International Journal of Engineering Trends and Technology (IJETT) – Volume 19 Number 3 – Jan 2015
Common Fixed Point Of Compatible Maps in
D-Metric Spaces
1
Dr.Amita Joshi, 2Dr. Bharat Singh
1
Assistant Professor, 2Associate Professor,
Department of Mathematics, IPS Acaedmy, Indore (M.P.), India
Abstract- In this paper, a transposition of these notions is being
D(x, y, z) = D(y, x, z) = …
(D-2)
made for 4-tuples A, B, C, D of self maps of a D-metric space (X,
(Symmetry),
D). Under suitable contractive conditions, some common fixed
point theorems involving such maps have been proved by several
(D-3) D(x,y,z)
authors.
in X.
D(x,y,a) + D(x,a,z) + D(a,y,z) for all x,y,z
Keywords- D-metric space, D-weakly compatible mappings
I INTRODUCTION
(Rectangle inequality)
The pair (X, D) is then called a D-metric space. As immediate
Dhage [1] introduced the concept of D-metric space. The
generalized metric D is a slight enlargement of the 2-metric
examples of such functions we quote, cf. Dhage [2]:
(a)
D(x,y,z) = max{d(x,y),d(y,z),d(z,x)} and
concept. He also gave a generalization of Banach’s
contraction principle for such structures; cf Rhoades [8].
Further Dhage [2], Raju, Rao and Rao [7], Veerapandi and
Rao [12], Sharma and Dawar [10] and many others have used
this concept in fixed point frame work.
(b)
D(x,y,z) =
d(x,y) + d(y,z) + d(z,x) .
Here, d is an ordinary metric on X. This notion is closely
related to the one of the 2-metric, introduced by Gähler [3] as
follows:
In 1976, Jungck [4] proved a common fixed point theorem for
commuting mappings. Sessa [9] defined a generalization of
commutativity, which is called weak commutativity. Further
Definition 2. Let X be a non-empty set. A 2- metric on X is a
function d from X X X to R+ which satisfies the following
conditions:
Jungck [5], [6] introduced more generalized commutativity so
called compatibility, which is more general than that of weak
(d-1)
commutativity.
such that
II PRELIMINARIES
for every distinct pair of x, y
d (x, y, z)
Definition 1 [1]. Let X be a non-empty set. A generalized
(d-2)
metric (or D-metric) on X is a function D from X X X to R+
y, z are equal,
X there is a z
X
0,
d(x, y, z) = 0 if and only if any two of the triplet x,
that satisfies the following conditions:
(d-3)
(D-1)
d(x, y, z) = d(y, x, z) = …
D(x, y, z) = 0 if and only if x = y = z
(Symmetry),
(Sufficiency),
(d-4)
d(x,y,z)
d(x,y,a) + d(x,a,z) + d(a,y,z) for all
x,y,z,a in X
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International Journal of Engineering Trends and Technology (IJETT) – Volume 19 Number 3 – Jan 2015
Clearly, commutativity implies D-weak commutativity and Dweak commutativity implies D-compatibility; but implication
(Rectangle inequality)
is not always reversible, see [1].
Geometrically a 2-metric d(x, y, z) represents the area, while
the D-metric D(x, y, z) represents the perimeter of a triangle
Definition 8.
Two self mappings A and B of a D-metric
whose vertices are x, y and z.
space X are said to be D-weakly compatible if they commute
at their coincidence points.
Definition 3 [2]. A sequence {xn } of points in a D-metric
space (X, D) is said to be D-convergent to the point x
for each
> 0, there exists n0
X if,
N such that
D (xn , xm , x) < , for all m, n
MAIN RESULTS
Let
be the family of all continuous non-decreasing
functions
n0 ;
: R+
n
n=1
(t) <
R+ such that
for all t
(t) < t, for t > 0 and
R+ .
and convergent, if it converges to some point x of X.
Singh and Sharma [11] proved the following:
Definition 4. [1] A sequence {xn } of points in a D-metric
space (X, D) is said to be D-Cauchy if for
exists n0
> 0, there
complete and bounded D-metric space (X, D) satisfying:
N such that
D(xn , xm , xp) <
Definition 5.
Theorem A: Let A, B, F and G be self – mappings of a
, for all m,p > n
n0.
[1] A D-metric space (X, D) is said to be
complete if every D-Cauchy sequence in X converges (to a
point in X).
(i)
A(X)
G(X) and B(X)
F(X),
(ii)
One of A, B, F or G is continuous,
(iii)
(A, F) and (B, G) are D-compatible pairs of
mappings,
Definition 6. Two self mappings A and B of a D-metric space
X are said to be D-weakly commuting if
(iv)
D(Ax,By,z)
[ max{D(Fx,Gy,z),
D(Fx,By,z),
D(Gy,Ax,z)}]
D(ABx ,BAx ,y)
D(Ax ,Bx ,z), where
y = ABx (or
for all x,y,z
BAx )
z = Ax (or Bx), for all x
X are said to be D-compatible if
D(ABxn ,BAxn , z ) = 0, where
ISSN: 2231-5381
We prove Theorem A without taking any function continuous
and replacing the condition of D- compatible mappings by
z = ABxn
Theorem 1. Let A, B, F and G be self – mappings of a
complete and bounded D-metric space (X,D) satisfying
whenever {xn } is a sequence in X such that limn
Bxn = y, for some y in X.
1/3.
D-weakly compatible mappings. We prove the following.
(or BAxn )
limn
and 0 <
Then A, B, F and G have a unique common fixed point in X.
X.
Definition 7. Two self mappings A and B of a D-metric space
limn
X, where
Axn =
conditions (i) and (iv) for all x,y,z
<
X, where
and 0
1/3 and
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International Journal of Engineering Trends and Technology (IJETT) – Volume 19 Number 3 – Jan 2015
(1)
(A, F) and (B, G) are D-weakly compatible pairs of
mappings
D(u, Bp,u)
=
D(Aq,Bp, u)
[ max{D(u,Gp,u),
D(u,Bp,u), D(Gp,u,u)}]
Then A, B, F and G have a unique common fixed point in X.
D(u, Bp,u)
Proof.
D (u, Bp,u) < D(u,Bp,u) and this yields Bp = u . Therefore
By assumption (i), since A(X)
arbitrary x0
=
X there exists a point x1
Gx1 . Since B(X)
choose a point x2
G(X), for an
X such that Ax0
[ max{D(u,u,u), D(u,Bp,u), D(u,u,u)}]
Bp = u = Gp.
F(X), for this point x1 we can
X such that Bx1 = Fx2 . Continuing
this way, we can construct a sequence {yn } in X such that
y2n = Fx2n = Bx2n-1 and
y2n +1 = Gx2n+1 = Ax2n ,
Hence Aq = Fq. = Bp = Gp = u.
Since the maps A and F are D-weakly compatible, then AFq =
FAq i.e.
Au = Fu. Now we show that u is a common fixed point of A.
for every n = 1,2,…
By (iv), we have
Since the conditions (i) and (iv) of our theorem are similar to
that of Theorem A of Singh and Sharma [11] therefore, we
conclude that {yn} is a D-Cauchy sequence in X. By
D(Au,Bx2n+1,u)
[ max{D(Fu,Gx2n+1,u), D(Fu,Bx2n+1,u),
D(Gx2n+1,Au,u)}]
completeness of X, {yn} and also its subsequences {Ax2n},
{Bx2n-1}, {Fx2n} and {Gx2n+1} converge to u in X.
D(Au,Bx2n+1,u)
Since B(X)
F(X), there exists a point q
D(Au,Bx2n+1,u),
Fq. Then using (iv), we have
Taking the limit n
D(Aq,Bx2n+1,u)
[ max{D(Au,Gx2n+1,u),
X, such that u =
[ max{D(Fq,Gx2n+1,u), D(Fq,Bx2n+1,u),
D(Gx2n+1,Au,u)}]
,
we get
D (Gx2n+1, Aq, u)}]
D (Au, u, u) < D(Au,u,u), which yields
Au = u. Therefore Au = u = Fu.
D(Aq,Bx2n+1,u)
[ max{D(u,Gx2n+1,u),
D(u,Bx2n+1,u),
Similarly, since B and G are D-weakly compatible, we can
D (Gx2n+1, Aq, u)}]
have Bu = u = Gu.
Taking the limit n
, we get D (Aq, u,u)
<
D(Aq,u,u)
Hence Bu = Gu = Au = Bu = u. Thus u is a common fixed
which yields
point of A, B, F and G.
Aq = u. Since Fq = u. Therefore Aq = u = Fq.
For uniqueness let w be another common fixed point of A, B,
Again since A(X)
G(X), there exists a point p
X such
F and G. By (iv), we have
that Gp = u.
D(u,w,u)
Using (iv), we have
ISSN: 2231-5381
=
D(Fu,Bw,u),
D(Au, Bw, u)
[ max{D(Fu,Gw,u),
D (Gw, Au, u)}]
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International Journal of Engineering Trends and Technology (IJETT) – Volume 19 Number 3 – Jan 2015
[4] Jungck G.,Commuting mappings and fixed points, Amer. Math. Monthly,
Vol.83, pp.261-263,1976.
< D (u, w, u),
Which yields u = w. This completes the proof.
[5] Jungck, G.,Compatible mappings and common fixed points, Internat. J.
Math. Math. Sci.,Vol. 9, pp.771-779, 1986.
Corollary 1: Let A and F be self – mappings of a complete
and bounded D-metric space (X, D) satisfying:
[6] Jungck, G.,Compatible mappings and common fixed points (2), Internat. J.
Math. and Math. Sci.,Vol. 11, pp.285-288,1988.
(2) A(X)
[7] Raju, V.C.C., Rao, K.P.R. and Rao, G.S.S. B.,Fixed point theorems in Dcompact metric spaces, Ind. J. math.,Vol. 43, pp.197-202, 2001.
F(X),
(3) (A, F) is D-weakly compatible pair of mappings,
(4)
D(Ax,Ay,z)
[ max{D(Fx,Fy,z),
D(Fx,Ay,z),
[8] Rhoades, B.E.,A fixed point theorem for generalized metric spaces,
Internat. J. Math. Math. Sci.,Vol. 19(3),pp.457-460,1996.
[9] Sessa, S., On weak commutativity condition of mappings in a fixed points
D(Fy,Ax,z)}]
considerations, Publ. Inst. Mat.,Vol. 32(46), pp.149-153, 1982.
for all x, y, z
X, where
and 0 <
1/3.
[10] Sharma, Sushi and Dawar, R. , Some results on common fixed points in
D-metric space, J. Acta Ciencia Indica, Vol. 23, No. 3,pp.181-
Then A and F have a unique common fixed point in X.
Finally, when F = I (identity mapping on X) in Corollary 1,
we get
186,1997.
[11] Singh, B. and Sharma, R.K., Common fixed point via compatible maps
in D-metric spaces, Radovi Math. , Vol.11, pp.145-153,2002.
[12] Veerapandi, T. and Rao, K.C.,Fixed point theorems of some multi-valued
mappings in D-metric space, Bull. Cal. Math. Soc.,Vol. 87, pp.549-556,1995.
Corollary 2: Let A be a self mapping of a complete and
bounded D-metric space (X, D) satisfying:
(5)
D(Ax,Ay,z)
[ max{D(x,y,z),
D(x,Ay,z),
D(y,Ax,z)}]
for all x, y, z
X, where
and 0 <
1/3.
Then A has a unique fixed point in X.
CONCLUSIONS
We prove Theorem without taking any function continuous
and replacing the condition of D- compatible mappings by Dweakly compatible mappings
REFRENCES:
[1] Dhage, B.C., Generalized metric spaces and mappings with fixed points,
Bull. Cal. Math. Soc.,Vol. 84, pp.329-336,1992.
[2] Dhage, B.C., A common fixed point principle in D-metric spaces, Bull.
Cal. Math. Soc., Vol.91, pp.475-480,1999.
[3] Gähler, S., 2-metric Raume und ihre topologiche Struktur, Math. Nachr,
Vol.26 , pp.115-148,1963.
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