXsatisfying the condition

d

Tx, Ty

≤hmax d

x, y

2 , dx, Tx, d y, Ty

,d x, Ty

d y, Tx q

, 1.5

for some 0< h <1,q≥12h, and for allx, yinC. LetT have the additional property that for each x∈∂C,the boundary ofC,Tx∈C. ThenT has the unique fixed point.

Recently Imdad and Kumar9extended this result of Rhoades by considering a pair of maps in the following way.

Theorem 1.4. LetXbe a Banach space, letCbe a nonempty closed subset ofX,and letF, T :C → X be two mappings satisfying the condition

d

Fx, Fy

≤hmax d

Tx, Ty

2 , dTx, Fx, d

Ty, Fy ,d

Tx, Fy d

Ty, Fx q

, 1.6

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for some 0< h <1,q≥12h, and for allx, y∈Cand suppose i∂C⊆TC,FC∩C⊂TC,

iiTx∈∂C⇒Fx∈C, iiiTCis closed inX.

Then there exists a coincidence pointzofF, T inX. Moreover, ifFandT are coincidentally commuting, thenzis the unique common fixed point ofFandT.

Recall that a pairf, gof mappings is coincidentally commutingsee, e.g.,2if they commute at their coincidence point, that is, iffxgxfor somex∈X, impliesfgxgfx.

In10, 11 these results were extended using complete metric spaces of hyperbolic type, instead of Banach spaces.

2. Results

2.1. Main Result

In12, assuming only that intP /∅, Theorems1.3and1.4are extended to the setting of cone metric spaces. Thus, proper generalizations of the results of Rhoades6 for one mapand of Imdad and Kumar9 for two mapswere obtained.Example 1.1of a nonnormal cone shows that the method of proof used in6,8,9cannot be fully applied in the new setting.

The purpose of this paper is to extend the previous results to the cone metric spaces, but with new contractive conditions. This is worthwhile, since from 2,13 we know that self-mappings that satisfy the new conditionsgiven belowdo have a unique common fixed point. Let us note that the questions concerning common fixed points for self-mappings in metric spaces, under similar conditions, were considered in14. It seems that these questions were not considered for nonself mappings. This is an additional motivation for studying these problems.

We begin with the following definition.

Definition 2.1. LetX, dbe a cone metric space, letCbe a nonempty closed subset ofX, and letf, g:C → X. Denote, forx, y∈C,

M^f,g₁ C;x, y

d gx, gy

, d fx, gx

, d fy, gy

,d fx, gy

d fy, gx 2

. 2.1

Thenfis called a generalizedgM1-contractive mapping of Cinto Xif for someλ∈0,√ 2−1 there exists

u x, y

∈M^f,g₁ C;x, y

, 2.2

such that for allx, yinC

d fx, fy

≤λ·u x, y

. 2.3

Our main result is the following.

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Theorem 2.2. LetX, dbe a complete cone metric space, letCbe a nonempty closed subset ofXsuch that, for eachx∈Cand eachy /∈C,there exists a pointz∈∂Csuch that

dx, z d z, y

d x, y

. 2.4

Suppose thatfis a generalizedgM1-contractive mapping ofCintoXand i∂C⊆gC,fC∩C⊂gC,

iigx∈∂C⇒fx∈C, iiigCis closed inX.

Then there exists a coincidence pointzoff, ginC. Moreover, if the pairf, gis coincidentally commuting, thenzis the unique common fixed point offandg.

Proof. We prove the theorem under the hypothesis that neither of the mappingsf andg is necessarily a self-mapping. We proceed in several steps.

Step 1 construction of three sequences. The following construction is the same as the construction used in10in the case of hyperbolic metric spaces. It diﬀers slightly from the constructions in6,9.

Letx∈∂Cbe arbitrary. We construct three sequences:{xn}and{zn}inCand{yn}in fC ⊆ X in the following way. Setz0 x. Sincez0 ∈ ∂C, byithere exists a pointx0 ∈ C such thatz0 gx0. Sincegx0 ∈∂C, fromiiwe conclude thatfx0 ∈C∩fC. Then fromi, fx₀ ∈gC. Thus, there existsx₁ ∈Csuch thatgx₁ fx₀ ∈ C. Setz₁ y₁ fx₀ gx₁and y2fx1.

Ify2 ∈fC∩C, then fromi,y2 ∈gCand so there is a pointx2 ∈ Csuch thatgx2 y₂z₂fx₁.

If y₂ fx₁/∈C, then z₂ is a point in ∂C,z2/y₂ such that dy1, z₂ dz2, y₂ dy1, y2 dfx0, fx1. Byi, there isx2∈Csuch thatgx2z2. Thusz2∈∂Canddy1, z2

dz2, y₂ dy1, y₂ dfx0, fx₁.

Now we sety₃ fx₂ z₃. Sincefx₂ ∈fC∩C⊆gC, fromiithere is a pointx₃ ∈C such thatgx3y3.

Note that in the casez₂/y₂ fx₁, we havez₁y₁fx₀andz₃ y₃fx₂.

Continuing the foregoing procedure we construct three sequences:{xn} ⊆C,{zn} ⊂C and{yn} ⊆fC⊂Xsuch that:

aynfx_n−1; bz_ngx_n;

cznynif and only ifyn∈C;

dz_n/y_nwhenevery_n/∈Cand thenz_n∈∂Canddy_n−1, z_n dzn, y_n dy_n−1, y_n. Step 2 {zn} is a Cauchy sequence. First, note that if zn/yn, then zn ∈ ∂C, which then implies, byb,ii, anda, thatz_n1 y_n1 ∈C.Also,zn/ynimplies thatz_n−1 y_n−1 ∈C, since otherwisez_n−1 ∈∂C, which then impliesz_ny_n∈C.

Proof ofStep 2

Now we have to estimatedzn, z_n1. Ifdzn, z_n1 0 for somen, then it is easy to show that dzn, z_nk 0 for allk≥1.

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Suppose thatdzn, z_n1>0 for alln. There are three possibilities:

1znyn∈Candz_n1y_n1∈C;

2z_ny_n∈C, butz_n1/y_n1; and

3z_n/y_n, in which case z_n ∈ ∂C and dyn−1, z_n dzn, y_n dyn−1, y_n dfxn−2, fx_n−1.

Note that the estimate ofdzn, z_n1in this cone version diﬀers from those from6,8–

11. In the case of convex metric spaces it can be used that, for eachx, y, u ∈ X and each λ∈0,1, it isλdu, x 1−λdu, y≤max{du, x, du, y}. In cone spaces the maximum of the set{du, x, du, y}needs not to exist. Therefore, besides2.4, we have to use here the relation “∈”, and to consider several cases. In cone metric spaces as well as in metric spaces the key step is Assad-Kirk’s induction.

Case 1. Letz_n y_n∈C,and letz_n1y_n1∈C. Thenz_ny_nfx_n−1,z_n1 y_n1fx_nand z_n−1gx_n−1observe that it is not necessarilyz_n−1y_n−1. Then from2.3,

dzn, z_n1 d

y_n, y_n1 d

fx_n−1, fx_n

≤λ·u_n, 2.5 where

u_n∈

d

gx_n−1, gx_n , d

fx_n−1, gx_n−1 , d

fx_n, gx_n ,d

fx_n−1, gxn

d

fxn, gx_n−1 2

dz_n−1, z_n, d

y_n, z_n−1 , d

y_n1, z_n ,d

yn, zn

d

y_n1, z_n−1 2

dz_n−1, z_n, dzn, z_n1,dz_n−1, z_n1 2

.

2.6

Clearly, there are infinitely manyn’s such that at least one of the following cases holds:

Idzn, z_n1≤λ·dzn−1, zn,

IIdzn, z_n1 ≤ λ·dzn, z_n1 ⇒ dzn, z_n1 0,contradicting the assumption that dzn, z_n1>0 for eachn. Hence,Iholds,

IIIdzn, z_n1 ≤ λ · dzn−1, z_n1/2 ≤ λ/2dzn−1, zn 1/2dzn, z_n1 ⇒ dzn, z_n1≤λdz_n−1, zn, that is,Iholds.

FromI,II, andIIIit follows that in Case1

dzn, z_n1≤λ·dz_n−1, z_n. 2.7 Case 2. Letz_n y_n ∈ C butz_n1/y_n1. Then z_n1 ∈ ∂C and dyn, z_n1 dzn1, y_n1 dyn, y_n1. It follows that

dzn, z_n1 d

yn, z_n1 d

yn, y_n1

−d

z_n1, y_n1

< d

yn, y_n1

, 2.8

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that is, according to2.3,dyn, y_n1 dfxn−1, fxn≤h·un, where

un∈

d

gx_n−1, gxn

,d

fx_n−1, gx_n−1 , d

fxn, gxn

,d

fx_n−1, gx_n d

fx_n, gx_n−1 2

dzn−1, zn, d

yn, z_n−1 , d

y_n1, yn

,d

y_n1, z_n−1 2

dzn−1, zn, dzn, z_n−1, d y_n1, yn

,d

y_n1, z_n−1 2

dzn−1, zn, d y_n1, yn

,d

y_n1, z_n−1 2

.

2.9

Again, we obtain the following three cases Idyn, y_n1≤λ·dzn−1, zn.

IIdyn, y_n1 ≤ λ·dyn, y_n1 ⇒ dyn, y_n1 0, contradicting the assumption that dzn, z_n1>0 for eachn. It follows thatIholds.

IIIdyn, y_n1 ≤ λ · dy_n1, z_n−1/2 ≤ λ/2dy_n1, y_n λ/2dyn, z_n−1 ≤ 1/2dyn1, yn λ/2dzn, z_n−1, that isdyn, y_n1≤λ·dzn−1, zn.

From2.8,I,II, andIII, we have

dzn, z_n1≤λ·dz_n−1, z_n, 2.10

in Case2.

Case 3. Letz_n/y_n.Thenz_n∈∂C, dy_n−1, z_n dzn, y_n dy_n−1, y_nand we havez_n1y_n1 andz_n−1y_n−1. From this and using2.4we get

dzn, z_n1 d

z_n, y_n1

≤d z_n, y_n

d

y_n, y_n1 d

y_n−1, yn

−dzn−1, zn d

yn, y_n1

. 2.11

We have to estimatedy_n−1, y_nand dyn, y_n1. Sincey_n−1 z_n−1, one can conclude that

d y_n−1, yn

≤λ·dzn−2, z_n−1, 2.12

in view of Case2. Further,

d

yn, y_n1 d

fx_n−1, fxn

≤λ·un, 2.13

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where

un∈

d

gx_n−1, gxn

, d

fx_n−1, gx_n−1 , d

fxn, gxn

,d

fx_n−1, gx_n d

fx_n, gx_n−1 2

dzn−1, zn, d

yn, y_n−1

, dzn, z_n1,d y_n, z_n

d

y_n1, z_n−1 2

.

2.14

Since

d yn, zn

d

y_n1, z_n−1

2 d

yn, zn

dzn1, z_n−1 2

d

y_n, y_n−1

−dz_n−1, z_n dz_n1, z_n−1 2

≤ d

yn, y_n−1

−dzn−1, zn dzn−1, zn dzn, z_n1 2

d

y_n, y_n−1

dzn, z_n1

2 ,

2.15

y_n−1z_n−1,y_n1z_n1, anddyn−1, y_n≤λ·dzn−2, z_n−1, we have that

d

y_n, y_n1

≤λ·u_n, 2.16

where

un∈ dzn−1, zn, λ·dzn−2, z_n−1, dzn1, zn,λ·dz_n−2, z_n−1 dzn, z_n1 2

. 2.17

Substituting2.12and2.16into2.11we get

dzn, z_n1≤λ·dzn−2, z_n−1−dzn−1, z_n λ·u_n. 2.18

We have now the following four cases:

I

dzn, z_n1≤λ·dz_n−2, z_n−1−dz_n−1, z_n λ·dz_n−1, z_n

λ·dzn−2, z_n−1−1−λdzn−1, zn≤λ·dzn−2, z_n−1; 2.19

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II

dzn, z_n1≤λdzn−2, z_n−1−dzn−1, zn λ²dzn−2, z_n−1

λλ²

dz_n−2, z_n−1−dz_n−1, z_n≤ λλ²

dz_n−2, z_n−1; 2.20

III

dzn, z_n1≤λ·dzn−2, z_n−1−dzn−1, zn λ·dzn, z_n1 ⇒1−λdzn, z_n1

≤λdzn−2, z_n−1 ⇒dzn, z_n1≤ λ

1−λdzn−2, z_n−1; 2.21

IV

dzn, z_n1≤λdzn−2, z_n−1−dzn−1, zn λ

2λdzn−2, z_n−1 dzn, z_n1

≤

λ λ² 2

dzn−2, z_n−1 1

2dzn, z_n1 ⇒dzn, z_n1

≤

2λλ²

dz_n−2, z_n−1.

2.22

It follows fromI,II,III, andIVthat

dzn, z_n1≤μ·dzn−2, z_n−1, where μmax λ, λλ², λ

1−λ,2λλ²

max λ

1−λ,2λλ²

. 2.23

Thus, in all Cases1–3,

dzn, z_n1≤μ·wn, 2.24

wherewn∈ {dz_n−2, z_n−1, dz_n−1, zn}and

μmax λ, λ

2−λ, λλ², λ

1−λ,2λλ²

max λ

1−λ,2λλ²

. 2.25

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It is not hard to conclude that for 0< λ <√ 2−1,

max λ

1−λ,2λλ²

2λλ²<1. 2.26

Now, following the procedure of Assad and Kirk8, it can be shown by induction that, forn >1,

dzn, z_n1≤μ^n−1/2·w₂, 2.27

wherew₂∈ {dz0, z₁, dz1, z₂}.

From2.27and using the triangle inequality, we have forn > m dzn, z_m≤dzn, z_n−1 dzn−1, z_n−2 · · ·dzm1, z_m

≤

μ^n−1/2μ^n−2/2· · ·μ^m−1/2

·w2

≤

√μ^m−1

1− √μ·w₂−→0, asm−→ ∞.

2.28

According to the propertyp₈from the Introduction,dzn, zmc; that is,{zn}is a Cauchy sequence.

Step 3 Common fixed point for f and g. In this step we use only the definition of convergence in the terms of the relation “”. The only assumption is that the interior of the coneP is nonempty; so we use neither continuity of vector metricd, nor the Sandwich theorem.

Sincezn gxn ∈ C∩gC andC∩gCis complete, there is some point z ∈ C∩gC such thatz_n → z. Letw ∈ Cbe such thatgw z. By the construction of {zn}, there is a subsequence{znk}such thatz_nky_nkfx_nk−1and hencefx_nk−1 → z.

We now prove thatfwz. We have

d fw, z

≤d

fw, fx_nk−1 d

fx_nk−1, z

≤λ·u_nkd

fx_nk−1, z

, 2.29

where

u_nk∈

d

gx_nk−1, gw ,d

fx_nk−1, gx_nk−1 ,d

fw, gw ,d

fx_nk−1, gw d

fw, gx_nk−1 2

.

2.30

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From the definition of convergence and the fact thatz_nk y_nk fx_nk−1 → z, as k → ∞, we obtainfor the givenc∈Ewith 0c

1d fw, z

≤λ·d

gx_nk−1, z d

fx_nk−1, z

λ· c 2λ c

2 c;

2d fw, z

≤λ·d

fx_nk−1, gx_nk−1 d

fx_nk−1, z

≤λ· d

fx_nk−1, z d

z, gx_nk−1 d

fx_nk−1, z λ1·d

fx_nk−1, z

λ·d

z, gx_nk−1 λ1· c

2λ1λ· c 2λ c;

3d fw, z

≤λ·d fw, z

d

fx_nk−1, z ⇒d

fw, z

≤ 1 1−λ·d

fx_nk−1, z

1

1−λ· c

1/1−λ c;

4d fw, z

≤λ·d

fx_nk−1, z d

fw, gx_nk−1

2 d

fx_nk−1, z

≤λ·d

fx_nk−1, z d

z, gx_nk−1

2 1

2d fw, z

d

fx_nk−1, z

; i.e.,

d fw, z

≤λ2·d

fx_nk−1, z

λ·d

z, gx_nk−1 λ2 c

2λ2λ c 2λ c.

2.31

In all the cases we obtaindfw, zcfor eachc ∈intP. According to the propertyp₂, it follows thatdfw, z 0, that is,fwz.

Suppose now thatfandgare coincidentally commuting. Then

zfwgw⇒fzfgwgfwgz. 2.32

Then from2.3,

d fz, z

d

fz, fw

≤λ·u, 2.33

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where

u∈

d

gz, gw , d

fz, gz , d

fw, gw ,d

fz, gw d

fw, gz 2

d fz, z

, d fz, gz

, dz, z,d fz, z

d z, fz 2

d fz, z

,0 .

2.34

Hence, we obtain the following cases:

d fz, z

≤λ·d fz, z

⇒d fz, z

0, d

fz, z

≤λ·00⇒d fz, z

0, 2.35

which implies thatfzz, that is,zis a common fixed point offandg.

Uniqueness of the common fixed point follows easily. This completes the proof of the theorem.

2.2. Examples

We present now two examples showing thatTheorem 2.2is a proper extension of the known results. In both examples, the conditions of Theorem 2.2 are fulfilled, but in the first one because of nonnormality of the conethe main theorems from6,9cannot be applied. This shows thatTheorem 2.2is more general, that is, the main theorems from6,9can be obtained as its special casesfor 0< λ <√

2−1taking · | · |,ERandP 0,∞.

Example 2.3The case of a nonnormal cone. LetX R, letC 0,1, andEC¹_R0,1, and letP {ϕ∈E:ϕt≥0, t∈0,1}. The mappingd:X×X → Eis defined in the following way:dx, y |x−y|ϕ, whereϕ∈Pis a fixed function, for example,ϕt e^t. Take functions fx ax,gx bx, 0 < a <1< b, so thata/b ≤λ <√

2−1, which map the setC 0,1 intoR. We have thatX, dis a complete cone metric space with a nonnormal cone having the nonempty interior. The topological and “metric” notions are used in the sense of definitions from15,16. For example, one easily checks the condition2.4, that is, that forx ∈ 0,1, y /∈0,1the following holds

dx,1 d 1, y

d x, y

⇐⇒ |1−x|ϕy−1ϕ y−xϕ⇐⇒1−xϕ

y−1 ϕ

y−x

ϕ. 2.36

The mappingsfandgare weakly compatible, that is, they commute in their fixed pointx0.

All the conditions ofTheorem 2.2are fulfilled, and so the nonself mappingsf andghave a unique common fixed pointx0.

Example 2.4The case of a normal cone. LetX 0,∞, letC 0,1⊂X, letER², and let P {x, y:x≥0, y≥0}. The mappingd:X×X → Eis defined asdx, y |x−y|, α|x−y|, α≥ 0. Take the functionsfx ax,gx bx, 0< a <1 < b, so thata/b < √

2−1, which

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map the set C 0,1into R. We have that X, d is a complete cone metric space with a normal cone having the normal coeﬃcientK 1, whose interior is obviously nonempty. All the conditions ofTheorem 2.2are fulfilled. We check again the condition2.4, that is, that for x∈C 0,1,y /∈C 0,1the following holds

dx,1 d 1, y

d x, y

⇐⇒|1−x|, α|1−x| y−1, αy−1 y−x, αy−x⇐⇒1−x

y−1 y−x,

α1−x α y−1

α y−x

.

2.37

The mappingsfandgare weakly compatible, that is, they commute in their fixed pointx0.

All the conditions ofTheorem 2.2are again fulfilled. The pointx0 is the unique common fixed point for nonself mappingsfandg.

2.3. Further Results

Remark 2.5. The following definition is a special case of Definition 2.1 when X, d is a metric space. But when X, d is a cone metric space, which is not a metric space, this is not true. Indeed, there may exist x, y ∈ X such that the vectors dfx, gx, dfy, gyand 1/2dfx, gx dfy, gy are incomparable. For the same reason Theorems2.2and 2.7 given beloware incomparable.

Definition 2.6. LetX, dbe a cone metric space, letCbe a nonempty closed subset ofX, and letf, g:C → X. Denote

M^f,g₂ C;x, y

d gx, gy

,d fx, gx

d fy, gy

2 ,d

fx, gy d

fy, gx 2

. 2.38

Thenfis called a generalizedgM2-contractive mapping fromCintoXif for someλ∈0,√ 2−1 there exists

u x, y

∈M^f,g₂ C;x, y

, 2.39

such that for allx, yinC

d fx, fy

≤λ·u x, y

. 2.40

Our next result is the following.

Theorem 2.7. LetX, dbe a complete cone metric space, and letCbe a nonempty closed subset ofX such that for eachx∈Candy /∈Cthere exists a pointz∈∂Csuch that

dx, z d z, y

d x, y

. 2.41

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Suppose thatfis a generalizedgM2-contractive mapping ofCintoXand i∂C⊆gC,fC∩C⊂gC,

iigx∈∂C⇒fx∈C, iiigCis closed inX.

Then there exists a coincidence point z of f and g in C. Moreover, if the pair f, g is coincidentally commuting, thenzis the unique common fixed point offandg.

The proof of this theorem is very similar to the proof ofTheorem 2.2and it is omitted.

We now list some corollaries of Theorems2.2and2.7.

Corollary 2.8. LetX, dbe a complete cone metric space, and letCbe a nonempty closed subset of Xsuch that, for eachx∈Cand eachy /∈C,there exists a pointz∈∂Csuch that

dx, z d z, y

d x, y

. 2.42

Letf, g:C → Xbe such that

d fx, fy

≤λ·d gx, gy

, 2.43

for some 0< λ <√

2−1 and for allx, y∈C.

Suppose, further, thatf, g,andCsatisfy the following conditions:

i∂C⊆gC,fC∩C⊂gC, iigx∈∂C⇒fx∈C, iiigCis closed inX.

Then there exists a coincidence pointzoffandginC. Moreover, iff, gis a coincidentally commuting pair, thenzis the unique common fixed point offandg.

dx, z d z, y

d x, y

. 2.44

Letf, g:C → Xbe such that d

fx, fy

≤λ· d

fx, gx d

fy, gy

, 2.45

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dx, z d z, y

d x, y

. 2.46

Letf, g:C → Xbe such that d

fx, fy

≤λ· d

fx, gy d

fy, gx

, 2.47

Remark 2.11. Corollaries2.8–2.10are the corresponding theorems of Abbas and Jungck from 2in the case thatf, gare nonself mappings.

Remark 2.12. If X, d is a metrically convex cone metric space, that is, if for each x, y ∈ X, x /ythere isz∈X, x /z /ysuch thatdx, z dx, y dy, z, we do not know whether 2.4holds for every nonempty closed subsetCinXsee8.

Acknowledgment

This work was supported by Grant 14021 of the Ministry of Science and Environmental Protection of Serbia.

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