Volume 2010, Article ID 978121,15pages doi:10.1155/2010/978121

*Research Article*

**Common Fixed Point Results in**

**Metric-Type Spaces**

**Mirko Jovanovi ´c,**

**1**

_{Zoran Kadelburg,}

_{Zoran Kadelburg,}

**2**

_{and Stojan Radenovi ´c}

_{and Stojan Radenovi ´c}

**3**

*1 _{Faculty of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73,}*

*11000 Beograd, Serbia*

*2 _{Faculty of Mathematics, University of Belgrade, Studentski Trg 16, 11000 Beograd, Serbia}*

*3*

_{Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16,}*11120 Beograd, Serbia*

Correspondence should be addressed to Stojan Radenovi´c,sradenovic@mas.bg.ac.rs

Received 16 October 2010; Accepted 8 December 2010

Academic Editor: Tomonari Suzuki

Copyrightq2010 Mirko Jovanovi´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.

Several fixed point and common fixed point theorems are obtained in the setting of metric-type spaces introduced by M. A. Khamsi in 2010.

**1. Introduction**

Symmetric spaces were introduced in 1931 by Wilson1, as metric-like spaces lacking the triangle inequality. Several fixed point results in such spaces were obtained, for example, in

2–4. A new impulse to the theory of such spaces was given by Huang and Zhang5when
they reintroduced cone metric spaces replacing the set of real numbers by a cone in a Banach
space, as the codomain of a metricsuch spaces were known earlier under the name of
K-metric spaces, see6. Namely, it was observed in7that if*dx, y*is a cone metric on the
set*X*in the sense of5, then*Dx, y* *dx, y*is symmetric with some special properties,
particularly in the case when the underlying cone is normal. The space*X, D*was then called
the symmetric space associated with cone metric space*X, d*.

The last observation also led Khamsi8to introduce a new type of spaces which he
called metric-type spaces, satisfying basic properties of the associated space*X, D*,*Dd*.
Some fixed point results were obtained in metric-type spaces in the papers7–10.

**2. Preliminaries**

Let *X* be a nonempty set. Suppose that a mapping *D* : *X* ×*X* → 0*,*∞ satisfies the
following:

s1*Dx, y* 0 if and only if*xy*;

s2*Dx, y* *Dy, x*, for all*x, y*∈*X*.

Then*D*is called a*symmetric*on*X*, and*X, D*is called a*symmetric space*1.

Let*E*be a real Banach space. A nonempty subset*P /*{0}of*E*is called a*cone*if*P* is
closed, if*a, b*∈R,*a, b*≥0, and*x, y*∈*P*imply*axby*∈*P*, and if*P*∩−*P* {0}. Given a cone

*P* ⊂*E*, we define the partial ordering with respect to*P* by*x* *y*if and only if*y*−*x*∈*P*.
Let *X* be a nonempty set. Suppose that a mapping *d* : *X* × *X* → *E* satisfies the
following:

co 10 *dx, y*for all*x, y*∈*X*and*dx, y* 0 if and only if*xy*;

co 2*dx, y* *dy, x*for all*x, y*∈*X*;

co 3*dx, y* *dx, z* *dz, y*for all*x, y, z*∈*X*.

Then*d*is called a*cone metric*on*X*and*X, d*is called a*cone metric space*5.

If*X, d*is a cone metric space, the function*Dx, y* *dx, y*is easily seen to be a
symmetric on*X*7,8. Following7, the space*X, D*will then be called*associated symmetric*
*space*with the cone metric space*X, d*. If the underlying cone*P*of*X, d*is normali.e., if, for
some*k*≥1, 0 *x* *y*always implies*x* ≤*ky*, the symmetric*D*satisfies some additional
properties. This led M.A. Khamsi to introduce a new type of spaces which he called metric
type spaces. We will use the following version of his definition.

*Definition 2.1* see8. Let*X* be a nonempty set, let*K* ≥ 1 be a real number, and let the
function*D*:*X*×*X* → Rsatisfy the following properties:

a*Dx, y* 0 if and only if*xy*;

b*Dx, y* *Dy, x*for all*x, y*∈*X*;

c*Dx, z*≤*KDx, y* *Dy, z*for all*x, y, z*∈*X*.

Then*X, D, K*is called a*metric-type space*.

Obviously, for*K*1, metric-type space is simply a metric space.

A metric type space may satisfy some of the following additional properties:

d*Dx, z* ≤ *KDx, y*1 *Dy*1*, y*2 · · · *Dyn, z* for arbitrary points *x, y*1*,*

*y*2*, . . . , yn, z*∈*X*;

efunction*D*is continuous in two variables; that is,

*xn*−→*x,* *yn*−→*y*in*X, D, K*imply*D*

*xn, yn*

−→*Dx, y.* 2.1

Conditiondwas used instead ofcin the original definition of a metric-type space
by Khamsi8. Both conditionsdandeare satisfied by the symmetric*Dx, y* *dx, y*

which is associated with a cone metric*d*with a normal cone see7–9.
Note that the weaker version of propertye:

e*xn* → *x*and*yn* → *x*in*X, D, K*imply that*Dxn, yn* → 0

is satisfied in an arbitrary metric type space. It can also be proved easily that the limit of a
sequence in a metric type space is unique. Indeed, if*xn* → *x*and*xn* → *y*in*X, D, K*and

*Dx, y* *ε >*0, then

0≤*Dx, y*≤*KDx, xn* *D*

*xn, y*

*< K* *ε*

2*K*
*ε*

2*K*

*ε* 2.2

for suﬃciently large*n*, which is impossible.

The notions such as*convergent sequence*,*Cauchy sequence,*and*complete space*are defined
in an obvious way.

We prove in this paper several versions of fixed point and common fixed point results in metric type spaces. We start with versions of classical Banach, Kannan and Zamfirescu results then proceed with Hardy-Rogers-type theorems, and with quasicontractions of ´Ciri´c and Das-Naik, and results for four mappings of Fisher and finally conclude with a result for strict contractions.

Recall also that a mapping*f* : *X* → *X* is said to have property P11if Fix*fn*

Fix*f*for each*n*∈N, where Fix*f*stands for the set of fixed points of*f*.

A point*w* ∈*X* is called a*point of coincidence*of a pair of self-maps*f, g*:*X* → *X*and

*u*∈*X*is its*coincidence point*if*fuguw*. Mappings*f*and*g*are*weakly compatible*if*fgu*
*gfu*for each of their coincidence points*u*12,13. The notion of*occasionally weak compatibility*

is also used in some papers, but it was shown in14that it is actually superfluous.

**3. Results**

We begin with a simple, but useful lemma.

**Lemma 3.1.** *Let*{*yn}be a sequence in a metric type spaceX, D, Ksuch that*

*Dyn, yn*1

≤*λDyn*−1*, yn*

3.1

*for someλ,*0*< λ <*1*/K, and eachn*1*,*2*, . . . .Then*{*yn}is a Cauchy sequence inX, D, K.*

*Proof.* Let *m, n* ∈ N and *m < n*. Applying the triangle-type inequality c to triples

*Dym, yn*

≤*KDym, ym*1

*Dym*1*, yn*

≤*KDym, ym*1

*K*2*Dy _{m}*1

*, ym*2

*Dy _{m}*2

*, yn*

≤ · · · ≤*KDym, ym*1

*K*2*Dym*1*, ym*2

· · ·

*Kn*−*m*−1*Dyn*−2*, yn*−1

*Dyn*−1*, yn*

≤*KDym, ym*1

*K*2*Dym*1*, ym*2

· · ·

*Kn*−*m*−1*Dyn*−2*, yn*−1

*Kn*−*mDyn*−1*, yn*

*.*

3.2

Now3.1and*Kλ <*1 imply that

*Dym, yn*

≤*KλmK*2*λm*1· · ·*Kn*−*mλn*−1*Dy*0*, y*1

*Kλm*1 *Kλ* · · · *Kλn*−*m*−1*Dy*0*, y*1

≤ *Kλm*

1−*KλD*

*y*0*, y*1

−→0 when*m*−→ ∞*.*

3.3

It follows that{*yn}*is a Cauchy sequence.

*Remark 3.2.* If, instead of triangle-type inequality c, we use stronger conditiond, then a
weaker condition 0*< λ <*1 can be used in the previous lemma to obtain the same conclusion.
The proof is similar.

Next is the simplest: Banach-type version of a fixed point result for contractive mappings in a metric type space.

**Theorem 3.3.** *LetX, D, Kbe a complete metric type space, and letf*:*X* → *Xbe a map such that*
*for someλ,*0*< λ <*1*/K,*

*Dfx, fy*≤*λDx, y* 3.4

*holds for allx, y*∈*X. Thenfhas a unique fixed pointz, and for everyx*0∈*X, the sequence*{*fnx*0}

*converges toz.*

*Proof.* Take an arbitrary*x*0∈*X*and denote*ynfnx*0. Then

*Dyn, yn*1

*Dfyn*−1*, fyn*

≤*λDyn*−1*, yn*

3.5

for each*n*1*,*2*. . . .*Lemma3.1implies that{*yn}*is a Cauchy sequence, and since*X, D, K*

is complete, there exists*z*∈*X*such that*yn* → *z*when*n* → ∞. Then

*Dfz, z*≤*KDfz, fyn*

*Dyn*1*, z*

≤*Kλ*D*z, yn*

*Dyn*1*, z*

−→0*,* 3.6

If *z*1 is another fixed point of *f*, then *Dz, z*1 *Dfz, fz*1 ≤ *λDz, z*1 which is

possible only if*zz*1.

*Remark 3.4.* In a standard way we prove that the following estimate holds for the sequence

{*fn _{x}*
0}:

*Dfmx*0*, z*

≤ *K*2*λm*

1−*KλD*

*x*0*, fx*0

_{}_{3.7}_{}

for each*m*∈N. Indeed, for*m < n*,

*Dfmx*0*, z*

≤*KDfmx*0*, fnx*0

*Dfnx*0*, z*

≤ *K*2*λm*

1−*KλD*

*x*0*, fx*0

*KDfnx*0*, z*

*,*

3.8

and passing to the limit when*n* → ∞, we obtain estimate3.7.
Note that continuity of function*D*propertyewas not used.

The first part of the following result was obtained, under the additional assumption of boundedness of the orbit, in8, Theorem 3.3.

**Theorem 3.5.** *LetX, D, Kbe a complete metric type space. Letf* : *X* → *X* *be a map such that*
*for everyn* ∈ N*there isλn* ∈ 0*,*1*such thatDfnx, fny* ≤ *λnDx, yfor all* *x, y* ∈ *X* *and let*

lim*n*→ ∞*λn*0*. Thenfhas a unique fixed pointz. Moreover,fhas the property P.*

*Proof.* Take*λ*such that 0 *< λ <* 1*/K*. Since*λn* → 0,*n* → ∞, there exists*n*0 ∈ Nsuch that

*λn* *< λ*for each *n* ≥ *n*0. Then *Dfn*x*, fny* ≤ *λDx, y* for all*x, y* ∈ *X* whenever *n* ≥ *n*0.

In other words, for any *m* ≥ *n*0, *g* *fm* satisfies *Dgx, gy* ≤ *λDx, y* for all*x, y* ∈ *X*.

Theorem 3.3implies that*g* has a unique fixed point, say *z*. Then*fm _{z}*

_{}

_{z}_{, implying that}

*fm*1*zfmfz* *fz*and*fz*is a fixed point of*gfm*. Since the fixed point of*g*is unique, it
follows that*fzz*and*z*is also a fixed point of*f*.

From the given condition we get that*Dfx, f*2_{x}_{D}_{}_{fx, ffx}_{}_{≤}_{λ}

1*Dx, fx*for some

*λ*1 *<*1 and each*x*∈*X*. This property, together with Fix*f/*∅, implies, in the same way as in

11, Theorem 1.1, that*f*has the property P.

*Remark 3.6.* If, in addition to the assumptions of previous theorem, we suppose that the series

_{∞}

*n*1*λn*converges and that*D*satisfies propertyd, we can prove that, for each*x* ∈ *X*, the

respective Picard sequence{*fn _{x}*

_{}}

_{converges to the fixed point}

_{z}_{.}

Indeed, let*m, n*∈Nand*n > m*. Then

*Dfmx, fnx*≤*KDfmx, fm*1*x*· · ·*Dfn*−1*xfnx*

*KDfmx, fmfx*· · ·*Dfn*−1*x, fn*−1*fx*

≤*Kλm*· · ·*λn*−1*D*

*x, fx*−→0*,*

3.9

when*m* → ∞due to the convergence of the given series. So,{*fn _{x}*

_{}}

_{is a Cauchy sequence}

and*gn _{x}*

_{→}

_{z}_{when}

_{n}_{→ ∞}

_{, but}

_{{}

_{f}mn_{x}_{}}

_{is a subsequence of}

_{{}

_{f}n_{x}_{}}

_{which is convergent;}

hence, the latter converges to*z*.

The next is a common fixed point theorem of Hardy-Rogers typesee, e.g., 15 in metric type spaces.

**Theorem 3.7.** *LetX, D, Kbe a metric type space, and letf, g*:*X* → *Xbe two mappings such that*
*fX*⊂*gXand one of these subsets ofXis complete. Suppose that there exist nonnegative coeﬃcients*
*ai,i*1*, . . . ,*5*such that*

2*Ka*1 *K*1*a*2*a*3

*K*2*Ka*4*a*5*<*2 3.10

*and that for allx, y*∈*X*

*Dfx, fy*≤*a*1*D*

*gx, gya*2*D*

*gx, fxa*3*D*

*gy, fya*4*D*

*gx, fya*5*D*

*gy, fx*

3.11

*holds. Then* *f* *and* *g* *have a unique point of coincidence. If, moreover, the pair* *f, g* *is weakly*
*compatible, thenfandghave a unique common fixed point.*

Note that condition3.10is satisfied, for example, when5_{i}_{1}*ai* *<* 1*/K*2. Note also

that when*K*1 it reduces to the standard Hardy-Rogers condition in metric spaces.

*Proof.* Suppose, for example, that*gX*is complete. Take an arbitrary *x*0 ∈*X* and, using that

*fX*⊂*gX*, construct a Jungck sequence{*yn}*defined by*yn* *fxn* *gxn*1,*n*0*,*1*,*2*, . . . .*Let

us prove that this is a Cauchy sequence. Indeed, using3.11, we get that

*Dyn, yn*1

*Dfxn, fxn*1

≤*a*1*D*

*gxn, gxn*1

*a*2*D*

*gxn, fxn*

*a*3*D*

*gxn*1*, fxn*1

*a*4*D*

*gxn, fxn*1

*a*5*D*

*gxn*1*, fxn*

*a*1*D*

*yn*−1*, yn*

*a*2D

*yn*−1*, yn*

*a*3*D*

*yn, yn*1

*a*4*D*

*yn*−1*, yn*1

*a*5·0
≤*a*1*a*2*D*

*yn*−1*, yn*

*a*3*D*

*yn, yn*1

*a*4*K*

*Dyn*−1*, yn*

*Dyn, yn*1

*a*1*a*2*Ka*4*D*

*yn*−1*, yn*

*a*3*Ka*4*D*

*yn, yn*1

*.*

3.12

Similarly, we conclude that

*Dy _{n}*1

*,*y

*n*

*Dfx _{n}*1

*, fxn*

≤*a*1*a*3*Ka*5*D*

*yn*−1*, yn*

*a*2*Ka*5*D*

*yn, yn*1

*.* 3.13

Adding the last two inequalities, we get that

2*Dyn, yn*1

≤2*a*1*a*2*a*3*Ka*4*Ka*5*D*

*yn*−1*, yn*

*a*2*a*3*Ka*4*Ka*5*D*

*yn, yn*1

*,*

that is,

*Dyn, yn*1

≤ 2*a*1*a*2*a*3*Ka*4*Ka*5

2−*a*2−*a*3−*Ka*4−*Ka*5

*Dyn*−1*, yn*

*λDyn*−1*, yn*

*.* 3.15

The assumption3.10implies that

2*Ka*1*Ka*2*Ka*3*K*2*a*4*a*5*<*2−*a*2−*a*3−*Ka*4−*Ka*5*,*

*λ* 2*a*1*a*2*a*3*Ka*4*Ka*5

2−*a*2−*a*3−*Ka*4−*Ka*5 *<*

1

*K.*

3.16

Lemma3.1implies that{*yn}* is a Cauchy sequence in*gX* and so there is*z* ∈ *X* such that

*fxngxn*1 → *gz*when*n* → ∞. We will prove that*fzgz*.

Using3.11we conclude that

*Dfxn, fz*

≤*a*1*D*

*gxn, gz*

*a*2*D*

*gxn, fxn*

*a*3*D*

*g*z*, fz*

*a*4*D*

*gxn, fz*

*a*5*D*

*gz, fxn*

≤*a*1*D*

*gxn, gz*

*a*2*D*

*gxn, fxn*

*a*3*K*

*Dgz, fxn*

*Dfxn, fz*

*a*4*K*

*Dgxn, fxn*

*Dfxn, fz*

*a*5*D*

*gz, fxn*

*a*1*D*

*gxn, gz*

*a*2*Ka*4*D*

*gxn, fxn*

*Ka*3*a*5*D*

*gz, fxn*

*Ka*3*a*4*D*

*fxn, fz*

*.*

3.17

Similarly,

*Dfz, fxn*

≤*a*1*D*

*gxn, gz*

*Ka*2*a*4*D*

*gz, fxn*

*Ka*2*a*5*D*

*fxn, fz*

*a*3*Ka*5*D*

*fxn, gxn*

*.* 3.18

Adding up, one concludes that

2−*Ka*2*a*3*a*4*a*5*D*

*fxn,*f*z*

≤2*a*1*D*

*gxn, gz*

*a*2*a*3*Ka*4*a*5*D*

*fxn, gxn*

*Ka*2*a*3 *a*4*a*5*D*

*fxn, gz*

*.*

3.19

The right-hand side of the last inequality tends to 0 when*n* → ∞. Since*Ka*2*a*3*a*4*a*5*<*

2*Ka*1*K*1*a*2*a*3*K*2*Ka*4*a*5*<*2because of3.10, it is 2−*Ka*2*a*3*a*4*a*5*>*0,

and so also the left-hand side tends to 0, and*fxn* → *fz*. Since the limit of a sequence is

Suppose that*w*1 *fz*1 *gz*1is another point of coincidence for*f*and*g*. Then3.11

implies that

*Dw, w*1 *D*

*fz, fz*1

≤*a*1*D*

*gz, gz*1

*a*2*D*

*gz, fza*3*D*

*gz*1*, fz*1

*a*4*D*

*gz, fz*1

*a*5*D*

*gz*1*, fz*

*a*1*Dw, w*1 *a*2·0*a*3·0*a*4*Dw, w*1 *a*5*Dw*1*, w*

*a*1*a*4*a*5*Dw, w*1*.*

3.20

Since*a*1*a*4*a*5 *<*1because of3.10, the last relation is possible only if*w* *w*1. So, the

point of coincidence is unique.

If*f, g*is weakly compatible, then13, Proposition 1.12implies that*f*and*g*have a
unique common fixed point.

Taking special values for constants*ai*, we obtain as special cases Theorem3.3as well

as metric type versions of some other well-known theoremsKannan, Zamfirescu, see, e.g.,

15:

**Corollary 3.8.** *LetX, D, Kbe a metric type space, and letf, g*:*X* → *Xbe two mappings such that*
*fX*⊂*gXand one of these subsets ofXis complete. Suppose that one of the following three conditions*
*holds:*

1◦*Dfx, fy*≤*a*1*Dgx, gyfor somea*1*<*1*/Kand allx, y*∈*X;*

2◦*Dfx, fy*≤*a*2*Dgx, fx* *Dgy, fyfor somea*2*<*1*/K*1*and allx, y*∈*X;*

3◦*Dfx, fy*≤*a*4*Dgx, fy* *Dgy, fxfor somea*4*<*1*/K*2*Kand allx, y*∈*X.*

*Then* *f* *and* *g* *have a unique point of coincidence. If, moreover, the pair* *f, g* *is weakly*
*compatible, thenfandghave a unique common fixed point.*

Putting*g* *iX* in Theorem3.7, we get metric type version of Hardy-Rogers theorem

which is obviously a special case for*K*1.

**Corollary 3.9.** *LetX, D, Kbe a complete metric type space, and letf*:*X* → *Xsatisfy*

*Dfx, fy*≤*a*1*D*

*x, ya*2*D*

*x, fxa*3*D*

*y, fya*4*D*

*x, fya*5*D*

*y, fx* 3.21

*for someai,i* 1*, . . . ,*5 *satisfying* 3.10*and for allx, y* ∈ *X. Thenf* *has a unique fixed point.*

*Proof.* We have only to prove the last assertion. For arbitrary*x*∈*X*, we have that

*Dfx, f*2*xDfx, ffx*

≤*a*1*D*

*x, fxa*2*D*

*x, fxa*3*D*

*fx, f*2*xa*4*D*

*x, f*2*xa*5*D*

f*x, fx*

≤*a*1*a*2*Ka*4*D*

*x, fx* *a*3*Ka*4*D*

*fx, f*2_{x}_{,}

3.22

and similarly

*Df*2*x, fxDffx, fx*≤*a*1*a*3*Ka*5*D*

*x, fx* *a*2*Ka*5*D*

*fx, f*2*x.* 3.23

Adding the last two inequalities, we obtain

*Dfx, f*2*x*≤ 2*a*1*a*2*a*3*Ka*4*a*5

2−*a*2−*a*3−*Ka*4*a*5

*Dx, fxλDx, fx.* 3.24

Similarly as in the proof of Theorem3.7, we get that*λ <* 1*/K <* 1. Now11, Theorem 1.1
implies that*f*has property P.

*Remark 3.10.* If the metric-type function *D* satisfies both properties d and e, then it is
easy to see that condition 3.10in Theorem3.7and the last corollary can be weakened to

*a*1*a*2*a*3*Ka*4*a*5*<*1. In particular, this is the case when*Dx, y* *dx, y*for a cone

metric*d*on*X*over a normal cone, see7.

The next is a possible metric-type variant of a common fixed point result for ´Ciri´c and Das-Naik quasicontractions16,17.

**Theorem 3.11.** *LetX, D, Kbe a metric type space, and letf, g* :*X* → *X* *be two mappings such*
*thatfX*⊂ *gXand one of these subsets ofX* *is complete. Suppose that there existsλ,*0 *< λ <*1*/K*
*such that for allx, y*∈*X*

*Dfx, fy*≤*λ*max*Mf, g*;*x, y,* 3.25

*where*

*Mf, g*;*x, y*

*Dgx, gy, Dgx, fx, Dgy, fy,D*

*gx, fy*

2*K* *,*

*Dgy, fx*

2*K*

*.* 3.26

*Proof.* Let*x*0 ∈ *X*be arbitrary and, using condition*fX* ⊂ *gX*, construct a Jungck sequence
{*yn}*satisfying *yn* *fxn* *gxn*1,*n* 0*,*1*,*2*, . . . .*Suppose that*Dyn, yn*1 *>* 0 for each *n*

otherwise the conclusion follows easily. Using3.25we conclude that

*Dyn*1*, yn*

*Dfxn*1*, fxn*

≤*λ*max

*Dgxn*1*, gxn*

*, Dgxn*1*, fxn*1

*, Dgxn, fxn*

*,D*

*gx _{n}*1

*, fxn*

2*K* *,*

*Dgxn, fxn*1

2*K*

*λ*max

*Dyn, yn*−1

*, Dyn, yn*1

*, Dyn*−1*, yn*

*,*0*,D*

*yn*−1*, yn*1

2*K*

≤*λ*max *Dyn, yn*−1

*,*1

2

*Dyn*−1*, yn*

*Dyn, yn*1

*.*

3.27

If *Dyn, yn*−1 *< Dyn*1*, yn*, then *Dyn, yn*−1 *<* 1*/*2*Dyn*−1*, yn* *Dyn, yn*1 *<*

*Dyn, yn*1, and it would follow from 3.27 that *Dyn*1*, yn* ≤ *λDyn*1*, yn* which is

impossible since*λ <* 1. For the same reason the term*Dyn, yn*1 was omitted in the last

row of the previous series of inequalities. Hence, *Dyn, yn*−1 *> Dyn*1*, yn* and 3.27

becomes*Dyn*1*, yn* ≤ *λDyn, yn*−1. Using Lemma3.1, we conclude that{*yn}*is a Cauchy

sequence in*gX*. Supposing that, for example, the last subset of*X* is complete, we conclude
that*yn* *fxngxn*1 → *gz*when*n* → ∞for some*z*∈*X*.

To prove that*fzgz*, put*xxn*and*yz*in3.25to get

*Dfxn, fz*

≤*λ*max

*Dgxn, gz*

*, Dgxn, fxn*

*, Dgz, fz,D*

*gxn, fz*

2*K* *,*

*Dgz, fxn*

2*K*

*.*

3.28

Note that *fxn* → *gz* and *gxn* → *gz* when *n* → ∞, implying that *Dgxn, fxn* ≤

*KDgxn, gz* *Dgz, fxn* → 0 when*n* → ∞. It follows that the only possibilities are

the following:

1◦*Dfxn, fz* ≤ *λDgz, fz* ≤ *λKDgz, fxn* *Dfxn, fz*; in this case 1 −

*λKDfxn, fz* ≤ *λKDgz, fxn* → 0, and since 1− *λK >* 0, it follows that

*fxn* → *fz*.

2◦*Dfxn, fz*≤ *λ*1*/*2*KDgxn, fz*≤*λ/*2*Dgxn, fxn* *Dfxn, fz*; in this case,

1−*λ/*2*Dfxn, fz*≤*λ/*2*Dgxn, fxn* → 0, so again*fxn* → *fz*,*n* → ∞.

Putting*giX*, we obtain the first part of the following corollary.

**Corollary 3.12.** *LetX, D, Kbe a complete metric type space, and letf* :*X* → *Xbe such that for*
*someλ,*0*< λ <*1*/K, and for allx, y*∈*X,*

*Dfx, fy*≤*λ*max

*Dx, y, Dx, fx, Dy, fy,D*

*x, fy*

2*K* *,*

*Dy, fx*

2*K*

3.29

*holds. Thenfhas a unique fixed point, sayz. Moreover, the functionfis continuous at pointzand it*
*has the property P.*

*Proof.* Let*xn* → *z*when*n* → ∞. Then

*Dfxn, fz*

≤*λ*max

*Dxn, z, D*

*xn, fxn*

*, Dz, fz,D*

*xn, fz*

2*K* *,*

*Dfxn, z*

2*K*

*λ*max

*Dxn, z, D*

*xn, fxn*

*,D*

*fxn, z*

2*K*

*.*

3.30

Since*Dxn, z* → 0 and*Dxn, fxn*≤ *KDxn, z* *Dfz, fxn*, the only possibility is that

*Dfx,fz*≤*λKDxn, zDfz, fxn*, implying that1−*λKDfxn, fz*≤*λKDxn, z* → 0,

*n* → ∞. Since 0*< λK <*1, it follows that*fxn* → *fzz*,*n* → ∞, and*f*is continuous at the

point*z*.

We will prove that*f*satisfies

*Dfx, f*2*x*≤*hDx, fx* 3.31

for some*h*, 0*< h <*1 and each*x*∈*X*.

Applying3.29to the points*x*and*fx*for any*x*∈*X*, we conclude that

*Dfx, f*2*x*≤*λ*max

*Dx, fx, Dx, fx, Dfx, f*2*x,D*

*x, f*2*x*

2*K* *,*

*Dfx, fx*

2*K*

*λ*max

*Dx, fx, Dfx, f*2*x,D*

*x, f*2_{x}

2*K*

*.*

3.32

The following cases are possible:

1◦*Dfx, f*2_{x}_{}_{≤}_{λD}_{}_{x, fx}_{}_{, and}_{}_{3.31}_{}_{holds with}_{h}_{}_{λ}_{;}

2◦*Dfx, f*2_{x}_{}_{≤}_{λD}_{}* _{fx, f}*2

_{x}_{}

_{, which is only possible if}

_{D}_{}

*2*

_{fx, f}

_{x}_{0 and then}

_{}

_{3.31}

_{}

obviously holds.

3◦*Dfx, f*2_{x}_{}_{≤}_{}_{λ/}_{2}_{K}_{}_{D}_{}* _{x, f}*2

_{x}_{}

_{≤}

_{}

_{λ/}_{2}

_{K}_{}

_{K}_{}

_{D}_{}

_{x, fx}

_{D}_{}

*2*

_{fx, f}

_{x}_{}

_{, implying that}

1−*λ/*2*Dfx, f*2_{x}_{}_{≤}_{}_{λ/}_{2}_{}_{D}_{}_{x, fx}_{}_{and}_{D}_{}* _{fx, f}*2

_{x}_{}

_{≤}

_{hD}_{}

_{x, fx}_{}

_{, where 0}

_{< h}_{}

So, relation3.31holds for some*h*, 0 *< h <*1 and each*x* ∈*X*. Using the mentioned
analogue of11, Theorem 1.1, one obtains that*f*satisfies property P.

We will now prove a generalization and an extension of Fisher’s theorem on four
mappings from 18 to metric type spaces. Note that, unlike in 18, we will not use the
case when*f*and*S*, as well as*g*and*T*, commute, neither when*S*and*T*are continuous. Also,
function*D*need not be continuousi.e., we do not use propertye.

**Theorem 3.13.** *LetX, D, Kbe a metric type space, and letf, g, S, T* :*X* → *Xbe four mappings*
*such thatfX*⊂*TXandgX*⊂*SX, and suppose that at least one of these four subsets ofXis complete.*
*Let*

*Dfx, gy*≤*λDSx, Ty* 3.33

*holds for someλ,*0*< λ <*1*/Kand allx, y*∈*X. Then pairsf, Sandg, Thave a unique common*
*point of coincidence. If, moreover, pairsf, Sandg, Tare weakly compatible, thenf,g,S, andT*
*have a unique common fixed point.*

*Proof.* Let*x*0∈*X*be arbitrary and construct sequences{*xn}*and{y* _{n}*}such that

*fx*2*n*−2*Tx*2*n*−1*y*2*n*−1*,* *gx*2*n*−1*Sx*2*ny*2*n* 3.34

for*n*1*,*2*, . . . .*We will prove that condition3.1holds for*n*1*,*2*, . . . .*Indeed,

*Dy*2*n*1*, y*2*n*2

*Dfx*2*n, gx*2*n*1

≤*λDSx*2*n, Tx*2*n*1 *λD*

*y*2*n, y*2*n*1

*,*

*Dy*2*n*3*, y*2*n*2

*Dfx*2*n*2*, gx*2*n*1

≤*λDSx*2*n*2*, Tx*2*n*1 *λD*

*y*2*n*2*, y*2*n*1

*.*

3.35

Using Lemma3.1, we conclude that{*yn}*is a Cauchy sequence. Suppose, for example, that

*SX* is a complete subset of*X*. Then *yn* → *u* *Sv*,*n* → ∞, for some*v* ∈ X. Of course,

subsequences{*y*2*n*−1}and{*y*2*n}*also converge to*u*. Let us prove that*fv* *u*. Using3.33,

we get that

*Dfv, u*≤*KDfv, gx*2*n*−1

*Dgx*2*n*−1*, u*

≤*KλDSv, Tx*2*n*−1 *D*

*gx*2*n*−1*, u*

−→*Kλ*·00 0*.* 3.36

hence*fvuSv*. Since*u*∈*fX*⊂*TX*, we get that there exists*w*∈*X*such that*Twu*. Let
us prove that also*gwu*. Using3.33, again we conclude that

*Dgw, u*≤*KDgw, fx*2*n*

*Dfx*2*n, u*

≤*KλDSx*2*n, Tw* *D*

*fx*2*n, u*

−→*Kλ*·00 0*,*

3.37

If now these pairs are weakly compatible, then for example,*fufSv* *SfvSu*
*z*1 and *gu* *gTw* *Tgw* *Tu* *z*2for example, . Moreover, *Dz*1*, z*2 *Dfu,*g*u* ≤

*λDSu, Tu* *λDz*1*, z*2and 0 *< λ <* 1 implies that*z*1 *z*2. So, we have that*fu* *gu*

*Su* *Tu*. It remains to prove that, for example, *u* *gu*. Indeed,*Du, gu* *Dfv, gu* ≤
*λDSv, Tu* *λDu, gu*, implying that*u* *gu*. The proof that this common fixed point of

*f, g, S*, and*T* is unique is straightforward.

We conclude with a metric type version of a fixed point theorem for strict contractions. The proof is similar to the respective proof, for example, for cone metric spaces in5. An example follows showing that additional condition ofsequential compactness cannot be omitted.

**Theorem 3.14.** *Let a metric type spaceX, D, Kbe sequentially compact, and letDbe a continuous*
*function (satisfying property (e)). Iff* :*X* → *Xis a mapping such that*

*Dfx, fy< Dx, y,* *forx, y*∈*X, x /y,* 3.38

*thenfhas a unique fixed point.*

*Proof.* According to 9, Theorem 3.1, sequential compactness and compactness are
equivalent in metric type spaces, and also continuity is a sequential property. The given
condition3.38of strict continuity implies that a fixed point of*f* is uniqueif it existsand
that both mappings*f*and*f*2are continuous. Let*x*0∈*X*be an arbitrary point, and let{*xn}*be

the respective Picard sequencei.e.,*xnfnx*0. If*xnxn*1for some*n*, then*xn*is aunique

fixed point. If*xn/xn*1for each*n*0*,*1*,*2*, . . .*, then

*Dn*:*Dxn*1*, xn* *D*

*fn*1* _{x}*
0

*, fnx*0

*< Dfn _{x}*

0*, fn*−1*x*0

*Dn*−1*.* 3.39

Hence, there exists*D*∗, such that 0 ≤ *D*∗ ≤ *Dn* for each *n*and*Dn* → *D*∗,*n* → ∞. Using

sequential compactness of *X*, choose a subsequence{*xni*} of {*xn}* that converges to some

*x*∗∈*X*when*i* → ∞. The continuity of*f*and*f*2_{implies that}

*fxni* −→*fx*∗*,* *f*
2_{x}

*ni* → *f*

2* _{x}*∗

_{when}

_{i}_{−→ ∞}

_{,}_{}

_{3.40}

_{}

and the continuity of the symmetric*D*implies that

*Dfxni, xni*

−→*Dfx*∗*, x*∗*,* *Df*2*xni, fxni*

−→*Df*2*x*∗*, fx*∗ when*i*−→ ∞*.* 3.41

It follows that*Dfxni, xni* *Dni* → *D*∗ *Dfx*∗*, x*∗. It remains to prove that *fx*∗ *x*∗. If

*fx*∗*/x*∗, then*D*∗*>*0 and3.41implies that

*D*∗ lim

*i*→ ∞*Dni*1*i*lim→ ∞*D*

*f*2*xni, fxni*

*Df*2*x*∗*, fx*∗*< Dfx*∗*, x*∗*D*∗*.* 3.42

*Example 3.15.* Let*E* R2_{,}_{P}_{} _{{}_{}_{x, y}_{} _{∈}_{E}_{:} _{x, y}_{≥} _{0}_{}}_{,}_{X}_{1}_{,}_{}_{∞}_{}_{, and}_{d}_{:} _{X}_{×}_{X}_{→} _{E}_{be}

defined by*dx, y* |*x*−*y*|*,*|*x*−*y*|. Then*X, d*is a cone metric space over a normal cone
with the normal constant*K*1see, e.g.,5. The associated symmetric is in this case simply
the metric*Dx, y* *dx, y*|*x*−*y*|√2.

Let*f*:*X* → *X*be defined by*fxx*1*/x*. Then

*Dfx, fyx*−*y*

1− 1

*xy*

_{√}

2*<x*−*y*√2*Dx, y* 3.43

for all *x, y* ∈ *X*. Hence, *f* satisfies condition 3.38 but it has no fixed points. Obviously,

*X, D, K*is notsequentiallycompact.

**Acknowledgment**

The authors are thankful to the Ministry of Science and Technological Development of Serbia.

**References**

1 W. A. Wilson, “On semi-metric spaces,”*American Journal of Mathematics*, vol. 53, no. 2, pp. 361–373,
1931.

2 J. Zhu, Y. J. Cho, and S. M. Kang, “Equivalent contractive conditions in symmetric spaces,”*Computers*
*& Mathematics with Applications*, vol. 50, no. 10-12, pp. 1621–1628, 2005.

3 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.
4 S.-H. Cho, G.-Y. Lee, and J.-S. Bae, “On coincidence and fixed-point theorems in symmetric spaces,”

*Fixed Point Theory and Applications*, vol. 2008, Article ID 562130, 9 pages, 2008.

5 L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,”

*Journal of Mathematical Analysis and Applications*, vol. 332, no. 2, pp. 1468–1476, 2007.

6 P. P. Zabrejko, “*K*-metric and*K*-normed linear spaces: survey,”*Collectanea Mathematica*, vol. 48, no.
4–6, pp. 825–859, 1997.

7 S. Radenovi´c and Z. Kadelburg, “Quasi-contractions on symmetric and cone symmetric spaces,”

*Banach Journal of Mathematical Analysis*, vol. 5, no. 1, pp. 38–50, 2011.

8 M. A. Khamsi, “Remarks on cone metric spaces and fixed point theorems of contractive mappings,”

*Fixed Point Theory and Applications*, vol. 2010, Article ID 315398, 7 pages, 2010.

9 M. A. Khamsi and N. Hussain, “KKM mappings in metric type spaces,”*Nonlinear Analysis: Theory,*
*Methods & Applications*, vol. 73, no. 9, pp. 3123–3129, 2010.

10 E. Karapınar, “Some nonunique fixed point theorems of ´Ciri´c type on cone metric spaces,”*Abstract*
*and Applied Analysis*, vol. 2010, Article ID 123094, 14 pages, 2010.

11 G. S. Jeong and B. E. Rhoades, “Maps for which*FT* *FTn*_{,”}_{Fixed Point Theory and Applications}_{,}

vol. 6, pp. 87–131, 2005.

12 G. Jungck, “Commuting mappings and fixed points,”*The American Mathematical Monthly*, vol. 83, no.
4, pp. 261–263, 1976.

13 G. Jungck, S. Radenovi´c, S. Radojevi´c, and V. Rakoˇcevi´c, “Common fixed point theorems for weakly
compatible pairs on cone metric spaces,”*Fixed Point Theory and Applications*, vol. 2009, Article ID
643840, 13 pages, 2009.

14 D. Djori´c, Z. Kadelburg, and S. Radenovi´c, “A note on occasionally weakly compatible mappings and
common fixed points,” to appear in*Fixed Point Theory*.

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

17 K. M. Das and K. V. Naik, “Common fixed-point theorems for commuting maps on a metric space,”

*Proceedings of the American Mathematical Society*, vol. 77, no. 3, pp. 369–373, 1979.