Abstract:Dhage introduced
the concept of D-metric spaces. Following this Naidu, Rao, Srinivasa Rao
and Asim, Aslam, Zafer discussed about the notion of balls in D-metric
spaces.Wataru Takahashi introduced a convex structure in
metric spaces and formulated some fixed point theorems for nonexpansive
mappings. The purpose of this paper is to formulate some fixed point
theorems in different convex structures in metric spaces.

Key words:
Fixed points, D-metric and nonexpansive mappings.

MSC 2010:
47H10, 54E50

1.Introduction and Preliminaries

The concept of D-metric space has been investigated initially
by Dhage[3], D-metric spaces are further studied by Naidu, Rao,
Srinivasa Rao[4] and Asim, Aslam, Zafer[2]. In 1970 Wataru Takahashi
introduced the concept of convex structure in metric spaces and
established some fixed point theorems.

Following this the authors[5, 6] introduced the concept of E-convex
structure in metric spaces and established some fixed point theorems.
Further the authors[7] studied the concepts of Strong, Weak and Quasi
convex structures in metric spaces and ultra metric spaces. In this
paper we introduce the concepts of Strong, Weak and Quasi convex
structures in D-metric spaces and discuss their basic properties.
Further, we establish some fixed point theorems in these structures.

For
the notations and terminology we refer Wataru Takahashi[9] and
Shyamala Malini et al.[5, 6].

Definition 1.1: Let (X, d) be a metric
space. A mapping W:X´X´[0,
1]®X
is said to be a convex structure[9] on (X, d) if for each (x, y,
l)ÎX´X´[0,
1] and for all uÎX,
the condition d(u, W(x, y, l))£ld(u,
x )+(1-l)d(u,
y)
holds.
The metric space (X, d) together with the convex structure W is called a
convex metric space denoted by (X, d, W).

The
following definition is due to Dhage.

Definition 1.2:
Let X be a nonempty set. Let D:X´X´X®[0,
¥)
be a function satisfying the following conditions for all x, y, zÎX.

D(x,
y, z) = 0 if and only if x = y= z.

D(x,
y, z) = D(sy,
sx,
sz)
for every permutation s
of x, y, z in X.

D(x,
y, z) ≤ D(x, y, a) + D(x, a, z) + D(a, y, z) for every aÎX.

Then
the function D is called a D-metric[3] on X. The set X together with a
D-metric D is called a D-metric space denoted by (X, D).

The notion of open balls in a metric space plays a dominant role
in Analysis. Various Mathematicians extended the notion of this ball to
D-metric spaces. The following definition is due to Dhage[3].

Definition 1.3: Let (X, r)
be a D-metric space and x_{0}ÎX
then for every r>0

S*(x_{0}, r) = {yÎX:
r(x_{0},
y, y) <r} Eqn. 4.1

and
S(x_{0}, r) = y, zÎX:
r(x_{0},
y, z) <r} Eqn. 4.2

The
above definition was not appropriate and it was rectified by Naidu et
al[11].

The
next definition is due to Naidu et.al.

Definition 1.4: Let (X, r)
be a D-metric space and x_{0}ÎX
then for every r>0

S*(x_{0}, r) = {xÎX:
r(x_{0},
x, x) <r} Eqn. 4.3

S(x_{0},
r) = xÎS*(x_{0},
r):r(x_{0},
x, y) < r for all yÎS*(x_{0},
r)}
Eqn. 4.4

There is a uniformity in the definition of the ball S*(x_{0},
r), in[2],[3],[4]. However the definition for S(x_{0}, r)(Eqn.
4.2) is not meaningful as was said in[4][page no. 134, line 9].

Also
Asim, Aslam and Zafer expressed their dissatisfaction in the definition
of Ŝ(x_{0}, r)(Eqn. 4.5), quoting thatr(x_{0},
x, y)< r is not always possible for all xÎX.
But their assumption is wrong. For even if
r(x_{0},
x, y)< r does not exist or ifr(x_{0},
x, y) is not less than r for all xÎX
then also the definition is meaningful. More precisely if the above
condition prevails for some x then we can take Ŝ (x_{0}, r) ={x_{0}}.

Thus
we have the following three types of open balls in D-metric
spaces.

Definition
1.6:Let (X, r)
be a D-metric space, x_{0}ÎX
and r>0.

S_{1}(x_{0},
r) = {xÎS*(x_{0},
r):r(x_{0},
x, y)< r for all yÎS*(x_{0},
r)}.(in the sense of Naidu)

Ŝ(x_{0},
r) ={x_{0}}È{xÎX:r(x_{0},
x, y) < r}}.(in the sense of Naidu)

S_{2}(x_{0},
r)={x_{0}}È{xÎX:r(x_{0},
x, y)< r }}.(in the sense of Asim)

2.Convex D-Metric Spaces

We
introduce the concept of convex structures in D-metric spaces and
discuss its basic properties. Some fixed point theorems are also
established in this structure.

Definition 2.1:
Let (X, D) be a metric space. A mapping W: X´X´X´(0,
1]®
X is said to be a convex structure on (X, D) if for each (x, y, z,
l)ÎX´X´X´(0,
1] and for all u, vÎX
the condition

If W
is convex on a D-metric space (X, D), then the triplet (X, D, W) is
called a convex D-metric space.

Example 2.2:
Consider a linear space L which is also a D-metric space with the
following properties:

(i)For x, y, zÎ
L, D(x, y, z) = D(x – y – z, 0, 0);

(ii)For x, y, z ÎL
and lÎ(0,
1],

D(x +y+, 0,0)£
D(x,0,0)+D(y,0,0)

+ D(z, 0, 0).

Let
W(x, y, z, l)
= x +
y +
z for all x, y, zÎX
and lÎ(0,
1]. Then (L, D, W) is a convex D-metric space.

Definition 2.3:
A subset M of convex D-metric space. (X, D, W) is said to
be a convex set in (X, D, W) if W(x, y, z, l)ÎM
for all x, y, zÎM
and for all l
with 0<l≤1.

Proposition 2.4:
If {K_{a}:aÎD}
is a family of convex subsets of the convex D-metric space (X, D, W),
then_{ }is also a convex subset in (X, D,
W).

Proposition 2.5: Let (X, D, W) be a convex
D-metric space, uÎX,
r >0. The balls S_{1}(u, r) (in the sense of Naidu), Ŝ(u, r) (in
the sense of Naidu) and S_{2}(u, r) (in the sense of Asim) in
(X, D, W) are convex subsets of (X, D, W).The balls
S*(u, r) and S*[u, r] are also convex in (X, D, W).

Definition 2.6:
Let (X, D, W) be a convex D-metric space. Let AÍX.

R_{x}
(A) = sup{ D(x, y, y): yÎA}.

R(A)
= inf{ R_{x}(A): xÎA}.

A_{c
}={xÎA:
R_{x}(A) = R(A)}.

Diameter of A = d(A)
= sup {D(x, y, y): x, yÎA}.

Definition 2.7:
Let (X, D, W) be a convex D-metric space. Let A be a subset of X. A
point xÎA
is a diametral point of A if
D(x, y, y) = d(A).

Definition 2.8:
A convex D-metric space (X, D, W) is said to have the Property(DC) if
every bounded decreasing sequence of nonempty closed convex subsets of
(X, D, W) has nonempty intersection.

Proposition 2.9
Let (X, D, W) be a convex D-metric space. Let AÍX.
If (X, D, W) has the Property(DC). Then A_{c} is nonempty,
closed and convex.

Proof: Let xÎA.
For each positive integer n, define A_{n}(x) = {yÎA:
D(x, y, x) ≤ R(A) + } and C_{n}.
Since xÎA_{n}(x),
A_{n}(x) is nonempty and closed. Let zÎA_{n}(x)
and lÎ(0,
1].

D(x,x,W(x, y, z,l))≤D(x, x, x)+D(x,x,y)+D(x,x,z)

≤(R(A) +
) + (R(A) +
)+ (R(A) +
)

<(R(A) + ).

Therefore W(x, y, z,l)ÎA_{n}(x)
which implies that A_{n}(x) is a convex set.

We
claim that {C_{n}} is a bounded decreasing sequence of nonempty,
closed and convex subsets.

Let
e
>0. Then by the Definition of R(A),there exists yÎA
with R_{y}(A)£R(A)+e
and by the Definition of R_{y}(A), D(x, y, y)
£R_{y}(A)
for every xÎA.

<R(A)+e
for every xÎA

As
e
is arbitrary for every xÎA,
D(x, y, y)£R(A)+e<
R(A)+ . Hence yÎA_{n}(x)
for every xÎA
and hence yÎ=
C_{n}. Thus C_{n} is nonempty. Since each A_{n}
is a closed set C_{n} is also closed. By Proposition 2.5, C_{n}
is a convex set. Hence C_{n} is a nonempty, closed and convex
set for all n.

_{
}Let zÎC_{n+1
.}Then zÎA_{n+1}(x)
for every xÎX,
D(x, x, z)≤R(A) +≤R(A)+. Therefore zÎA_{n}(x)
for every xÎX
which implies that zÎC_{n}.
Now we prove that that A_{c }=. Since the space (X, D, W) has the Property
(DC), is nonempty. Let xÎ.
Then xÎA_{n}(y)
for every yÎA
and for every n. Now D(x, y, y)≤R(A)+for every yÎA
and for every n. Thus R_{x}(A)≤R(A). By the Definition of R(A),
we have R(A)≤ R_{x}(A). Hence R(A)=R_{x}(A) implies xÎA_{c}.

Conversely, let xÎA_{c}._{
}Then R_{x}(A)=R(A). Now, for every yÎA
D(x, y, y)£R_{x}(A)=R(A)£R(A)+for every n. This shows that xÎA_{n}(y)
for every yÎX
and for every n. This completes the proof.

Proposition 2.10: Let M be a nonempty compact
subset of a convex D-metric space (X, D, W) and let K be the least
closed convex set containing M. If the diameter d(M)
is positive,then there exists an element uÎK
such that sup{D(y, y, u):x, yÎM}<d
(M).

Proof: Since M
is compact, we may find x_{1}, x_{2}, x_{3}ÎM
such that D(x_{1}, x_{2}, x_{3}) = d(M).
Let M_{0 }ÍM
be maximal so that M_{0 }Ê{x_{1},
x_{2}, x_{3}} and D(x, y, y) = 0 or d(M)
for all x, yÎM_{0}.
Since M is compact, M_{0} is finite. Let us assume that M_{0}={x_{1},x_{2},…,x_{n}}.

Let y_{1 }= W(x_{1}, x_{2}, x_{3,})

y_{2 }= W(x_{3}, x_{4} y_{1},
)

….

y_{n-2 }= W(x_{n-2}, x_{n-1,
}y_{n-3}, )

y_{n-1}= W(x_{n-1, }x_{n},
y_{n-2}, ) = u.

As K
is a convex set, uÎK.
Since M is compact, we can find y_{0}ÎM
such that D(y_{0}, y_{0},u)= sup{D(y, y, u): yÎM}.

Suppose D(y_{0}, y_{0,} u) = d(M).
Then D(y_{0}, y_{0, } x_{k}) = d(M)
>0 for all k=1,2,3…,n. Since y_{0}ÎM_{0}
we have y_{0}= x_{k} for some k=1,2,…,n. This implies
d(M)
=0 which is a contradiction. Therefore sup{D(x, y, y): xÎM}=D(y_{0},
y_{0,}u) <d(M).

Definition 2.11:A
convex D-metric space (X, D, W) is said to have normal structure if for
each closed bounded convex subset A of (X, D, W) which contains at least
two points, there exists xÎA
which is not a diametral point of A.

Definition 2.12: Let (X, D, W) be a convex
D-metric space and K be a subset of (X, D, W). A mapping T of K into X
is said to be nonexpansive if for any three elements x, y
and z of K, we have D(T(x), T(y), T(z))≤D(x, y, z).

Theorem 2.13:
Let (X, D, W) be a convex D-metric space. Suppose that (X, D, W) has the
Property(DC). Let K be a nonempty bounded closed convex subset of (X, D,
W) with normal structure. If T is a nonexpansive mapping of K into
itself, then T has a fixed point in K.

Proof: Let
F
be a family of all nonempty closed and convex subsets of K, such that
each of which is mapped into itself by T. By the Property(DC) and
Zorn’s Lemma, F
has a minimal element A. We show that A consists of a single point. Let
xÎA_{c}.
Since T is nonexpansive map, D(T(x), T(y), T(y)))≤D(x, y, y) for all yÎA.

Again
for every yÎA,
D(x, y, y)≤R_{x}(A), which implies D(T(x), T(y), T(y))≤ D(x, y,
y)≤R_{x}(A) = R(A) for all yÎA
and hence T(A) is contained in the closed ball S*[T(x), R(A)]. Since T(A)ÇS*[T(x),
R(A)])ÍAÇS*[T(x),
R(A)] the minimality of A implies AÍS*[T(x),
R(A)]. Hence R_{T(x)}(A)≤R(A). We know that R(A)≤R_{T(x)
}(A) for all xÎA.
Thus R(A)=R_{T(x)}(A). Hence T(x)ÎA_{c}
and T(A_{c})ÍA_{c}.
By using Proposition 2.9, A_{c}ÎF.
If zÎA_{c},
then D(y, y, z)≤ R_{z}(A) =R(A) that implies d(A_{c})≤R(A).
By the Definition of normal structure R(A)<d(A).
This proves that d(A_{c})≤R(A)<d(A).
This is a contradiction to the minimality of A. Therefore we must have
d(A)=0
that implies A consists of single point.

Definition 2.14: Let K be a compact convex
D-metric space. Then a family F of nonexpansive mappings T of K into
itself is said to have invariant property in K if for any compact and
convex subset A of K such that TAÍA
for each TÎF,
there exists a compact subset MÍA
such that TM=M for each TÎF.

Theorem 2.15:
Let (X, D, W) be a convex D-metric space and K be a compact convex
D-metric space. If F is a family of nonexpansive mappings with invariant
property in K, then the family F has a common fixed point.

Proof: By
applying Zorn’s Lemma, we can find a minimal nonempty convex compact set
AÍK
such that A is invariant under each TÎF.
If A consists of a single point, then the theorem is proved. Suppose
that A contains more than one point, then by the hypothesis, there
exists a compact subset M of A such that M= {T(x): xÎM}
for each TÎF.

If M
contains more than one point, by Proposition 2.10, there exists an
element u in the least convex set containing M such that r
= sup{D(x, x, u): xÎM}<
d
(M), where d(M)
is the diameter of M.

Let
us define A_{0 }=. Clearly A_{0} is nonempty and closed.
By using Proposition 2.4, A_{0 }is convex. Since u is not in A_{0},
A_{0} is a proper subset of A invariant under each T in F. This
is a contradiction to the minimality of A.

3.Strong convex D-metric spaces

We
introduce the concept of strong convex structure on D-metric spaces and
extend some fixed point theorems of convex D-metric spaces to strong
convex D-metric spaces.

Definition 3.1:
Let (X, D) be a metric space. A mapping W:X´X´X´(0,
1]®X
is said to be a strong convex structure on (X, D) if for each (x, y, z,
l)ÎX´X´X´(0,
1] and for all u, vÎX
the condition

D(u,
v, W(x, y, z,l))≤max{D(u, v, x),D(u, v, y),

D(u, v, z)} holds.

If W
is strong convex on a D-metric space (X, D), then the triplet (X, D, W)
is called a strong convex D-metric space.

Example 3.2:
Consider a linear space L which is also a D-metric space
with the following properties:

(i)For x, y, zÎ
L, D(x, y, z) = D(x – y – z, 0, 0);

(ii)For x, y, z ÎL
and lÎ(0,
1],

D(x +y +z, 0, 0)£max{D(x,0,0) ,D(y,0,0),

D(z, 0, 0)}.

Let
W(x, y, z, l)
=x+y +z for all x, y, zÎX
and lÎ(0,
1]. Then (L, D, W) is a strong convex D-metric space.

Proposition 3.3: If(X, D, W) is a
strong convex D-metric space then it is a convex D-metric space.

Definition 3.4:
A subset M of a strong convex D-metric space (X, D, W) is said to be a
convex set (X, D, W) if W(x, y, z, l)ÎM
for all x, y, zÎM
and for all l
with 0<l≤1.

Proposition 3.5: Let (X, D, W) be a strong
convex D-metric space, uÎX,
r >0. Then

(i)If {K_{a}:aÎD}
is a family of convex subsets of the strong convex D-metric space (X, D,
W), thenis also a convex subset in
(X, D, W).

(ii)The balls S_{1}(u, r) .(in the sense of Naidu), Ŝ(u,
r) .(in the sense of Naidu) and S_{2}(u, r) (in the sense of
Asim) in (X, D, W) are convex subsets of (X, D, W).The balls
S*(u, r) and S*[u, r] are convex in (X, D, W).

Proposition 3.6: Let (X, D, W) be a strong
convex D-metric space. Let AÍX.
If (X, D, W) has the Property(DC), then A_{c} is nonempty,
closed and convex.

Proof: Analogous
to Proposition 2.9.

Proposition 3.7: Let M be a nonempty compact
subset of a convex D-metric space (X, D, W) and let K be the least
closed convex set containing M. If the diameter d(M)
is positive, then there exists an element uÎK
such that sup{D(y, y, u):x, yÎM}<d(M).

Proof: Since (X,
D, W) is a strong convex D-metric space, by using Proposition 3.3, (X,
D, W) is a convex D-metric space. Then K is also a convex set containing
M in the convex D-metric space (X, D, W). By applying Proposition 2.10,
there exists an element uÎK
such that sup{d(x, u):xÎM}<
d(M).

Definition 3.8:
A strong convex D-metric space (X, D, W) is said to have normal
structure if for each closed bounded convex subset A of (X, D, W) which
contains at least two points, there exists xÎA
which is not a diametral point of A.

Theorem 3.9: Let
(X, D, W) be a strong convex D-metric space. Suppose that (X, D, W) has
the Property(DC). Let K be a nonempty bounded closed convex subset of
(X, D, W) with normal structure. If T is a nonexpansive mapping of K
into itself, then T has a fixed point in K.

Proof: Since (X,
D, W) is a strong convex D-metric space, by using Proposition 3.3, (X,
D, W) is a convex D-metric space. Then K is also a bounded closed convex
subset of convex D-metric space (X, D, W) with normal structure. If T is
a nonexpansive mapping of K into itself, then by applying Theorem 2.13,
T has a fixed point in K.

Definition 3.10: Let K be a compact strong
convex D-metric space. Then a family F of
nonexpansive mappings T of K into itself is said to have invariant
property in K if for any compact and strong convex subset A of K such
that TAÍA
for each TÎF,
there exists a compact subset MÍA
such that TM=M for each TÎF.

Theorem 3.11:
Let (X, D, W) be a strong convex D-metric space.Let K be a
compact strong convex D-metric space. If F is a family of nonexpansive
mappings with invariant property in K, then the family F has a common
fixed point.

Proof: Since (X,
D, W) is a strong convex D-metric space, by using Proposition 3.3, (X,
D, W) is a convex D-metric space. Then K is also a convex D-metric space
(X, D, W). If F is a family of nonexpansive mappings with invariant
property in K, then by applying Theorem 2.15 the family has a common
fixed point in K.

4.J-convex D-metric spaces

When
maximum is replaced by minimum in the definition of strong D-convex
structures we have the new convex structure called the J-convex
structure. In this section we introduce the concept of J-convex
structure and extend some properties of convex D-metric spaces and
strong convex D-metric spaces to J- convex D-metric spaces.

Definition 4.1:
Let (X, D) be a metric space. A mapping W:X´X´X´(0,
1)®X
is said to be a J-convex structure on a D-metric space (X, D) if for
each (x, y, z, l)Î
X´
X ´
X ´
I and for all u, vÎX
the condition D(u, v, W(x, y, z, l))≤
min{D(u, v, x),
D(u, v, y),
D(u, v, z)} holds.

If W
is a J- convex on a D-metric space (X, D), then the triplet (X, D, W) is
called a J-convex D-metric space.

Example 4.2:
Consider a linear space L which is also a D-metric space
with the following properties:

(i)For x, y, zÎ
L, D(x, y, z) = D(x – y –z, 0, 0);

(ii)For x, y, z ÎL
and lÎ(0,
1),

D(x +y+z,0,0)£min{D(x,0,0),D(y,0,0),

D(z, 0, 0)}.

Let
W(x, y, z, l)=x+y +z for all x, y, zÎX
and lÎ(0,
1). Then (L, D, W) is a J-convex D-metric space.

Definition 4.3:
A subset M of a J-convex D-metric space (X, D, W) is said to be a
convex set if W(x, y, z,l)ÎM
for all x, y, zÎM
and for all l
with 0<l<1.

Proposition 4.4: Let (X, D, W) be a J-
convex D-metric space, uÎX,
r >0. Then

(i)If {K_{a}:aÎD}
is a family of convex subsets of the strong convex D-metric
space (X, D, W), then_{ }is also a convex subset in (X, D,
W).

(ii)The balls S_{1}(u, r) .(in the sense of Naidu), Ŝ(u,
r) .(in the sense of Naidu) and S_{2}(u, r) (in the sense of
Asim) in (X, D, W) are convex subsets of (X, D, W).The balls
S*(u, r) and S*[u, r] are convex in (X, D, W).

Definition 4.5:
A J-convex D-metric space (X, D, W) is said to have the Property(JDC) if
every bounded decreasing sequence of nonempty closed convex subsets of
(X, D, W) has nonempty intersection.

Proposition 4.6: Let (X, D, W) be a
J-convex D-metric space. Let AÍX.
If (X, D, W) has the Property(JDC), then A_{c} is nonempty,
closed and convex.

Proof: Analogous
to Proposition 2.9.

Proposition 4.7: Let M be a nonempty compact
subset of a J-convex D-metric space (X, D, W) and let K be the least
closed J-convex set containing M. If the diameter d(M)
is positive, then there exists an element uÎK
such that sup{D(y, y, u): yÎM}<d(M).

Proof: Analogous
to Proposition 2.10.

Definition 4.8:
A J-convex D-metric space (X, D, W) is said to have normal structure if
it has the same normal structure if for each closed bounded J-convex
subset A of (X, D, W) which contains at least two points, there exists xÎA
which is not a diametral point of A.

Theorem 4.9: Let
(X, D, W) be a J-convex D-metric space. Suppose that (X, D, W) has the
Property(JDC). Let K be a nonempty bounded closed convex subset of (X,
D, W) with normal structure. If T is a nonexpansive mapping of K into
itself, then T has a fixed point in K.

Proof: Analogous
to Proposition 2.13.

Definition 4.9:
Let K be a compact J-convex D-metric space. Then a family F of
nonexpansive mappings T of K into itself is said to have invariant
property in K if for any compact and J-convex subset A of K such that
TAÍA
for each TÎF,
there exists a compact subset MÍA
such that TM=M for each TÎF.

Theorem 4.12:
Let (X, D, W) be a J-convex D-metric space.Let K be a compact
convex D-metric space. If F is a family of nonexpansive mappings with
invariant property in K, then the family F has a common fixed point.

Proof: Analogous
to Proposition 2.15.

5.Weak convex D-metric spaces

Definition 5.1:
Let (X, D) be a D-metric space. A mapping W:X´X´X´(0,
1]®
X is said to be a weak convex structure on (X, D) if for each (x, y, z,
l)ÎX´X´X´(0,
1] and for all u, vÎX
the condition

If W
is a weak convex structure on (X, D), then the triplet (X, D, W) is
called a weak convex D-metric space.

Example 5.2:
Consider a linear space L which is also a D-metric space with the
following properties:

(i)For x, y, zÎ
L, D(x, y, z) = D(x – y – z, 0, 0);

(ii)For x, y, z ÎL
and lÎ(0,
1],

D(x +y+z, 0, 0)£D(x,
0, 0)+D(y, 0, 0)+D(z, 0, 0).

Let
W(x, y, z, l)=x+ y +z for all x, y, zÎX
and lÎ(0,
1],. Then (L, D, W) is a weak convex D-metric space.

Proposition 5.3: Every convex D-metric space
(X, D, W) is a weak convex D-metric space.

Definition 5.4:
A subset M of weak convex D-metric space (X, D, W) is said to be a weak
convex if W(x, y, z, l)ÎM
for all x, y, zÎM
and for all l
with 0<l≤1.

Proposition 5.5:Let
(X, D, W) be a weak convex D-metric space, uÎX,
r >0. Then

(i)If {K_{a}:aÎD}
is a family of convex subsets of the weak convex D-metric space (X, D,
W), then_{ }is also a convex subset in (X, D,
W).

(ii)The balls S_{1}(u, r).(in the sense of Naidu),Ŝ(u, r)
.(in the sense of Naidu) and S_{2}(u, r) (in the sense of Asim)
in (X, D, W) are convex subsets of (X, D, W).The balls S*(u, r)
and S*[u, r] are convex in (X, D, W).

Definition 5.6: A weak convex D-metric
space (X, D, W) is said to have the Property(WDC) if every bounded
decreasing sequence of nonempty closed weak convex subsets of
(X, D, W) has nonempty intersection.

Proposition 5.7: Let (X, D, W) be a weak
convex D-metric space. Let AÍX.
If (X, D, W) has the Property(DC), then A_{c} is nonempty,
closed and weak convex.

Proof:
Analogous to Proposition 2.9.

Theorem 5.8: Let
M be a nonempty compact subset of a Quasi convex D-metric space (X, D,
W) and let K be the least closed quasi convex set containing M. If the
diameter d(M)
is positive, then there exists an element uÎK
such that sup{D(y, y, u): yÎM}<d(M).

Proof:
Analogous to Proposition 2.10.

Theorem 5.9: Let
(X, D, W) be a convex D-metric space. Suppose that (X, D, W) has the
Property(DC). Let K be a nonempty bounded closed convex subset of (X, D,
W) with normal structure. If T is a nonexpansive mapping of K into
itself, then T has a fixed point in K.

Proof: Analogous
to Proposition 2.13.

Definition 5.10: Let K be a compact weak
convex D-metric space. Then a family F of nonexpansive mappings T of K
into itself is said to have invariant property in K if for any compact
and weak convex subset A of K such that TAÍ
A for each TÎF,
there exists a compact subset MÍA
such that TM=M for each TÎF.

Theorem 5.11:
Let (X, D, W) be a weak convex D-metric space.Let K be a compact
weak convex D-metric space. If F is a family of nonexpansive mappings
with invariant property in K, then the family F has a common fixed
point.

Proof: Analogous
to Proposition 2.15.

6.Quasi convex D-Metric Spaces

A new
convex structure in a metric space can be introduced by taking l= in the definition of convex structure
introduced by Wataru Takahashi. This new structure is named as quasi
convex structure in D-metric spaces. In this section we introduce such a
structure and establish some basic properties of quasi convex D-metric
spaces.

Definition 6.1:
Let (X, D) be a metric space. A mapping W: X´X´X´(0,
1]®
X is said to be a Quasi convex structure on (X, D) if for each (x, y, z,
l)ÎX´X´X´(0,
1] and for all u, vÎX
the condition

D(u,v,W(x,y,z,l))≤D(u,v, x)+D(u,v,y)+D(u,v,z).

If W
is a Quasi convex on a D-metric space (X, D), then the triplet (X, D, W)
is called a Quasi convex D-metric space.

Example 6.2:
Consider a linear space L which is also a D-metric space with the
following properties:

(i)For x, y, zÎL,
D(x, y, z) = D(x – y – z, 0, 0);

(ii)For x, y, zÎL
and lÎ(0,
1].

D(x+y+z, 0, 0)£D(x,
0, 0)+ D(y, 0, 0) +

D(z,
0, 0).

Let
W(x, y, z,l)
=x+y+z for all x, y, zÎX
and lÎ(0,
1]. Then (L, D, W) is a Quasi convex D-metric space.

Definition 6.3:
A subset M of a Quasi convex D-metric space (X, D, W) is said to be a
Quasi convex if W(x, y, z, l)ÎM
for all x, y, zÎM
and for all l
with 0<l≤1.

Proposition 6.4: Every Quasi convex D-metric
space (X, D, W) is a weak convex D-metric space.

Proposition 6.5: Let (X, D, W) be a Quasi
convex D-metric space, uÎX,
r >0. Then

(i)If {K_{a}:aÎD}
is a family of convex subsets of the Quasi convex D-metric space (X, D,
W), then_{ }is also a convex subset in
(X, D, W).

(ii)The balls S_{1}(u, r) .(in the sense of Naidu),
Ŝ(u, r) .(in the sense of Naidu) and S_{2}(u, r) (in
the sense of Asim) in (X, D, W) are convex subsets of (X, D, W).
The balls S*(u, r) and S*[u, r] are convex in (X, D, W).

Definition 6.6: A Quasi convex D-metric
space (X, D, W) is said to have the Property(QDC) if every bounded
decreasing sequence of nonempty closed quasi convex subsets of (X, D,
W) has nonempty intersection.

Proposition 6.7: Let (X, D, W) be a Quasi
convex D-metric space. Let AÍX.
If (X, D, W) has the Property(QDC), then A_{c} is nonempty,
closed and Quasi convex.

Proof: Analogous
to Proposition 2.9.

Theorem 6.8:Let
M be a nonempty compact subset of a Quasi convex D-metric space (X, D,
W) and let K be the least closed quasi convex set containing M. If the
diameter d(M)
is positive, then there exists an element uÎK
such that sup{D(y, y, u): yÎM}<
d(M).

Proof: Since (X,
D, W) is a quasi convex D-metric space, by using Proposition 6.4, (X, D,
W) is a weak convex D-metric space. Then K is also a weak convex set
containing M in the weak convex D-metric space (X, D, W). By applying
Proposition 5.8, there exists an element uÎK
such that sup{d(x, u):xÎM}<d(M).

Definition 6.9:
A quasi convex D-metric space (X, D, W) is said to have normal structure
if for each closed bounded quasi convex subset A of (X, D, W) which
contains at least two points, there exists xÎA
which is not a diametral point of A.

Theorem 6.10:
Let (X, D, W) be a quasi convex D-metric space. Suppose that (X, D, W)
has the Property(QDC). Let K be a nonempty bounded closed quasi convex
subset of (X, D, W) with normal structure. If T is a
nonexpansive mapping of K into itself, then T has a fixed point in K.

Proof: Since (X,
D, W) is a strong convex D-metric space, by using Proposition 6.4, (X,
D, W) is a convex D-metric space. Then K is also a bounded closed convex
subset of convex D-metric space (X, D, W) with normal structure. If T is
a nonexpansive mapping of K into itself, then by applying Theorem 5.9, T
has a fixed point in K.

Definition 6.11: Let K be a compact Quasi
convex D-metric space. Then a family F of nonexpansive mappings T of K
into itself is said to have invariant property in K if for any compact
and quasi convex subset A of K such that TAÍA
for each TÎF,
there exists a compact subset MÍA
such that TM=M for each TÎF.

Theorem 6.12:
Let (X, D, W) be a Quasi convex D-metric space.Let K be a
compact Quasi convex D-metric space. If F is a family of nonexpansive
mappings with invariant property in K, then the family F has a common
fixed point.

Proof: Since (X,
D, W) is a quasi convex D-metric space, by using Proposition 6.4, (X, D,
W) is a weak convex D-metric space. Then K is also a weak convex
D-metric space (X, D, W). If F is a family of nonexpansive mappings with
invariant property in K, then by applying Theorem 5.11 the family has a
common fixed point in K.

7.Conclusion

Thus
we have introduced and studied four types of convex structures in a
D-metric space. We established some fixed point theorems in these
structures. The link between the convex structures is given below:

(i)If (X, D, W) is a Strong convex D-metric space then (X, D, W) is
a convex D-metric space.

(ii)If (X, D, W) is a J- convex D-metric space then (X, D, W) is a
Strong convex D-metric space.

(iii)If (X, D, W) is a convex D-metric space then (X, D, W) is a weak
convex D-metric space.

(iv)If (X, d, W) is a Quasi convex D-metric space then (X, D, W) is a
Weak convex D-metric space.

8.References

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[5] Shyamala Malini
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[6] Shyamala Malini
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[7] Shyamala Malini
S. Thangavelu P and Jeyanthi P., Convexity in Metric Spaces and its
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[8] Shyamala Malini
S. Thangavelu P and Jeyanthi P., Convexity in Ultra metric Spaces and
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[9] Wataru
Takahashi, A convexity in metric space and nonexpansive mappings, I
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