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Convexity in D-Metric Spaces and its Applications Fixed Point Theorems

P.Thangavelu1 ,S.Shyamala Malini*, and P.Jeyanthi2

1Department of Mathematics, Kaurnya University, Coimbatore- 641 114, India.

*4/1 Vijay Bhavanam, N.G.O Colony 6th street extn 2, Melagaram,Tenkasi- 627 818,India.

2Department of Mathematics,Govindammal Aditanar College for Women, Tiruchendur-628 215,India.

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:XX[0, 1]X is said to be a convex structure[9] on    (X, d) if for each (x, y, l)XX[0, 1] and for all uX, 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:XXX[0, ) be a function satisfying the following conditions for all x, y, zX.

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  aX.

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 x0X then for every r>0

S*(x0, r) = {yX: r(x0, y, y) <r}     Eqn. 4.1

and S(x0, r) = y, zX: r(x0, 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 x0X then for every r>0

S*(x0, r) = {xX: r(x0, x, x) <r}           Eqn. 4.3

S(x0, r) = xS*(x0, r):r(x0, x, y) < r for all yS*(x0, r)}                                                                                                                                   Eqn. 4.4

Ŝ(x0, r) ={x0}{xX:r(x0, x, y) < r}               Eqn. 4.5

Following this Asim, Aslam and Zafer[2] modified the above definition assuming that r(x0, x, y) < r is not always possible.

Definition 1.5: Let (X, r) be a D-metric space and x0X then for every r>0

S*(x0, r)= {xX: r(x0, x, x) <r}             Eqn. 4.6

S(x0, r)={x0}{xX:r(x0, x, y)< r}      Eqn. 4.7

There is a uniformity in the definition of the ball S*(x0, r), in[2],[3],[4]. However the definition for S(x0, 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(x0, r)(Eqn. 4.5), quoting thatr(x0, x, y)< r is not always possible for all xX. But their assumption is wrong. For even if r(x0, x, y)< r does not exist or ifr(x0, x, y) is not less than r for all xX then also the definition is meaningful. More precisely if the above condition prevails for some x then we can take Ŝ (x0, r) ={x0}.

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, x0X and r>0.

S1(x0, r) = {xS*(x0, r):r(x0, x, y)< r for all yS*(x0, r)}.(in the sense of Naidu)

Ŝ(x0, r) ={x0}{xX:r(x0, x, y) < r}}.(in the sense of Naidu)

S2(x0, r)={x0}{xX:r(x0, 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: XXX(0, 1] X is said to be a convex structure on  (X, D) if for each (x, y, z, l)XXX(0, 1] and for all       u, vX the condition

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

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, z L, D(x, y, z) = D(x � y � z, 0, 0);

(ii)               For x, y, z L and l�(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, zX and l�(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)M for all x, y, zM and for all l with 0<l≤1.

Proposition 2.4: If {Ka:a�D} 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, uX, r >0. The balls S1(u, r) (in the sense of Naidu), Ŝ(u, r) (in the sense of Naidu) and S2(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 AX.

Rx (A) = sup{ D(x, y, y): yA}.

R(A) = inf{ Rx(A): xA}.

Ac ={xA: Rx(A) = R(A)}.

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

Definition 2.7: Let (X, D, W) be a convex D-metric space. Let A be a subset of X. A point xA 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 AX. If (X, D, W) has the Property(DC).  Then Ac is nonempty, closed and convex.

Proof: Let xA. For each positive integer n, define An(x) = {yA: D(x, y, x) ≤ R(A) + }  and Cn . Since xAn(x), An(x) is nonempty and closed. Let zAn(x) and l�(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)An(x) which implies that An(x) is a convex set.

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

Let e >0. Then by the Definition of R(A),there exists yA with Ry(A)R(A)+e and by the Definition of Ry(A),                  D(x, y, y) Ry(A) for every xA.

<R(A)+e for every xA

As e is arbitrary for every xA, D(x, y, y)R(A)+e< R(A)+ . Hence yAn(x) for every xA and hence y= Cn. Thus Cn is nonempty. Since each An is a closed set Cn is also closed. By Proposition 2.5, Cn is a convex set. Hence Cn is a nonempty, closed and convex set for all n.

Let zCn+1 .Then zAn+1(x) for every xX, D(x, x, z)≤R(A) +≤R(A)+. Therefore zAn(x) for every xX which implies that zCn. Now we prove that that Ac =. Since the space (X, D, W) has the Property (DC), is nonempty. Let x. Then xAn(y) for every yA and for every n. Now D(x, y, y)≤R(A)+for every yA and for every n. Thus Rx(A)≤R(A). By the Definition of R(A), we have R(A)≤ Rx(A). Hence R(A)=Rx(A) implies xAc.

Conversely, let xAc. Then Rx(A)=R(A). Now, for every yA D(x, y, y)Rx(A)=R(A)R(A)+for every n. This shows that xAn(y) for every yX 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 uK such that sup{D(y, y, u):x, yM}<d (M).

Proof: Since M is compact, we may find x1, x2, x3M such that D(x1, x2, x3) = d(M). Let M0 M be maximal so that            M0 {x1, x2, x3} and D(x, y, y) = 0 or d(M) for all  x, yM0. Since M is compact, M0 is finite. Let us assume that M0={x1,x2,�,xn}.

Let                   y1 = W(x1, x2, x3,)

y2 = W(x3, x4 y1, )

�.

yn-2 = W(xn-2, xn-1, yn-3, )

yn-1= W(xn-1, xn, yn-2, ) = u.

As K is a convex set, uK. Since M is compact, we can find y0M such that D(y0, y0,u)= sup{D(y, y, u): yM}.

Now by the Definition 2.1, we have

D(y0, y0, u) = D(y0, y0, W(xn-1, xn, yn-2, )

D(y0, y0,xn-1)  +D(y0, y0, xn) + D(y0, y0, yn-2)

=D(y0,y0,xn-1) +D(y0,y0,xn)+

D(y0,y0,W(xn-2, xn-1, yn-3; ))

≤(y0, y0, xk) ≤ d (M)

Suppose D(y0, y0, u) = d(M).  Then D(y0, y0,  xk) = d(M) >0 for all k=1,2,3�,n. Since y0M0 we have y0= xk for some k=1,2,�,n. This implies d(M) =0 which is a contradiction. Therefore  sup{D(x, y,  y): xM}=D(y0, y0,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 xA 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 xAc. Since T is nonexpansive map, D(T(x), T(y), T(y)))≤D(x, y, y) for all yA.

Again for every yA, D(x, y, y)≤Rx(A), which implies D(T(x), T(y), T(y))≤ D(x, y, y)≤Rx(A) = R(A) for all yA and hence T(A) is contained in the closed ball S*[T(x), R(A)]. Since T(A)S*[T(x), R(A)])AS*[T(x), R(A)]  the minimality of A implies AS*[T(x), R(A)]. Hence RT(x)(A)≤R(A). We know that R(A)≤RT(x) (A) for all xA. Thus R(A)=RT(x)(A). Hence T(x)Ac and T(Ac)Ac. By using Proposition 2.9, Ac�F. If zAc, then D(y, y, z)≤ Rz(A) =R(A) that implies d(Ac)≤R(A). By the Definition of normal structure R(A)<d(A). This proves that d(Ac)≤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 TAA for each TF,  there exists a compact subset MA such that TM=M for each TF.

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 AK such that A is invariant under each TF. 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): xM} for each TF.

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): xM}< d (M), where d(M) is the diameter of M.

Let us define A0 =. Clearly A0 is nonempty and closed. By using Proposition 2.4, A0 is convex. Since u is not in A0, A0 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:XXX(0, 1]X is said to be a strong convex structure on (X, D) if for each  (x, y, z, l)XXX(0, 1]  and  for all u, vX 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, z L, D(x, y, z) = D(x � y � z, 0, 0);

(ii)               For x, y, z L and l�(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, zX and l�(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)M for all x, y, zM and for all l with 0<l≤1.

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

(i)                 If {Ka:a�D} 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 S1(u, r) .(in the sense of Naidu),      Ŝ(u, r) .(in the sense of Naidu) and S2(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 AX. If (X, D, W) has the Property(DC), then Ac 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 uK such that sup{D(y, y, u):x, yM}<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 uK such that sup{d(x, u):xM}< 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 xA 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  TAA for each TF,  there exists a compact subset MA such that TM=M for each TF.

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:XXX(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) X X X I and for all u, vX 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, z L, D(x, y, z) = D(x � y �z, 0, 0);

(ii)               For x, y, z L and l�(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, zX and l�(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)M for all   x, y, zM and for all l with 0<l<1.

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

(i)                 If {Ka:a�D} 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 S1(u, r) .(in the sense of Naidu),         Ŝ(u, r) .(in the sense of Naidu) and S2(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 AX. If (X, D, W) has the Property(JDC), then Ac 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 uK such that sup{D(y, y, u): yM}<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 xA 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 TAA for each TF,  there exists a compact subset MA such that TM=M for each TF.

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:XXX(0, 1] X is said to be a weak convex structure on (X, D) if for each (x, y, z, l)XXX(0, 1] and for all u, vX the condition

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

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, z L, D(x, y, z) = D(x � y � z, 0, 0);

(ii)               For x, y, z L and l�(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, zX and l�(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)M for all  x, y, zM and for all l with 0<l≤1.

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

(i)                 If {Ka:a�D} 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 S1(u, r).(in the sense of Naidu),Ŝ(u, r) .(in the sense of Naidu) and S2(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 AX. If (X, D, W) has the Property(DC), then Ac 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 uK such that sup{D(y, y, u): yM}<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 TA A for each TF, there exists a compact subset MA such that TM=M for each TF.

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: XXX(0, 1] X is said to be a Quasi convex structure on (X, D) if for each (x, y, z, l)XXX(0, 1] and for all u, vX 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, zL, D(x, y, z) = D(x � y � z, 0, 0);

(ii)               For x, y, zL and l�(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, zX and l�(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)M for all  x, y, zM 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, uX, r >0. Then

(i)                 If {Ka:a�D} 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 S1(u, r) .(in the sense of Naidu),            Ŝ(u, r) .(in the sense of Naidu) and S2(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 AX. If (X, D, W) has the Property(QDC), then Ac 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 uK such that   sup{D(y, y, u): yM}< 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 uK such that                                        sup{d(x, u):xM}<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 xA 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 TAA for each TF, there exists a compact subset MA such that TM=M for each TF.

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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[7]        Shyamala Malini S. Thangavelu P and Jeyanthi P., Convexity in Metric Spaces and its applications to Fixed Point Theorems.(submitted).

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[9]        Wataru Takahashi, A convexity in metric space and nonexpansive mappings, I Kodai Math.Sem.Rep.22(1970),142-149.